State whether the following variables are Qualitative or Quantitative.
\nNote that you will be deducted one mark for every wrong choice. However the minimum mark is 0.
", "tags": ["ACE2013", "checked2015", "MAS1403", "MAS1604"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "01/02/2013:
\nFinished first draft.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Choosing whether given random variables are qualitative or quantitative.
"}, "advice": ""}, {"name": "Classify sampling methods", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"displayType": "radiogroup", "layout": {"type": "all", "expression": ""}, "marks": 0, "choices": ["{ch1}", "{ch2}", "{ch3}"], "matrix": "w", "prompt": "\nIdentify each of the following scenarios as one of the following:
\nNote that you will lose 1 mark for every incorrect answer, however the minimum mark for this part of the question is 0.
\n\n ", "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": false, "warningType": "none", "scripts": {}, "maxMarks": 0, "type": "m_n_x", "minMarks": 0, "shuffleAnswers": false, "showCorrectAnswer": true, "answers": ["Simple Random Sampling", "Stratified Sampling", "Systematic Sampling", "Judgemental Sampling"]}], "variables": {"w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "map(switch(chlist[x]=0,[1,-1,-1,-1],chlist[x]=1,[-1,1,-1,-1],chlist[x]=2,[-1,-1,1,-1],[-1,-1,-1,1]),x,0..2)", "name": "w", "description": ""}, "ch3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(chlist[2]=0,random(a except [ch1,ch2]),chlist[2]=1,random(b except [ch1,ch2]),chlist[2]=2,random(c except [ch1,ch2]),random(d except [ch1,ch2]))", "name": "ch3", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[\"A local bus company is planning a new route to serve four housing estates. Random samples of households are taken from each estate and sample members are asked to rate on a scale of 1 (strongly opposed) to 5 (strongly in favour) their reaction to the proposed service.\",\"A company has three divisions, and auditors are attempting to estimate the total amounts of the company's accounts receivable. Simple random samples of these accounts were taken for each of the three divisions.\",\"A company has three divisions, and auditors are attempting to estimate the total amounts of the company's accounts receivable. Simple random samples of these accounts were taken for each of the three divisions.\",\"This form of sampling reflects the major groupings within a population.\"]", "name": "b", "description": ""}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "map(switch(chlist[x]=0,[1,-1,-1],chlist[x]=1,[1,-1,-1],chlist[x]=2,[-1,1,-1],[-1,-1,1]),x,0..2)", "name": "v", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[\"A company director believes she knows what characteristics make up the target population for a new product her company intends to launch. The company's team of market researchers check the viability of this new product by eliciting the opinions of the target population as specified by the director.\",\"Specific members of a population are sampled because of their known honesty and integrity.\",\"This form of sampling can provide a coherent and focussed sample by asking people with experience and relevant knowledge to provide their opinions.\",\"With this form of sampling, the researcher decides what he or she constitutes a representative sample.\"]", "name": "d", "description": ""}, "ch1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(chlist[0]=0,random(a),chlist[0]=1,random(b),chlist[0]=2,random(c),random(d))", "name": "ch1", "description": ""}, "ch2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(chlist[1]=0,random(a except ch1),chlist[1]=1,random(b except ch1),chlist[1]=2,random(c except ch1),random(d except ch1))", "name": "ch2", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[\"One hundred small businesses in Newcastle are placed in alphabetical order and then numbered 1-100. The random number generator is then used to select twenty of these businesses.\",\"Under this form of sampling, if there are five hundred elements in the population, each element has a one-in-five hundred chance of being selected. \",\"One advantage of this form of sampling is that every element in the population has an equal chance of being selected.\",\"We are interested in the employment status of 25-40 year olds in South Tyneside. The names of all such people are obtained from the electoral roll and put into a hat; one hundred of these are then selected without replacement.\",\"One of six branches of a large retail outlet is to be selected for an audit. Each outlet is assigned a number from one to six, and then a fair, six-sided die is rolled to select the branch which will be audited.\"]", "name": "a", "description": ""}, "chlist": {"group": "Ungrouped variables", "templateType": "anything", "definition": "repeat(random(0,1,2,3),3)", "name": "chlist", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[\"The first item to be checked for faults on a production line is chosen at random, thereafter, every 100th item is checked.\",\"A credit card company wants to investigate the spending habits if its customers. From its lists, the first customer is selected at random; thereafter, every 25th customer is selected.\",\"In an inquiry on heating costs, we decide to sample every 4th house on the street.\",\"To sample 1% of its target population, consisting of 5000 members, a market research company chooses the first member at random; after that, every 100th member is also selected.\",\"This form of sampling could produce an unrepresentative sample because of patterns in the sampling frame.\"]", "name": "c", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "chlist", "ch1", "ch2", "ch3", "w", "v"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n
Answer the following questions on the sampling methods used in these situations.
\n\n ", "tags": ["ACE2013", "checked2015"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "
Deciding whether or not three sampling methods are simple random sampling, stratified sampling, systematic or judgemental sampling.
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\nSample Standard Deviation = [[1]] {shortform}. Give your answer to $2$ decimal places (include trailing zeros if required).
\nSample Median = [[2]] (Input as an exact decimal).
\nThe interquartile range= [[3]] (Input as an exact decimal).
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\n{table(tble,t)}
\nAnswer the following questions:
\n\n
", "tags": ["ACE2013", "checked2015", "interquartile range", "lower quartile", "MAS1403", "mean", "mean ", "median", "quartiles", "sample data", "sample mean", "sample standard deviation", "standard deviation", "statistics", "upper quartile"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "
Note that the uquartile and lquartile are calculated as given by the functions below these may change!
\n21/12/2012:
\nThree user defined functions. Added tag udf.
\nflattenint, takes an array of arrays with integers leaves and converts to an integer array by flattening the array. Other two functions, uquartile and lquartile find the lower and upper quartiles.
\nScenarios possible, added sc.
\n22/10/2013:
\n
Redefined functions uquartile and lquartile to fit new definitions. Added helper udf interpolate.
Given sample data find mean, standard deviation, median, interquartile range,
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "As we have to find the median and the interquartile range it is a good idea to order the data and also to total up the data (for the mean) and find the total of the squares of the data (for the variance).
\n{table([['Data']+sort(r),['Squared data']+map(x^2,x,sort(r)),['Index']+map(x,x,1..m*n)],[])}
\nNote that from the above table:
\n$n=\\var{m*n}$.
\n$\\displaystyle \\sum x_i = \\var{sum(r)}$ and
\n$\\displaystyle \\sum x^2_i = \\var{sum(map(x^2,x,r))}$ .
\nThe sample mean is $\\bar{x}=\\displaystyle \\frac{ \\sum x_i}{n}=\\frac{\\var{sum(r)}}{\\var{m*n}}=\\var{mean(r)}=\\var{av}$ to 2 decimal places.
\nThe sample deviation is the square root of the sample variance.
\nSample variance:\\[\\begin{eqnarray*}\\frac{1}{ n -1}\\left(\\sum x_i ^ 2 - n \\bar{x} ^ 2\\right)&=& \\frac{1}{\\var{m*n-1}}\\left(\\var{sum(map(x^2,x,r))}-\\var{m*n}\\times\\var{mean(r)^2}\\right)\\\\&=&\\var{variance(r,true)}\\end{eqnarray*}\\] {Note}
\nSo the sample standard deviation = $\\sqrt{\\var{variance(r,true)}}=\\var{std}$ to 2 decimal places.
\nThe median is $\\var{median(r)} $.
\nThe lower quartile is : $\\var{lquartile(r)}$.
\nThe upper quartile is : $\\var{uquartile(r)}$.
\nThe interquartile range is the difference between these quartiles =$\\var{uquartile(r)}-\\var{lquartile(r)}=\\var{uquartile(r)-lquartile(r)}$
\n\n
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"[[round(tc1/random(7.5..8.5#0.1)),0,round(tc1/random(6.5..7.5#0.1)),round(tc1/random(19.5..20.5#0.1))],[round(tc2/random(3.5..4.5#0.1)),0,round(tc2/random(19.5..20.5#0.1)),round(tc2/random(19.5..20.5#0.1))],[round(tc3/random(2.5..3.5#0.1)),0,random(5..15),round(tc3/random(14.5..15.5#0.1))]]", "name": "a", "description": ""}, "w1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..6)", "name": "w1", "description": ""}, "tc2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "n-tc1-tc3", "name": "tc2", "description": ""}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..3)", "name": "t", "description": ""}, "tc3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "round(n/random(6.5..7.5#0.1))", "name": "tc3", "description": ""}, "sumr": {"group": "Ungrouped variables", "templateType": "anything", "definition": "map(sum(list(r[y])),y,0..3)", "name": "sumr", "description": ""}, "oneof": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(v=1,cat[0]+\" or \"+ cat[1] ,v=2,cat[0]+\" or \"+cat[2],v=3,cat[0]+\" or \"+cat[3],v=4,cat[1]+\" or \"+cat[2],v=5,cat[1]+\" or \"+cat[3],cat[2]+\" or \"+cat[3])", "name": "oneof", "description": ""}, "othercats": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(t=0,cat[1]+\" or \"+ cat[2] ,t=1,cat[0]+\" or \"+cat[3],t=2,cat[0]+\" or \"+cat[1],t=3,cat[0]+\" or \"+cat[2],cat[0]+\" or \"+cat[2])", "name": "othercats", "description": ""}, "ans": {"group": "Ungrouped variables", "templateType": "anything", "definition": "map(precround(x,3),x,[sumr[t]/n,tc[u]/n,ve/n,1-tc[u]/n,ce1/n,ce2/n,sumr[t]*(sumr[t]-1)/(n*(n-1)),tc[u1]*(tc[u1]-1)/(n*(n-1)),r[t][u]/tc[u],we2/tc[u]])", "name": "ans", "description": ""}, "n": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1200..2300#2)", "name": "n", "description": ""}, "tc": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[tc1,tc2,tc3]", "name": "tc", "description": ""}, "at": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[\"teetotal\",\"drinking 1-20 units/week\",\"drinking 21+ units/week\"]", "name": "at", "description": ""}, "we2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(u=0, if(t=0,r[1][0]+r[2][0], if(t=1,r[0][0]+r[3][0], if(t=2,r[0][0]+r[1][0],r[0][0]+r[2][0]))),if(u=1, if(t=0,r[1][1]+r[2][1], if(t=1,r[0][1]+r[3][1], if(t=2,r[0][1]+r[1][1],r[0][1]+r[2][1]))),if(t=0,r[1][2]+r[2][2], if(t=1,r[0][2]+r[3][2], if(t=2,r[0][2]+r[1][2], r[0][2]+r[2][2])))))", "name": "we2", "description": ""}, "u1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..2)", "name": "u1", "description": ""}, "ce1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(w=1, r[0][0],w=2, r[0][1],w=3, r[0][2],w=4,r[1][0],w=5, r[1][1],r[1][2])", "name": "ce1", "description": ""}, "ce2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(w1=1, r[2][0],w1=2, r[2][1],w1=3, r[2][2],w1=4,r[3][0],w1=5, r[3][1],r[3][2])", "name": "ce2", "description": ""}, "drk1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "At[u1]", "name": "drk1", "description": ""}, "tc1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "round(n/random(2.5..3.5#0.1))", "name": "tc1", "description": ""}, "tot": {"group": "Ungrouped variables", "templateType": "anything", "definition": "sum(tc)", "name": "tot", "description": ""}, "things": {"group": "Ungrouped variables", "templateType": "anything", "definition": "\"male\"", "name": "things", "description": ""}, "cats": {"group": "Ungrouped variables", "templateType": "anything", "definition": "\"Marital Status\"", "name": "cats", "description": ""}, "catattrib1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(w=1,cat[0]+\" and \"+At[0],w=2,cat[0]+\" and \"+At[1],w=3,cat[0]+\" and \"+At[2],w=4,cat[1]+\" and \"+At[0],w=5,cat[1]+\" and \"+At[1],cat[1]+\" and \"+At[2])", "name": "catattrib1", "description": ""}, "ve": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(v=1,sumr[0]+ sumr[1] ,v=2,sumr[0]+sumr[2],v=3,sumr[0]+sumr[3],v=4,sumr[1]+sumr[2],v=5,sumr[1]+sumr[3],sumr[2]+sumr[3])", "name": "ve", "description": ""}, "this": {"group": "Ungrouped variables", "templateType": "anything", "definition": "\"alcohol consumption\"", "name": "this", "description": ""}, "catattrib2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(w1=1,cat[2]+\" and \"+At[0],w1=2,cat[2]+\" and \"+At[1],w1=3,cat[2]+\" and \"+At[2],w1=4,cat[3]+\" and \"+At[0],w1=5,cat[3]+\" and \"+At[1],cat[3]+\" and \"+At[2])", "name": "catattrib2", "description": ""}, "somecat": {"group": "Ungrouped variables", "templateType": "anything", "definition": "cat[t]", "name": "somecat", "description": ""}, "drkpair": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(u=0,\"drinks alcohol\",u=1,At[0]+' or ' + At[2],At[0]+\" or \" +At[1])", "name": "drkpair", "description": ""}, "r": {"group": "Ungrouped variables", "templateType": "anything", "definition": "transpose(matrix(map(norm(a[y],1,tc[y]),y,0..2)))", "name": "r", "description": ""}, "drk": {"group": "Ungrouped variables", "templateType": "anything", "definition": "At[u]", "name": "drk", "description": ""}, "u": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..2)", "name": "u", "description": ""}}, "ungrouped_variables": ["somecat", "ve", "othercats", "at", "ce1", "ans", "cat", "drk1", "things", "u1", "tot", "catattrib2", "cats", "tc", "w1", "oneof", "tc2", "tc3", "tc1", "catattrib1", "a", "drkpair", "this", "n", "we2", "drk", "r", "u", "t", "w", "v", "sumr", "ce2"], "rulesets": {}, "showQuestionGroupNames": false, "functions": {"norm": {"type": "list", "language": "javascript", "definition": "\n var b=a;\n var s=-b[x];\n for(i=0;i
1) is {somecat}: Probability =? [[0]]
\n2) is {drk}: Probability = ? [[1]]
\n3) is either {oneof}: Probability =? [[2]]
\n4) {drkpair}: Probability =? [[3]]
\n5) {catattrib1}: Probability =? [[4]]
\n6) {catattrib2}: Probability=? [[5]]
\n ", "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans[6]", "minValue": "ans[6]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans[7]", "minValue": "ans[7]", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\nFind the probability (to 3 decimal places) that two randomly selected {things}s in this survey are
\n7) both {somecat}: Probability = ? [[0]]
\n8) both {drk1}: Probability =? [[1]]
\n ", "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans[8]", "minValue": "ans[8]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans[9]", "minValue": "ans[9]", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\nGiven that a randomly selected {things} in this survey is {drk}, what is the probability that he:
\n9) is {somecat}: Probability = ? [[0]]
\n10) is {othercats}: Probability =? [[1]]
\n\n ", "marks": 0}], "statement": "\n
A survey was conducted to obtain information on {this}. A random sample of {things}s gave :
\n\n
| {Cats} | \n{At[0]} | \n{At[1]} | \n{At[2]} | \nTotal | \n
| {cat[0]} | \n{r[0][0]} | \n{r[0][1]} | \n{r[0][2]} | \n{sumr[0]} | \n
| {cat[1]} | \n{r[1][0]} | \n{r[1][1]} | \n{r[1][2]} | \n{sumr[1]} | \n
| {cat[2]} | \n{r[2][0]} | \n{r[2][1]} | \n{r[2][2]} | \n{sumr[2]} | \n
| {cat[3]} | \n{r[3][0]} | \n{r[3][1]} | \n{r[3][2]} | \n{sumr[3]} | \n
| Totals | \n{tc[0]} | \n{tc[1]} | \n{tc[2]} | \n{tot} | \n
29/12/2012:
\n \t\tAdded tags and description.
\n \t\tThis is easily configurable to other surveys by changing the variables used for labelling. Added the tag sc to denote this. Also added tag, table, as there is a table included.
\n \t\tThe column labels in the table need to be centred.
\n \t\tThe presentation and layout of the questions should be improved and made consistent.
\n \t\tThere is a user-defined function norm(a,x,n) which takes a numeric list a and changes the entry a[x] so that the sum of entries in a is n. Added the tag udf.
\n \t\tCalculations correct on testing.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Finding probabilities from a survey giving a table of data on the alcohol consumption of males. This can be easily adapted to data from other types of surveys.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n1) The total number of {somecat} {things}s is $\\var{sumr[t]}$ hence the probability that a random {things} from this survey is {somecat} is $\\displaystyle \\frac{ \\var{sumr[t]}}{\\var{n}}=\\var{ans[0]}$ to 3 decimal places.
\n\n
2) The total number of {things}s who are {drk} is $\\var{tc[u]}$ hence the probability that a random {things} from this survey is {drk} is $\\displaystyle \\frac{ \\var{tc[u]}}{\\var{n}}=\\var{ans[1]}$ to 3 decimal places.
\n\n
3) Looking at the table there are $\\var{ve}$ {things}s that are {oneof}. Hence the probability is $\\displaystyle \\frac{ \\var{ve}}{\\var{n}}=\\var{ans[2]}$ to 3 decimal places.
\n\n
4) These are the {things}s that are not {drk}, and hence there are $\\var{n}-\\var{tc[u]}=\\var{n-tc[u]}$ of them (see answer to part b)), and the probability of randomly selecting one is $\\displaystyle \\frac{ \\var{n-tc[u]}}{\\var{n}}=\\var{ans[3]}$ to 3 decimal places.
\n\n
5)Looking at the table we see that the number corresponding to {catattrib1} is $\\var{ce1}$. Hence the probability of randomly selecting one is $\\displaystyle \\frac{ \\var{ce1}}{\\var{n}}=\\var{ans[4]}$ to 3 decimal places.
\n6) As in the last question, looking at the table we see that the number corresponding to {catattrib2} is $\\var{ce2}$. Hence the probability of randomly selecting one is $\\displaystyle \\frac{ \\var{ce2}}{\\var{n}}=\\var{ans[5]}$ to 3 decimal places.
\n\n
7) We know from question a) that the probability of selecting a {somecat} {things} is, $\\displaystyle \\frac{ \\var{sumr[t]}}{\\var{n}}$, after this we now have $\\var{sumr[t]-1} $ {somecat} {things}s amongst the $\\var{n-1}$ left, and the probability of yet again selecting one of these is $\\displaystyle \\frac{ \\var{sumr[t]-1}}{\\var{n-1}}$. So the probability of selecting two is $\\displaystyle \\frac{ \\var{sumr[t]}\\times \\var{sumr[t]-1}}{\\var{n}\\times\\var{n-1}}=\\var{ans[6]}$ to 3 decimal places.
\n8) The probability of selecting a {things} who is {drk1} is $\\displaystyle \\frac{ \\var{tc[u1]}}{\\var{n}}$, after this we now have $\\var{tc[u1]-1}$ {drk1} {things}s amongst the $\\var{n-1}$ left, and the probability of yet again selecting one of these is $\\displaystyle \\frac{ \\var{tc[u1]-1}}{\\var{n-1}}$. So the probability of selecting two is $\\displaystyle \\frac{ \\var{tc[u1]}\\times \\var{tc[u1]-1}}{\\var{n}\\times\\var{n-1}}=\\var{ans[7]}$ to 3 decimal places.
\n9) Since there are $\\var{r[t][u]}$ {somecat} {things}s from the $\\var{tc[u]}$ {things}s that are {drk} the probability of selecting one is $\\displaystyle \\frac{\\var{r[t][u]}}{\\var{tc[u]}}= \\var{ans[8]}$ to 3 decimal places.
\n10) Since there are $\\var{we2}$ {othercats} {things}s from the $\\var{tc[u]}$ {things}s that are {drk} the probability of selecting one is $\\displaystyle \\frac{\\var{we2}}{\\var{tc[u]}}= \\var{ans[9]}$ to 3 decimal places.
\n "}, {"name": "Find z-score for sample and calculate confidence interval", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"expb": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'Not at all important'", "name": "expb", "description": ""}, "top": {"group": "Ungrouped variables", "templateType": "anything", "definition": "7", "name": "top", "description": ""}, "score": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5#0.1)", "name": "score", "description": ""}, "sstdev": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0.9..2.0#0.01)", "name": "sstdev", "description": ""}, "upperbound": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(samplemean+1.96*sstdev/sqrt(samplesize),3)", "name": "upperbound", "description": ""}, "these": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'UK shoppers'", "name": "these", "description": ""}, "zscore": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround((score-samplemean)/sstdev,3)", "name": "zscore", "description": ""}, "this": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'the importance of price when making food choice decisions'", "name": "this", "description": ""}, "expt": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'Extremely important'", "name": "expt", "description": ""}, "samplemean": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(4.5..6.5#0.01)", "name": "samplemean", "description": ""}, "samplesize": {"group": "Ungrouped variables", "templateType": "anything", "definition": "1000", "name": "samplesize", "description": ""}, "bottom": {"group": "Ungrouped variables", "templateType": "anything", "definition": "1", "name": "bottom", "description": ""}, "lowerbound": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(samplemean-1.96*sstdev/sqrt(samplesize),3)", "name": "lowerbound", "description": ""}}, "ungrouped_variables": ["zscore", "lowerbound", "bottom", "this", "top", "upperbound", "samplemean", "these", "sstdev", "score", "samplesize", "expb", "expt"], "rulesets": {}, "showQuestionGroupNames": false, "functions": {}, "parts": [{"prompt": "\n\t\t\tWhat is the $z$-score for a score of $\\var{score}$?
\n\t\t\t \n\t\t\tEnter your answer to 3 decimal places.
\n\t\t\t", "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "zscore-0.001", "showCorrectAnswer": true, "marks": 2, "maxValue": "zscore+0.001"}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "lowerbound-0.001", "showCorrectAnswer": true, "marks": 1, "maxValue": "lowerbound+0.001"}, {"showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "upperbound-0.001", "showCorrectAnswer": true, "marks": 1, "maxValue": "upperbound+0.001"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n\t\t\tCalculate the $95$% confidence interval for the population mean $\\mu$:
\n\t\t\tLower bound: [[0]]
\n\t\t\tUpper bound: [[1]]
\n\t\t\t \n\t\t\tEnter your answers to 3 decimal places.
\n\t\t\t", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n\tA recent survey asked $\\var{samplesize}$ {these} to rate {this} on a scale from $\\var{bottom}$ ({expb}) to $\\var{top}$ ({expt}).
\n\tThe mean rating was $\\var{samplemean}$ with SD $\\var{sstdev}$.
\n\t \n\tEnter all values to 3 decimal places.
\n\t \n\t", "tags": ["95%", "ACE2013", "checked2015", "confidence interval", "mean", "mean ", "population mean", "sample", "sample mean", "scale", "standard deviation", "statistics", "z-score"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t17/10/2013:
\n\t\t
Created question.
Given mean and sd of 1000 sample returns on a scale of 1 to 7 together with a given score, find the z-score.
\n\t\tAlso find the 95% confidence interval for the population mean.
\n\t\t"}, "advice": "\n\ta)
\n\tThe $z$-score is given by
\n\t\\[z=\\frac{\\var{score}-\\var{samplemean}}{\\var{sstdev}}=\\var{zscore}\\]
\n\t(To 3 decimal places).
\n\tb)
\n\tThe lower bound for the 95% confidence interval is given by:
\n\tLower bound = $\\displaystyle \\var{samplemean}-1.96 \\times \\frac{ \\var{sstdev}}{\\sqrt{\\var{samplesize}}}=\\var{lowerbound}$
\n\t \n\tUpper bound = $\\displaystyle \\var{samplemean}+1.96 \\times \\frac{ \\var{sstdev}}{\\sqrt{\\var{samplesize}}}=\\var{upperbound}$
\n\t(Both to 3 decimal places.)
\n\tHence for the population mean $\\mu$ we can say that $\\var{lowerbound} \\le\\mu \\le \\var{upperbound}$ with $95$% confidence.
\n\t"}, {"name": "Probability sample mean is within d of the population mean", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"thismany1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "generateprobs[1]", "name": "thismany1", "description": ""}, "mm": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(prob1 > 0.95,[1,0,0,0], prob2>0.95,[0,1,0,0],prob3>0.95,[0,0,1,0],[0,0,0,1])", "name": "mm", "description": ""}, "message": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(prob1>=0.95, \"We take a sample of size \"+ thismany1+ \" as the probability in this case is $\\\\ge 0.95.$\",\nprob2>=0.95, \"We take a sample of size \"+ thismany2+ \" as the probability in this case is $\\\\ge 0.95.$\", \nprob3>=0.95, \"We take a sample of size \"+ thismany3+ \" as the probability in this case is $\\\\ge 0.95$.\",\n \"None of the probabilities is $\\\\ge 0.95$ so we need to take a larger sample.\")", "name": "message", "description": ""}, "popstdev": {"templateType": "anything", "group": "Ungrouped variables", "definition": "generateprobs[0]", "name": "popstdev", "description": ""}, "prob3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(1-2*normalcdf(-thismany3^0.5*thismuch/popstdev,0,1),3)", "name": "prob3", "description": ""}, "thismany3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "generateprobs[3]", "name": "thismany3", "description": ""}, "prob1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(1-2*normalcdf(-thismany1^0.5*thismuch/popstdev,0,1),3)", "name": "prob1", "description": ""}, "these": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random('petrol stations')", "name": "these", "description": ""}, "this": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random('price')", "name": "this", "description": ""}, "thismuch": {"templateType": "anything", "group": "Ungrouped variables", "definition": "generateprobs[5]", "name": "thismuch", "description": ""}, "thismany2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "generateprobs[2]", "name": "thismany2", "description": ""}, "prob2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(1-2*normalcdf(-thismany2^0.5*thismuch/popstdev,0,1),3)", "name": "prob2", "description": ""}, "units": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'pounds'", "name": "units", "description": ""}, "popmean": {"templateType": "anything", "group": "Ungrouped variables", "definition": "generateprobs[4]", "name": "popmean", "description": ""}, "that": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random('unleaded petrol in the UK per litre in 2013')", "name": "that", "description": ""}, "generateprobs": {"templateType": "anything", "group": "Ungrouped variables", "definition": "satisfy(\n [popstdev,thismany1,thismany2,thismany3,popmean,thismuch],\n [random(0.08..0.12#0.02),random(15..30#5),random(40..55#5),random(80..100#10),\n random(1.25..1.35#0.01),random(0.02..0.06)],\n [precround(1-2*normalcdf(-thismany1^0.5*thismuch/popstdev,0,1),3)<>0.95,\n precround(1-2*normalcdf(-thismany2^0.5*thismuch/popstdev,0,1),3)<>0.95,\n precround(1-2*normalcdf(-thismany3^0.5*thismuch/popstdev,0,1),3)<>0.95],\n 1000\n )\n \n \n ", "name": "generateprobs", "description": ""}}, "ungrouped_variables": ["these", "generateprobs", "that", "this", "popstdev", "thismany2", "mm", "thismany1", "thismany3", "units", "popmean", "message", "thismuch", "prob2", "prob3", "prob1"], "rulesets": {}, "functions": {}, "showQuestionGroupNames": false, "parts": [{"showCorrectAnswer": true, "correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "prob1+0.001", "minValue": "prob1-0.001", "prompt": "What is the probability that the mean {this} taken from a sample of $\\var{thismany1}$ {these} is within $\\var{thismuch}$ {units} of the population mean?
\nInput the probability to 3 decimal places.
", "marks": 1}, {"showCorrectAnswer": true, "correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "prob2+0.001", "minValue": "prob2-0.001", "prompt": "What is the probability that the mean {this} taken from a sample of $\\var{thismany2}$ {these} is within $\\var{thismuch}$ {units} of the population mean?
\nInput the probability to 3 decimal places.
", "marks": 1}, {"showCorrectAnswer": true, "correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "prob3+0.001", "minValue": "prob3-0.001", "prompt": "What is the probability that the mean {this} taken from a sample of $\\var{thismany3}$ {these} is within $\\var{thismuch}$ {units} of the population mean?
\nInput the probability to 3 decimal places.
", "marks": 1}, {"displayType": "radiogroup", "choices": ["{thismany1}
", "{thismany2}
", "{thismany3}
", "Increase the sample size
"], "displayColumns": 0, "prompt": "Given the probabilities you have found, what sample size would you recommend to have at least a $0.95$ probability that the sample mean is within $\\var{thismuch}$ {units} of the population mean?
", "distractors": ["", "", "", ""], "shuffleChoices": false, "scripts": {}, "maxMarks": 0, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "matrix": "mm", "marks": 0}], "statement": "The population mean of the {this} of {that} is $\\var{popmean}$ {units} with population standard deviation $\\var{popstdev}$ {units}.
\nFind the following probabilities.
\nInput all probabilities to 3 decimal places.
\n", "tags": ["ACE2013", "checked2015", "mean", "mean ", "normal distribution", "Normal distribution", "Probability", "probability", "sampling", "satisfy", "standard deviation", "statistics", "z score"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "There are variables set up to set up different scenarios. Present one is price of unleaded petrol in the UK.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Given data on population mean and population standard deviation and three sampling sizes, calculate the probabilities that the sample means are within a specified distance from the population mean.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a)
\nLet $M$ be the mean price of the sample
\nConverting to $N(0,1)$ we see that we require the probability that if $\\displaystyle Z=\\frac{|M-\\var{popmean}|}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany1}}}}$ then we require the probability that \\[\\frac{-\\var{thismuch}}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany1}}}} \\lt Z \\lt \\frac{\\var{thismuch}}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany1}}}} \\implies \\var{-thismany1^0.5*thismuch/popstdev}\\lt Z \\lt \\var{thismany1^0.5*thismuch/popstdev}\\].
\nThe probability that $Z \\lt \\var{-thismany1^0.5*thismuch/popstdev}$ is $p=\\var{normalcdf(-thismany1^0.5*thismuch/popstdev,0,1)}$.
\nHence the probability we want is $P_1=1-2p=\\var{prob1}$ to 3 decimal places.
\nb)
\nSimilarly we have for sample size $\\var{thismany2}$ we want the probability that:
\n\\[\\frac{-\\var{thismuch}}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany2}}}} \\lt Z \\lt \\frac{\\var{thismuch}}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany2}}}} \\implies \\var{-thismany2^0.5*thismuch/popstdev}\\lt Z \\lt \\var{thismany2^0.5*thismuch/popstdev}\\].
\nThe probability that $Z \\lt \\var{-thismany2^0.5*thismuch/popstdev}$ is $p=\\var{normalcdf(-thismany2^0.5*thismuch/popstdev,0,1)}$.
\nHence the probability we want is $P_2=1-2p=\\var{prob2}$ to 3 decimal places.
\nc)
\nFor sample size $\\var{thismany3}$ we want the probability that:
\n\\[\\frac{-\\var{thismuch}}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany3}}}} \\lt Z \\lt \\frac{\\var{thismuch}}{\\frac{\\var{popstdev}}{\\sqrt{\\var{thismany3}}}} \\implies \\var{-thismany3^0.5*thismuch/popstdev}\\lt Z \\lt \\var{thismany3^0.5*thismuch/popstdev}\\].
\nThe probability that $Z \\lt \\var{-thismany3^0.5*thismuch/popstdev}$ is $p=\\var{normalcdf(-thismany3^0.5*thismuch/popstdev,0,1)}$.
\nHence the probability we want is $P_3=1-2p=\\var{prob3}$ to 3 decimal places.
\nd) {message}
"}, {"name": "Find a confidence interval for the population mean with variance unknown, ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Shweta Sharma", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21418/"}], "tags": ["checked2015"], "metadata": {"description": "Finding the confidence interval at either 90%, 95% or 99% for the mean given the mean and standard deviation of a sample. The population variance is not given and so the t test has to be used. Various scenarios are included.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "The management of {sc[s]} wants to {dothis[s]}.
\nA random sample of {spec} $\\var{n}$ {t[s]} gave a mean and standard deviation of {p}$\\var{m[s]}$ {units} and {p}$\\var{sd[s]}$ {units} respectively.
", "advice": "1.
\nThe population variance is unknown. So we have to use the t tables to find the confidence interval.
\n2.
\nWe now calculate the $\\var{confl}$ confidence interval:
\nAs we have $\\var{n}-1=\\var{n-1}$ degrees of freedom, the interval is given by:
\n\\[ \\var{m[s]} \\pm t_{\\var{n-1}}\\sqrt{\\frac{\\var{sd[s]}^2}{\\var{n}}}\\]
\nLooking up the t tables for $\\var{confl}$% we see that $t_{\\var{n-1}}=\\var{invt}$ to 3 decimal places.
\nHence:
\nLower value of the confidence interval $=\\;\\displaystyle \\var{m[s]} -\\var{invt} \\sqrt{\\frac{\\var{sd[s]} ^ 2} {\\var{n}}} =\\var{p} \\var{lci}$ {units} to 2 decimal places.
\nUpper value of the confidence interval $=\\;\\displaystyle \\var{m[s]} +\\var{invt} \\sqrt{\\frac{\\var{sd[s]} ^ 2} {\\var{n}}} = \\var{p}\\var{uci}$ {units} to 2 decimal places.
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management and this\",\n \"clothing outlets \",\n \"workers\",\n \"tickets \",\n \"chocolate bars \",\n sc6ch+\" \"]\n \n ", "description": "", "templateType": "anything", "can_override": false}, "sc4ch": {"name": "sc4ch", "group": "Ungrouped variables", "definition": "random(\"the Caribbean\",\"the Mediterranean\",\"North East England\",\"South West England\",\"California\")", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(10..30)", "description": "", "templateType": "anything", "can_override": false}, "sc2ch": {"name": "sc2ch", "group": "Ungrouped variables", "definition": "random(\"local\",\"national\")", "description": "", "templateType": "anything", "can_override": false}, "dothis": {"name": "dothis", "group": "Ungrouped variables", "definition": "\n [\"estimate the mean cost per room of repairing damage caused by its customers during a bank holiday weekend\",\n \"estimate the mean 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\n[[0]]
\nCalculate a $\\var{confl}$% confidence interval $(a,b)$ for the population mean:
\n$a=\\;${p}[[1]] {units} $b=\\;${p}[[2]] {units}
\nEnter both to 2 decimal places.
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\nA company {sc[s]} {dothis[s]} $\\var{sd[s]}$ {units}.
\nA random sample of $\\var{n}$ {t[s]} gives a mean of $\\var{m[s]}$ {units}.
\n\n ", "advice": "
a)
\nWe use the z tables to find the confidence interval as we know the population variance.
\nWe now calculate the $\\var{confl}$% confidence interval.
\nNote that $z_{\\var{confl/100}}=\\var{zval}$ and the confidence interval is given by:
\n\\[ \\var{m[s]} \\pm z_{\\var{confl/100}}\\sqrt{\\frac{\\var{sd2}}{\\var{n}}}\\]
\nHence:
\nLower value of the confidence interval $=\\;\\displaystyle \\var{m[s]} -\\var{zval} \\sqrt{\\frac{\\var{sd2}} {\\var{n}}} = \\var{lci}${units} to 2 decimal places.
\nUpper value of the confidence interval $=\\;\\displaystyle \\var{m[s]} +\\var{zval} \\sqrt{\\frac{\\var{sd2}} {\\var{n}}} = \\var{uci}${units} to 2 decimal places.
\nb)
\nSince $\\var{aim}$ {doornot} {lies} in the confidence interval the answer is {Correct}.
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Is the process satisfactory?\",\n \"The vending machines are supposed to fill 100ml cups. Is the machine working satisfactorily?\",\n \"The company aims for an average salary of \u00a31500 per month per worker. Is the aim being met?\"]\n ", "description": "", "templateType": "anything", "can_override": false}, "correct": {"name": "correct", "group": "Ungrouped variables", "definition": "if(test=0, \"yes\", \"no\")", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(20..100)", "description": "", "templateType": "anything", "can_override": false}, "sc2ch": {"name": "sc2ch", "group": "Ungrouped variables", "definition": "random(\"bolts\",\"screws\")", "description": "", "templateType": "anything", "can_override": false}, "dothis": {"name": "dothis", "group": "Ungrouped variables", "definition": "\n [var1 + \" is\",\n \"with a \"+var2+\" of\",\n var3+ \" is\",\n \"knows that the population standard deviation for the wages of employees is\"]\n \n \n \n \n ", "description": "", "templateType": "anything", "can_override": false}, "var3": {"name": "var3", "group": "Ungrouped variables", "definition": "random(\"The variance of the filling process \",\"The process variance \")", "description": "", "templateType": "anything", "can_override": false}, "sc3ch": {"name": "sc3ch", "group": "Ungrouped variables", "definition": "random(\"hot water.\",\"tea.\",\"coffee.\",\"hot chocolate.\",\"cappuccino.\")", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "\n [random(700..745),\n random(95..98),\n random(90..99),\n random(1000..2500#50)]\n \n \n ", "description": "", "templateType": "anything", "can_override": false}, "mm": {"name": "mm", "group": "Ungrouped variables", "definition": "[1-test,test]", "description": "", "templateType": "anything", "can_override": false}, "sc1ch": {"name": "sc1ch", "group": "Ungrouped variables", "definition": "random(\"flour.\",\"sugar.\",\"dried milk.\",\"instant coffee.\")", "description": "", "templateType": "anything", "can_override": false}, "var2": {"name": "var2", "group": "Ungrouped variables", "definition": "random(\"process variance \",\"population variance \")", "description": "", "templateType": "anything", "can_override": false}, "lci": {"name": "lci", "group": "Ungrouped variables", "definition": "precround(tlci,2)", "description": "", "templateType": "anything", "can_override": false}, "sc": {"name": "sc", "group": "Ungrouped variables", "definition": "\n [\"packs sacks of \"+sc1ch,\n \"manufactures \"+sc2ch,\n \"produces vending machines which fill cups with \"+sc3ch,\n \"in charge of the accounts of a large chain of \"+sc4ch\n ]\n \n ", "description": "", "templateType": "anything", "can_override": false}, "uci": {"name": "uci", "group": "Ungrouped variables", "definition": "precround(tuci,2)", "description": "", "templateType": "anything", "can_override": false}, "units": {"name": "units", "group": "Ungrouped variables", "definition": "switch(s=0,\"g\",s=1,\"mm\",s=2,\"ml\",\"pounds\")", "description": "", "templateType": "anything", "can_override": false}, "tuci": {"name": "tuci", "group": "Ungrouped variables", "definition": "m[s]+zval*sqrt(sd1^2/n)", "description": "", "templateType": "anything", "can_override": false}, "zval": {"name": "zval", "group": "Ungrouped variables", "definition": "if(confl=90,1.645,if(confl=95,1.96,2.576))", "description": "", "templateType": "anything", "can_override": false}, "tlci": {"name": "tlci", "group": "Ungrouped variables", "definition": "m[s]-zval*sqrt(sd1^2/n)", "description": "", "templateType": "anything", "can_override": false}, "spec": {"name": "spec", "group": "Ungrouped variables", "definition": "if(s=2,\"the timecards of \", \" \")", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "random(0..abs(sc)-1)", "description": "", "templateType": "anything", "can_override": false}, "test": {"name": "test", "group": "Ungrouped variables", "definition": "if(aim
Calculate a $\\var{confl}$% confidence interval $(a,b)$ for the population mean:
\n$a=\\;$[[0]]{units} $b=\\;$[[1]]{units}
\nEnter both to 2 decimal places.
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{howwell[s]}
\n[[0]]
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"g"}, "tsqovern": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[t[0]^2/n1,t[1]^2/n2,t[2]^2/n3]", "description": "", "name": "tsqovern"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(v[0]=1,0,v[1]=1,1,v[2]=1,2,v[3]=1,3,4)", "description": "", "name": "u"}, "sd2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(pstdev(r2),2)", "description": "", "name": "sd2"}}, "ungrouped_variables": ["sd1", "sd2", "sd3", "vr", "mbt", "pmsg", "mu3", "stderror", "n", "m1", "tol", "m2", "btss", "stovern", "tss", "dfbt", "m3", "ssq", "sig1", "v1", "sig3", "sig2", "dfrs", "cmsg", "rss", "tsqovern", "mu1", "g", "r2", "mu2", "ss", "k", "r3", "mrs", "r1", "u", "t", "v", "n1", "n2", "n3", "pvalue"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {"pstdev": {"type": "number", "language": "jme", "definition": "sqrt(abs(l)/(abs(l)-1))*stdev(l)", "parameters": [["l", "list"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "m1", "minValue": "m1", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "sd1", "minValue": "sd1", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "t[0]", "minValue": "t[0]", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ssq[0]", "minValue": "ssq[0]", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "m2", "minValue": "m2", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "sd2", "minValue": "sd2", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "t[1]", "minValue": "t[1]", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ssq[1]", "minValue": "ssq[1]", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "m3", "minValue": "m3", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "sd3", "minValue": "sd3", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "t[2]", "minValue": "t[2]", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ssq[2]", "minValue": "ssq[2]", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "g", "minValue": "g", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ss", "minValue": "ss", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\nFirst fill in this table with the appropriate values, all decimals to 2 decimal places:
\n| \n | $\\overline{x}_i$ | \n$s_i$ | \n$T_i$ | \n$\\sum x^2$ | \n$n_i$ | \n
| Group A | \n[[0]] | \n[[1]] | \n[[2]] | \n[[3]] | \n6 | \n
|---|---|---|---|---|---|
| Group B | \n[[4]] | \n[[5]] | \n[[6]] | \n[[7]] | \n6 | \n
| Group C | \n[[8]] | \n[[9]] | \n[[10]] | \n[[11]] | \n6 | \n
| \n | \n | \n | $G=\\;$[[12]] | \nSum of Squares=[[13]] | \n$N=18$ | \n
Note that in doing this you will have supplied the sample means and sample standard deviations for the three groups.
\n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "tss", "minValue": "tss", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "btss", "minValue": "btss", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "rss", "minValue": "rss", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Now find the following, all to 2 decimal places:
\n$\\displaystyle SST\\;=\\;$[[0]], $\\displaystyle SSG\\;=\\;$[[1]], $\\displaystyle SSW\\;=\\;$[[2]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "dfbt", "minValue": "dfbt", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "btss", "minValue": "btss", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "mbt", "minValue": "mbt", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "vr", "minValue": "vr", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "dfrs", "minValue": "dfrs", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "rss", "minValue": "rss", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "mrs", "minValue": "mrs", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "n-1", "minValue": "n-1", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "tss", "minValue": "tss", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Now complete the ANOVA table using the values to 2 decimal places obtained above:
\n| Source | df | SS | MS | F |
|---|---|---|---|---|
| Between groups | \n[[0]] | \n[[1]] | \n[[2]] | \n[[3]] | \n
| Within groups | \n[[4]] | \n[[5]] | \n[[6]] | \n- | \n
| Total | \n[[7]] | \n[[8]] | \n- | \n- | \n
Note that F is found by taking the ratio of two of the values in this table.
\nInput all numbers to 2 decimal places.
", "showCorrectAnswer": true, "marks": 0}, {"displayType": "radiogroup", "choices": ["$p$ less than $0.1\\%$
", "$p$ lies between $0.1\\%$ and $1\\%$
", "$p$ lies between $1 \\%$ and $5\\%$
", "$p$ lies between $5 \\%$ and $10\\%$
", "$p$ is greater than $10\\%$
"], "displayColumns": 1, "prompt": "Give the value of $F$ you have found, choose the range for the $p$ value by looking up the critical values of $F_{2,15}$ (one-sided).
\n| $10\\%$ | \n$5\\%$ | \n$1\\%$ | \n$0.1\\%$ | \n
| $2.70$ | \n$3.68$ | \n$6.36$ | \n$11.34$ | \n
Given the $p$-value and the range you have found, what is the strength of evidence against the null hypothesis that the mean times taken are the same for the three groups?
", "distractors": ["", "", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "v", "marks": 0}, {"displayType": "radiogroup", "choices": ["We reject the null hypothesis at the $0.1\\%$ level", "We reject the null hypothesis at the $1\\%$ level.", "We reject the null hypothesis at the $5\\%$ level.", "We do not reject the null hypothesis but consider further investigation.", "We do not reject the null hypothesis."], "displayColumns": 1, "prompt": "Hence what is your decision based on the above ANOVA analysis?
", "distractors": ["", "", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "v", "marks": 0}], "statement": "The following data arose in a comparison of the effects of alcohol on the time taken to complete a task. There were three groups of subjects; Group A had no alcohol, Group B had two units over 1 hour and Group C had 4 units over 1 hour. The responses are the times (in seconds) taken to complete a word-matching task.
\n| Group A (0 units) | \n$\\var{r1[0]}$ | \n$\\var{r1[1]}$ | \n$\\var{r1[2]}$ | \n$\\var{r1[3]}$ | \n$\\var{r1[4]}$ | \n$\\var{r1[5]}$ | \n
|---|---|---|---|---|---|---|
| Group B (2 units) | \n$\\var{r2[0]}$ | \n$\\var{r2[1]}$ | \n$\\var{r2[2]}$ | \n$\\var{r2[3]}$ | \n$\\var{r2[4]}$ | \n$\\var{r2[5]}$ | \n
| Group C (4 units) | \n$\\var{r3[0]}$ | \n$\\var{r3[1]}$ | \n$\\var{r3[2]}$ | \n$\\var{r3[3]}$ | \n$\\var{r3[4]}$ | \n$\\var{r3[5]}$ | \n
", "tags": ["ACE2013", "ANOVA", "average", "checked2015", "data analysis", "degrees of freedom", "F-test", "hypothesis testing", "mean", "mean ", "one-way ANOVA", "one-way Anova", "standard deviation", "statistics", "stats", "variance"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t
11/07/2012:
\n \t\t \t\t
Added tags.
Calculation not yet tested.
\n \t\t \t\t23/07/2012:
\n \t\t \t\tAdded description.
\n \t\t \t\tChecked calculation.
\n \t\t \t\t3/08/2012:
\n \t\t \t\tAdded tags.
\n \t\t \t\tQuestion appears to be working correctly.
\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "One-way ANOVA example
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": ""}, {"name": "Perform a paired sample t-test, ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"pvalue": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ttest(0,d,2),3)", "name": "pvalue", "description": ""}, "r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu1,sig1)),12)", "name": "r1", "description": ""}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(tvalue>=t999,[1,0,0,0,0],if(tvalue>=t99,[0,1,0,0,0],if(tvalue>=t95,[0,0,1,0,0],if(tvalue>=t90,[0,0,0,1,0],[0,0,0,0,1]))))", "name": "v", "description": ""}, "pmsg": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[' is less than $0.001$',' lies between $0.001$ and $0.01$',' lies between $0.01$ and $0.05$',' lies between $0.05$ and $0.10$',' is greater than $0.10$']", "name": "pmsg", "description": ""}, "sig1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..3#0.2)", "name": "sig1", "description": ""}, "t90": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1.796", "name": "t90", "description": ""}, "tvalue": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(meandiff*sqrt(n)/stdiff),3)", "name": "tvalue", "description": ""}, "cmsg": {"templateType": "anything", "group": "Ungrouped variables", "definition": "['do reject','do reject','do not reject','do not reject','do not reject']", "name": "cmsg", "description": ""}, "t99": {"templateType": "anything", "group": "Ungrouped variables", "definition": "3.106", "name": "t99", "description": ""}, "mu1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(18..25#0.5)", "name": "mu1", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(v[0]=1,0,v[1]=1,1,v[2]=1,2,v[3]=1,3,4)", "name": "t", "description": ""}, "sig2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..4#0.2)", "name": "sig2", "description": ""}, "meandiff": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(mean(d),3)", "name": "meandiff", "description": ""}, "t999": {"templateType": "anything", "group": "Ungrouped variables", "definition": "4.437", "name": "t999", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "list(vector(r2)-vector(r1))", "name": "d", "description": ""}, "object": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'Individual'", "name": "object", "description": ""}, "stdiff": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(pstdev(d),3)", "name": "stdiff", "description": ""}, "t95": {"templateType": "anything", "group": "Ungrouped variables", "definition": "2.201", "name": "t95", "description": ""}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "12", "name": "n", "description": ""}, "r2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu2,sig2)),12)", "name": "r2", "description": ""}, "mu2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mu1+random(2..4#0.1)", "name": "mu2", "description": ""}, "cmsg1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "['very strong','strong','slight','no','no']", "name": "cmsg1", "description": ""}, "msg": {"templateType": "anything", "group": "Ungrouped variables", "definition": "['is greater than $\\\\var{t999}$','lies between $\\\\var{t99}$ and $\\\\var{t999}$','lies between $\\\\var{t95}$ and $\\\\var{t99}$','lies between $\\\\var{t90}$ and $\\\\var{t95}$','is less than $\\\\var{t90}$']", "name": "msg", "description": ""}}, "ungrouped_variables": ["pmsg", "msg", "cmsg1", "t999", "meandiff", "object", "sig1", "tvalue", "stdiff", "sig2", "t99", "t95", "t90", "cmsg", "d", "mu1", "r1", "r2", "mu2", "n", "t", "v", "pvalue"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {"pstdev": {"type": "number", "language": "jme", "definition": "sqrt(abs(l)/(abs(l)-1))*stdev(l)", "parameters": [["l", "list"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{meandiff}", "maxValue": "{meandiff}", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{stdiff}", "maxValue": "{stdiff}", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{tvalue}", "maxValue": "{tvalue}", "marks": 1}], "type": "gapfill", "prompt": "Find the mean and standard deviations of the difference between left and right {attempt}s.
\nCalculate differences for left {attempt} times – right {attempt} times. Make sure you take the differences this way round.
\nMean of difference = [[0]] (input to 3 decimal places )
\nStandard deviation of difference = [[1]] (input to 3 decimal places)
\nNow find the t-test statistic $T$ using the values you have just calculated and input the absolute value $|T|$ here: [[2]] (3 decimal places).
\n", "showCorrectAnswer": true, "marks": 0}, {"displayType": "radiogroup", "choices": ["
$p$ less than $0.1 \\%$
", "$p$ lies between $0.1\\%$ and $1 \\%$
", "$p$ lies between $1 \\%$ and $5\\%$
", "$p$ lies between $5 \\%$ and $10\\%$
", "$p$ is greater than $10\\%$
"], "displayColumns": 0, "prompt": "Give the value of the t-statistic you have found, choose the range for the $p$ value by looking up the t-statistic tables:
", "distractors": ["", "", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "v", "marks": 0}, {"displayType": "radiogroup", "choices": ["Very Strong Evidence
", "Strong Evidence
", "Evidence
", "Weak Evidence
", "No Evidence
"], "displayColumns": 0, "prompt": "Given the $p$-value and the range you have found, what is the strength of evidence against the null hypothesis that there is no difference in the average times for the left and right hands?
", "distractors": ["", "", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "v", "marks": 0}], "statement": "The following data was obtained from $12$ individuals. The observations consist of the time taken to complete a dexterity task using their left and right hands.
\n| {object} | A | B | C | D | E | F | G | H | I | J | K | L |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Right | \n$\\var{r1[0]}$ | \n$\\var{r1[1]}$ | \n$\\var{r1[2]}$ | \n$\\var{r1[3]}$ | \n$\\var{r1[4]}$ | \n$\\var{r1[5]}$ | \n$\\var{r1[6]}$ | \n$\\var{r1[7]}$ | \n$\\var{r1[8]}$ | \n$\\var{r1[9]}$ | \n$\\var{r1[10]}$ | \n$\\var{r1[11]}$ | \n
| Left | \n$\\var{r2[0]}$ | \n$\\var{r2[1]}$ | \n$\\var{r2[2]}$ | \n$\\var{r2[3]}$ | \n$\\var{r2[4]}$ | \n$\\var{r2[5]}$ | \n$\\var{r2[6]}$ | \n$\\var{r2[7]}$ | \n$\\var{r2[8]}$ | \n$\\var{r2[9]}$ | \n$\\var{r2[10]}$ | \n$\\var{r2[11]}$ | \n
Carry out by hand a paired t-test to test whether there is evidence of a difference in the average times for the left and right hands.
", "tags": ["ACE2013", "average", "checked2015", "data analysis", "differences", "elementary statistics", "hypothesis testing", "mean", "mean ", "mean of differences", "paired t-test", "PSY2010", "standard deviation", "standard deviation of differences", "statistics", "stats", "t-test", "variance"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t11/07/2012:
\n \t\t \t\t
Added tags.
Calculation not yet tested.
\n \t\t \t\t23/07/2012:
\n \t\t \t\tAdded description.
\n \t\t \t\tChecked calculation.
\n \t\t \t\tChanged display slightly in Advice.
\n \t\t \t\t3/08/2012:
\n \t\t \t\tAdded tags.
\n \t\t \t\tQuestion appears to be working correctly.
\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Paired t-test to see if there is a difference between times take in a task.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The table of differences is given by:
\n| {object} | A | B | C | D | E | F | G | H | I | J | K | L |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Right | \n$\\var{r1[0]}$ | \n$\\var{r1[1]}$ | \n$\\var{r1[2]}$ | \n$\\var{r1[3]}$ | \n$\\var{r1[4]}$ | \n$\\var{r1[5]}$ | \n$\\var{r1[6]}$ | \n$\\var{r1[7]}$ | \n$\\var{r1[8]}$ | \n$\\var{r1[9]}$ | \n$\\var{r1[10]}$ | \n$\\var{r1[11]}$ | \n
| Left | \n$\\var{r2[0]}$ | \n$\\var{r2[1]}$ | \n$\\var{r2[2]}$ | \n$\\var{r2[3]}$ | \n$\\var{r2[4]}$ | \n$\\var{r2[5]}$ | \n$\\var{r2[6]}$ | \n$\\var{r2[7]}$ | \n$\\var{r2[8]}$ | \n$\\var{r2[9]}$ | \n$\\var{r2[10]}$ | \n$\\var{r2[11]}$ | \n
| Differences | \n$\\var{d[0]}$ | \n$\\var{d[1]}$ | \n$\\var{d[2]}$ | \n$\\var{d[3]}$ | \n$\\var{d[4]}$ | \n$\\var{d[5]}$ | \n$\\var{d[6]}$ | \n$\\var{d[7]}$ | \n$\\var{d[8]}$ | \n$\\var{d[9]}$ | \n$\\var{d[10]}$ | \n$\\var{d[11]}$ | \n
We test the following hypothesis:
\n$H_0:\\;\\mu_d=0$ versus $H_1:\\;\\mu_d\\neq 0$
\n$n=\\var{n}$ and the mean of the differences is $\\overline{d}=\\var{meandiff}$.
\nThe variance $V$ of the differences is calculated to be $\\var{pstdev(d)^2}$
Hence we have the standard deviation $s_d= \\sqrt{V}=\\var{stdiff}$ to 3 decimal places.
The paired t-statistic is given by:
\n\\[\\begin{eqnarray*} T&=&\\frac{\\overline{d}-\\mu_d}{\\frac{s_d}{\\sqrt{n}}}\\\\&=&\\frac{\\var{meandiff}-0}{\\frac{\\var{stdiff}}{\\sqrt{\\var{n}}}}\\\\&=&\\var{tvalue}\\end{eqnarray*}\\]
\n(Using the null hypothesis that the means are the same i.e. $\\mu_d=0$.)
\nHence our test statistic $|T|=\\var{tvalue}$.
\nLooking up this value on the T-distribution table for $t_{11}$
\n\\[\\begin{array}{r|rrrrr}&0.20&0.10&0.05&0.01&0.001\\\\\\hline11&1.363&1.796&2.201&3.106&4.437\\end{array}\\]
\nWe see that the t-statistic {msg[t]} and the table tells us that the $p$ value {pmsg[t]}.
\nHence we conclude that we {cmsg[t]} the null hypothesis. There is {cmsg1[t]} evidence of a difference between the average scores of the two groups.
\n"}, {"name": "Perform a t-test to decide if two sample means differ, , ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "evi": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[\"None\",\"Slight\",\"Moderate\",\"Strong\"]", "description": "", "name": "evi"}, "n1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(5..15)", "description": "", "name": "n1"}, "confl": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[90,95,99]", "description": "", "name": "confl"}, "sd1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(65..90)", "description": "", "name": "sd1"}, "correctc": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(pval>1,\"There is evidence to suggest that average call times for male and female employees differ\",\"There is insufficent evidence to suggest that average call times for male and female employees differ\")", "description": "", "name": "correctc"}, "tval1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(m-m1)*sqrt(n1*n2)/(tpsd*sqrt(n1+n2))", "description": "", "name": "tval1"}, "crit": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(precround(x,3),x,[studenttinv((90+100)/200,n-1),studenttinv((95+100)/200,n-1),studenttinv((99+100)/200,n-1)])", "description": "", "name": "crit"}, "this": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"A call centre company wants to know if there is any difference between the average time spent on the telephone, per call to customers, between male and female employees.\"", "description": "", "name": "this"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "n1+n2-1", "description": "", "name": "n"}, "tval": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tval1,3)", "description": "", "name": "tval"}, "dothis": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(pval <2, 'retain','reject')", "description": "", "name": "dothis"}, "pm": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[\"is greater than 10%\",\"lies between 5% and 10%\",\"lies between 1% and 5%\",\"is less than 1%\"]", "description": "", "name": "pm"}, "n2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(20..30)-n1", "description": "", "name": "n2"}, "mm": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(pval=0,[1,0,0,0],pval=1,[0,1,0,0],pval=2,[0,0,1,0],[0,0,0,1])", "description": "", "name": "mm"}, "things": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"male employees\"", "description": "", "name": "things"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(220..380#10)", "description": "", "name": "m"}, "evi1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[\"no\",\"slight\",\"moderate\",\"strong\"]", "description": "", "name": "evi1"}, "units": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"seconds\"", "description": "", "name": "units"}, "sd": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(55..85)", "description": "", "name": "sd"}, "m1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(280..400#10)", "description": "", "name": "m1"}, "fac": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(pval<2,\"There is evidence to suggest that average call times for male and female employees differ\",\"There is insufficent evidence to suggest that average call times for male and female employees differ\")", "description": "", "name": "fac"}, "tpsd": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sqrt(((n1-1)*sd^2+(n2-1)*sd1^2)/(n-1))", "description": "", "name": "tpsd"}, "dmm": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(pval<2,[1,0],[0,1])", "description": "", "name": "dmm"}, "that": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"the average time spent on the telephone \"", "description": "", "name": "that"}, "pval": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(tval
If $\\mu_M$ is the mean for time spent by {things} and $\\mu_F$ is the mean for time spent by {things1} then you are given that:
\n$\\operatorname{H}_0\\;:\\;\\mu_M=\\mu_F$.
\nStep 2: Alternative Hypothesis
\n$\\operatorname{H}_1\\;:\\;\\mu_M \\neq \\mu_F$.
\n\n ", "showCorrectAnswer": true, "scripts": {}, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "t", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "psd+tol", "minValue": "psd-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "tval+tol", "minValue": "tval-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "
Step 3: Test statistic
\nShould we use the z or t test statistic? Input z or t.
\n[[0]]
\nNow calculate the pooled standard deviation: [[1]] (to 3 decimal places)
\n\n
Now calculate the test statistic = ? [[2]] (to 3 decimal places)
\n\n
(Note that in this calculation you should use a value for the pooled standard deviation which is accurate to at least 5 decimal places and not the value you found to 3 decimal places above).
", "showCorrectAnswer": true, "marks": 0}, {"stepsPenalty": 0, "scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["{pm[0]}", "{pm[1]}", "{pm[2]}", "{pm[3]}"], "displayColumns": 0, "distractors": ["", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "mm", "marks": 0}], "type": "gapfill", "prompt": "Step 4: p-value range
\nUse tables to find a range for your p -value.
\nChoose the correct range here for p : [[0]]
\n\n
Click on Show steps below to get more information on using the t tables to find the p-value range. You will not lose any marks.
\n", "steps": [{"type": "information", "prompt": "
Click here to get more information about using t tables.
\nYou will also find the critical values of the t tables in this link.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["{evi[0]}", "{evi[1]}", "{evi[2]}", "{evi[3]}"], "displayColumns": 0, "distractors": ["", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "mm", "marks": 0}, {"displayType": "radiogroup", "choices": ["Retain", "Reject"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "dmm", "marks": 0}, {"displayType": "radiogroup", "choices": ["{Correctc}", "{Fac}"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0], "marks": 0}], "type": "gapfill", "prompt": "\nGiven the $p$ - value and the range you have found, what is the strength of evidence against the null hypothesis?
\n[[0]]
\nYour Decision:
\n[[1]]
\n\n
Conclusion:
\n[[2]]
\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\n{this}
\nA random sample of $\\var{n1}$ {things} and $\\var{n2}$ {things1} gave the following results in {units}.
\n{table([['Male',{m},{sd}],['Female',{m1},{sd1}]],[' ','Mean','Standard deviation'])}
\nPerform an appropriate hypothesis test to see if there is any difference between {that} between {things} and {things1} (the null and alternative hypotheses have been set out for you).
\n ", "tags": ["ACC1012", "accept null hypothesis", "ACE2013", "alternative hypothesis", "checked2015", "comparing means", "degree of freedom", "diagram", "hypothesis testing", "link", "MAS1403", "null hypothesis", "p values", "pooled standard deviation", "population variance", "random sample", "reject null hypothesis", "sample mean", "sampling", "sc", "statistics", "Steps", "steps", "t tables", "t test", "test statistic", "two-tailed test"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t3/01/2012:
\n \t\tAdded tag sc as can be changed to other applications. Perhaps the tables used should be improved.
\n \t\tMissing a diagram from the original iassess question, hence tag diagram added.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Given two sets of data, sample mean and sample standard deviation, on performance on the same task, make a decision as to whether or not the mean times differ. Population variance not given, so the t test has to be used in conjunction with the pooled sample standard deviation.
\nLink to use of t tables and p-values in Show steps.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "
b)
Step 3 : Test statistic
\nWe should use the t statistic as the population variance is unknown.
\nThe pooled standard deviation is given by :
\n\\[s = \\sqrt{\\frac{\\var{n1 -1} \\times \\var{sd} ^ 2 + \\var{n2 -1} \\times \\var{sd1} ^ 2 }{\\var{n1} + \\var{n2} -2}} = \\var{tpsd} = \\var{psd}\\] to 3 decimal places.
\nThe test statistic is given by \\[t = \\frac{|\\var{m} -\\var{m1}|}{s \\sqrt{\\frac{1}{ \\var{n1} }+\\frac{1}{ \\var{n2}}}} = \\var{tval}\\] to 3 decimal places.
\n(Using $s=\\var{tpsd}$ in this formula.)
\nc)
\nStep 4: p value range.
\nAs the degree of freedom is $\\var{n1}+\\var{n2}-2=\\var{n-1}$ we use the $t_{\\var{n-1}}$ tables. We have the following data from the tables:
\n{table([['Critical Value',crit[0],crit[1],crit[2]]],['p value','10%','5%','1%'])}
\nWe see that the $p$ value {pm[pval]}.
\nd)
\nStep 5: Conclusion
\nHence there is {evi1[pval]} evidence against $\\operatorname{H}_0$ and so we {dothis} $\\operatorname{H}_0$.
\n{Correctc}.
"}, {"name": "Perform t-test for hypothesis given sample mean and standard deviation", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "dmm": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(pval<2,[1,0],[0,1])", "description": "", "name": "dmm"}, "thisamount": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(70..90)", "description": "", "name": "thisamount"}, "confl": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(90,95,99)", "description": "", "name": "confl"}, "pval": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(tvalStep 1: Null Hypothesis
\n$\\operatorname{H}_0\\;: \\; \\mu=\\;$[[0]]
\nStep 2: Alternative Hypothesis
\n$\\operatorname{H}_1\\;: \\; \\mu \\neq\\;$[[1]]
\n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "t", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "tval+tol", "minValue": "tval-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Step 3: Test statistic
\nShould we use the z or t test statistic? [[0]] (enter z or t).
\nNow calculate the test statistic = ? [[1]] (to 3 decimal places)
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["{pm[0]}", "{pm[1]}", "{pm[2]}", "{pm[3]}"], "displayColumns": 0, "distractors": ["", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "mm", "marks": 0}], "type": "gapfill", "prompt": "\nStep 4: p-value
\nUse tables to find a range for your $p$-value.
\nChoose the correct range here for $p$ : [[0]]
\n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["{evi[0]}", "{evi[1]}", "{evi[2]}", "{evi[3]}"], "displayColumns": 0, "distractors": ["", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "mm", "marks": 0}, {"displayType": "radiogroup", "choices": ["Retain", "Reject"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "dmm", "marks": 0}, {"displayType": "radiogroup", "choices": ["{Correctc}", "{Fac}"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0], "marks": 0}], "type": "gapfill", "prompt": "\nStep 5: Conclusion
\n\n
Given the $p$ - value and the range you have found, what is the strength of evidence against the null hypothesis?
\n[[0]]
\nYour Decision:
\n[[1]]
\n\n
Conclusion:
\n[[2]]
\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\n{this}
\n{claim}
\n{test}
\nA sample of {n} {things}
\n{resultis} £{m} with a standard deviation of £{stand}.
\nPerform an appropriate hypothesis test to see if the claim made by the online flight company is substantiated (use a two-tailed test).
\n ", "tags": ["checked2015", "MAS1403"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t2/01/2012:
\n \t\tAdded tag sc as has string variables in order to generate other scenarios.
\n \t\tThe jstat function studenttinv(critvalue,n-1) gives the critical p values correctly.
\n \t\tAdded tag diagram as the i-assess question advice has a nice graphic of the p-value and the appropriate decision.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Provided with information on a sample with sample mean and standard deviation, but no information on the population variance, use the t test to either accept or reject a given null hypothesis.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\na)
\nStep 1: Null Hypothesis
\n$\\operatorname{H}_0\\;: \\; \\mu=\\;\\var{thisamount}$
\nStep 2: Alternative Hypothesis
\n$\\operatorname{H}_1\\;: \\; \\mu \\neq\\;\\var{thisamount}$
\nb)
\nWe should use the t statistic as the population variance is unknown.
\nThe test statistic:
\n\\[t =\\frac{ |\\var{m} -\\var{thisamount}|} {\\sqrt{\\frac{\\var{stand} ^ 2 }{\\var{n}}}} = \\var{tval}\\]
\nto 3 decimal places.
\nc)
\nAs $n=\\var{n}$ we use the $t_{\\var{n-1}}$ tables. We have the following data from the tables:
\n{table([['Critical Value',crit[0],crit[1],crit[2]]],['p value','10%','5%','1%'])}
\nWe see that the $p$ value {pm[pval]}.
\n
d)
Hence there is {evi1[pval]} evidence against $\\operatorname{H}_0$ and so we {dothis} $\\operatorname{H}_0$.
\n{Correctc}
\n "}, {"name": "Find regression equation and correlation coefficient", "extensions": ["stats", "jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[random(0..6),random(1..7),random(4..10),random(7..13),random(12..18),random(4..20),random(16..22),random(19..25),random(17..23),random(13..19),random(8..14),random(3..9)]", "description": "", "name": "r1"}, "spxy": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sxy-t[0]*t[1]/n", "description": "", "name": "spxy"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(1/n*(t[1]-spxy/ss[0]*t[0]),3)", "description": "", "name": "a"}, "tcorr": {"templateType": "anything", "group": "Ungrouped variables", "definition": "spxy/sqrt(ss[0]*ss[1])", "description": "", "name": "tcorr"}, "ssq": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[sum(map(x^2,x,r1)),sum(map(x^2,x,r2))]", "description": "", "name": "ssq"}, "prediction": {"templateType": "anything", "group": "Ungrouped variables", "definition": "round(a+b*thisval)", "description": "", "name": "prediction"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[sum(r1),sum(r2)]", "description": "", "name": "t"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..20)", "description": "", "name": "a1"}, "sumr": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(sum(map(res[x]^2,x,0..n-1)),3)", "description": "", "name": "sumr"}, "tol1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.01", "description": "", "name": "tol1"}, "beverage": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(\"home-brewed beer\",\"home-brewed lager\",\"specially-brewed beer\",\"super-strength lager\",\"cold-filtered lager\",\"ice-filtered cider\",\"cherry cider\")", "description": "", "name": "beverage"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "12", "description": "", "name": "n"}, "ss": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[ssq[0]-t[0]^2/n,ssq[1]-t[1]^2/n]", "description": "", "name": "ss"}, "corr": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tcorr,2)", "description": "", "name": "corr"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(spxy/ss[0],3)", "description": "", "name": "b"}, "pub": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(\"Black Bull Inn\",\"County Inn\",\"Dog and Duck Pub\",\"Slug and Lettuce Pub\", \"Cross Keys Pub\",\"Newcastle Arms Pub\",\"Red Lion Pub\")", "description": "", "name": "pub"}, "res": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(precround(r2[x]-(a+b*r1[x]),2),x,0..n-1)", "description": "", "name": "res"}, "ch": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..11)", "description": "", "name": "ch"}, "sc": {"templateType": "anything", "group": "Ungrouped variables", "definition": "r1[ch]", "description": "", "name": "sc"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.25..0.45#0.05)", "description": "", "name": "b1"}, "sxy": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(r1[x]*r2[x],x,0..n-1))", "description": "", "name": "sxy"}, "thisval": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(15..22)", "description": "", "name": "thisval"}, "obj": {"templateType": "anything", "group": "Ungrouped variables", "definition": "['Jan','Feb','March','April','May','June','July','August','Sept','Oct','Nov','Dec']", "description": "", "name": "obj"}, "owner": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(\"Kevin\",\"Mary\",\"Bill\",\"Doreen\",\"Peter\",\"Helen\",\"Michael\",\"Samantha\")", "description": "", "name": "owner"}, "r2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(round(a1+b1*x+random(-9..9)),x,r1)", "description": "", "name": "r2"}, "ls": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(a+b*sc,2)", "description": "", "name": "ls"}, "tsqovern": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[t[0]^2/n,t[1]^2/n]", "description": "", "name": "tsqovern"}, "rsquared": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(spxy^2/(ss[0]*ss[1]),3)", "description": "", "name": "rsquared"}}, "ungrouped_variables": ["ch", "prediction", "b1", "owner", "sxy", "res", "spxy", "ls", "tol", "tcorr", "tsqovern", "ssq", "sumr", "thisval", "a1", "pub", "corr", "a", "b", "obj", "r1", "r2", "ss", "tol1", "n", "beverage", "t", "sc", "rsquared"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {"regressline": {"type": "html", "language": "javascript", "definition": "\n var div = Numbas.extensions.jsxgraph.makeBoard('600px','600px',\n{boundingBox:[-5,maxy,maxx,-5],\n axis:true,\n showNavigation:false,\n grid:false});\n var board = div.board; \nvar l1=board.create('text',[maxx/2,-2,'Temperature']);\nvar l2=board.create('text',[-2,maxy/2,'Sales']);\nvar names = ['Jan','Feb','Mar','Apr','May','Jun','Jul','Aug','Sep','Oct','Nov','Dec'];\nfor (j=0;j<12;j++){ board.create('point',[r1[j],r2[j]],{fixed:true, style:3,name:'\\\\\\\\['+names[j]+'\\\\\\\\]'})};\nvar a1 = board.create('point',[minx+5,miny+5],{color:'blue'});\nvar b1 = board.create('point',[minx+7,miny+5],{color:'blue'});\nfunction updr(a,b){\n var s=0;\n for(var i=0;i<12;i++){\ns=s+Math.pow(r2[i]-a*r1[i]-b,2);}\ns=Numbas.math.niceNumber(Numbas.math.precround(s,2));\n$('#rsquared').text(s);}\n var li=board.create('line',[a1,b1], {straightFirst:false, straightLast:false});\n var a=0;\n var b=0;\n function dr(p){\n p.on('drag',function(){\n a = Numbas.math.niceNumber((b1.Y()-a1.Y())/(b1.X()-a1.X()));\n b = Numbas.math.niceNumber((a1.Y()*b1.X()-a1.X()*b1.Y())/(b1.X()-a1.X()));\n Numbas.exam.currentQuestion.parts[1].gaps[0].display.studentAnswer(a);\n Numbas.exam.currentQuestion.parts[1].gaps[1].display.studentAnswer(b);\n updr(a,b);\n })};\n dr(a1);\n dr(b1);\n \nreturn div;\n\n \n", "parameters": [["r1", "list"], ["r2", "list"], ["minx", "number"], ["maxx", "number"], ["miny", "number"], ["maxy", "number"]]}, "regfun": {"type": "html", "language": "javascript", "definition": "\n var div = Numbas.extensions.jsxgraph.makeBoard('600px','600px',\n{boundingBox:[-5,maxy,maxx,-5],\n axis:true,\n showNavigation:false,\n grid:true});\n var board = div.board; \nvar l1=board.create('text',[maxx/2,-2,'Temperature']);\nvar l2=board.create('text',[-2,maxy/2,'Sales']);\n var names = ['Jan','Feb','Mar','Apr','May','Jun','Jul','Aug','Sep','Oct','Nov','Dec'];\n for (j=0;j<12;j++){ board.create('point',[r1[j],r2[j]],{fixed:true, style:3, strokecolor:\"#0000a0\", name:'\\\\\\\\['+names[j]+'\\\\\\\\]'})};\nvar regressionPolynomial = JXG.Math.Numerics.regressionPolynomial(1, r1, r2);\nvar reg = board.create('functiongraph',[regressionPolynomial],{strokeColor:'blue',name:'Regression Line.',withLabel:true}); \n //for(var i=0;i<12;i++){board.create(\"segment\",[[r1[i],r2[i]],[r1[i],regressionPolynomial(r1[i])]])};\nvar regExpression = regressionPolynomial.getTerm();\nvar regTeX = Numbas.jme.display.exprToLaTeX(regExpression,[],scope);\n\nvar t = board.create('text',[1,5,\nfunction(){ return \"\\\\[r(Y) = \" + regExpression +'\\\\]';}\n],\n{strokeColor:'black',fontSize:18}); \nvar t1 = board.create('text',[5,maxy,\nfunction(){ return \"\\\\[SSE = \" + sumr +'\\\\]';}\n],\n{strokeColor:'black',fontSize:18}); \nvar t2 = board.create('text',[20,maxy,\nfunction(){ return \"\\\\[R^2 = \" + rsquared +'\\\\]';}\n],\n{strokeColor:'black',fontSize:18}); \nreturn div;\n \n", "parameters": [["r1", "list"], ["r2", "list"], ["maxx", "number"], ["maxy", "number"], ["rsquared", "number"], ["sumr", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "corr+tol1", "minValue": "corr-tol1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Calculate the sample correlation coefficient $r$ for these data:
\n$r=\\;$[[0]] (enter to 2 decimal places).
", "showCorrectAnswer": true, "marks": 0}, {"stepsPenalty": 0, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "b+tol", "minValue": "b-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "a+tol", "minValue": "a-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Calculate the equation of the best fitting regression line.
\n\\[Y = \\beta_0 + \\beta_1X.\\] Find $\\beta_0$ and $\\beta_1$ to 5 decimal places, then input them below to 3 decimal places. You will use these approximate values in the rest of the question.
\n$\\beta_1=\\;$[[0]], $\\beta_0=\\;$[[1]] (enter both to 3 decimal places).
\nYou can experiment by dragging the points A and B around to see if you can get close to the regression line.
\n{regressline(r1,r2,min(r1)-10,max(r1)+10,min(r2)-10,max(r2)+10)}
\n\nClick on Show steps if you want more information on calculating $\\beta_0$ and $\\beta_1$. You will not lose any marks by doing so.
\n", "steps": [{"type": "information", "prompt": "
To find $\\beta_0$ and $\\beta_1$ you first find $\\displaystyle \\beta_1 = \\frac{SPXY}{SSX}$ where:
\n$\\displaystyle SPXY=\\sum xy - \\frac{(\\sum x)\\times (\\sum y)}{\\var{n}}$
\n$\\displaystyle SSX=\\sum x^2 - \\frac{(\\sum x)^2}{\\var{n}}$
\nThen $\\displaystyle \\beta_0 = \\frac{1}{\\var{n}}\\left[\\sum y-\\beta_1 \\sum x\\right]$
\nNow go back and fill in the values for $\\beta_0$ and $\\beta_1$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "prediction+1", "minValue": "prediction-1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Next month, the average temperature in {owner}'s town is forecast to be {thisval} Celsius. Use the regression equation in the second part to predict sales of the {beverage} in that month.
\nWhat is the predicted value of sales (in hundreds of pounds) ?
\nUse the values of $\\beta_0$ and $\\beta_1$ you input above to 3 decinal places.
\nEnter the predicted sales here: [[0]] (hundreds of pounds to the nearest whole number).
\n", "showCorrectAnswer": true, "marks": 0}], "statement": "{owner} owns the {pub}. {owner} believes that sales of {beverage} in the pub are linked to the average monthly temperature, with higher sales being recorded in months with higher temperatures. To investigate, {owner} records the average monthly temperature in the local town over a period of one year ($X$ degrees Celsius), along with total monthly sales of {beverage} ($Y$ hundred pounds). The results are shown in the table below:
\n| Month | $\\var{obj[0]}$ | $\\var{obj[1]}$ | $\\var{obj[2]}$ | $\\var{obj[3]}$ | $\\var{obj[4]}$ | $\\var{obj[5]}$ | $\\var{obj[6]}$ | $\\var{obj[7]}$ | $\\var{obj[8]}$ | $\\var{obj[9]}$ | $\\var{obj[10]}$ | $\\var{obj[11]}$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $X$ (temperature) | \n$\\var{r1[0]}$ | \n$\\var{r1[1]}$ | \n$\\var{r1[2]}$ | \n$\\var{r1[3]}$ | \n$\\var{r1[4]}$ | \n$\\var{r1[5]}$ | \n$\\var{r1[6]}$ | \n$\\var{r1[7]}$ | \n$\\var{r1[8]}$ | \n$\\var{r1[9]}$ | \n$\\var{r1[10]}$ | \n$\\var{r1[11]}$ | \n
| $Y$ (sales, £100s) | \n$\\var{r2[0]}$ | \n$\\var{r2[1]}$ | \n$\\var{r2[2]}$ | \n$\\var{r2[3]}$ | \n$\\var{r2[4]}$ | \n$\\var{r2[5]}$ | \n$\\var{r2[6]}$ | \n$\\var{r2[7]}$ | \n$\\var{r2[8]}$ | \n$\\var{r2[9]}$ | \n$\\var{r2[10]}$ | \n$\\var{r2[11]}$ | \n
You are given the following information:
\n| $X$ | \n$\\sum x=\\;\\var{t[0]}$ | \n$\\sum x^2=\\;\\var{ssq[0]}$ | \n
|---|---|---|
| $Y$ | \n$\\sum y=\\;\\var{t[1]}$ | \n$\\sum y^2=\\;\\var{ssq[1]}$ | \n
Also you are given $\\sum xy = \\var{sxy}$.
", "tags": ["ACE2013", "checked2015", "correlation", "data analysis", "fitted value", "linear regression", "regression", "statistics"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "04/02/2014:
\nNo advice as yet. Adapted from iassess question for ACE.
\n18/02/2014:
\nSlight changes in notation from Regression 3. No SSE
", "licence": "Creative Commons Attribution 4.0 International", "description": "Find a regression equation given 12 months data on temperature and sales of a drink.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "{regfun(r1,r2,max(r1)+10,max(r2)+10,rsquared,sumr)}
"}, {"name": "Interpret logistic regression output from Minitab", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ansa+ansb*thismuch/1000", "description": "", "name": "r1"}, "seb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.04..1.2#0.000001)", "description": "", "name": "seb"}, "sea": {"templateType": "anything", "group": "Ungrouped variables", "definition": "round((t*v1+(100-t)*v2)+0.000001)/100", "description": "", "name": "sea"}, "ansa": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(-2.24*sea,3)", "description": "", "name": "ansa"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..100)", "description": "", "name": "t"}, "tol1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.01", "description": "", "name": "tol1"}, "thismuch": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(15000..35000#1000)", "description": "", "name": "thismuch"}, "v2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "seb*thismuch/1000+0.9", "description": "", "name": "v2"}, "v1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "seb*thismuch/1000-0.9", "description": "", "name": "v1"}, "prob1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(e^(r1)/(1+e^r1),2)", "description": "", "name": "prob1"}, "ansb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(2.25*seb,3)", "description": "", "name": "ansb"}}, "ungrouped_variables": ["seb", "r1", "tol1", "v1", "v2", "t", "tol", "ansa", "ansb", "thismuch", "sea", "prob1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansa+tol", "minValue": "ansa-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansb+tol", "minValue": "ansb-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Find $A$ and $B$ each to 3 decimal places:
\n$A=\\;$[[0]] $B=\\;$[[1]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "prob1+tol1", "minValue": "prob1-tol1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Find the probability that an employee with an annual salary of £{thismuch} will be obese.
\nEnter your answer to two decimal places.
\nProbability = [[0]]
", "showCorrectAnswer": true, "marks": 0}], "statement": "It is claimed that there is a relationship between job type and obesity, with employees in more highly-paid, senior roles being more likely to be overweight than those in other roles.
\nYou are an analyst working for a large company and have been given the task to see if this is the case at your workplace.
\nYou randomly sample 15 employees and record their annual salary ($X$, in thousands of pounds) and whether or not they are obese (have a body mass index of more than 30: $Y=1$ if yes, $Y=0$ if no).
\nA logistic regression was performed in Minitab, with the following (edited) output shown below:
\n| \n\n | \n\n | \n\n | \n\n | \n\n | \n\n Odds \n | \n 95% CI \n | |
| \n Predictor \n | \n Coef \n | \n SE Coef \n | \n Z \n | \n P \n | \n\n Ratio \n | \n Lower \n | \n Upper |
| \n Constant \n | \n\n $A$ \n | \n\n {sea} \n | \n\n -2.24 \n | \n\n 0.025 \n | \n\n\n | \n\n\n | \n\n\n | \n
| \n $x$ \n | \n\n $B$ \n | \n\n {seb} \n | \n\n 2.25 \n | \n\n 0.025 \n | \n\n 1.18 \n | \n\n 1.02 \n | \n\n 1.37 \n | \n
09/02/2014:
\nBased on an i-assess question. First draft finished.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Interpreting the minitab output from a logistic regression model of salary against obesity as measured by BMI.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a)
\nA is given by A = -2.24 x SE Coef (A) = -2.24 x {sea} = {ansa} .
\nB is given by B = 2.25 x SE Coef (B) = 2.25 x {seb} = {ansb} .
\nBoth to 3 decimal places.
\nb)
\nThe probability is given by:
\n\\[\\begin{align}P(Y=1 | X=\\var{thismuch})&=\\frac{e^{A+B\\times\\var{thismuch/1000}}}{1+e^{A+B\\times\\var{thismuch/1000}}}\\\\&=\\frac{e^{\\var{r1}}}{1+e^{\\var{r1}}}=\\var{prob1}\\end{align}\\]
\nto 2 decimal places.
\n"}, {"name": "Interpret Minitab output of multiple regression", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "pred": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ansa +cb*thatmany+ansc*thismany/1000+ansd*q,1)", "description": "", "name": "pred"}, "thismany": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(5000..12000#1000)", "description": "", "name": "thismany"}, "thatmany": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "name": "thatmany"}, "thisrest": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(\"McDonald's\",\"Pizza Hut\",\"Kentucky Fried Chicken\",\"The Log Fire Pizza Co.\",\"The China Cook\",\"Aneesa's Indian Buffet\",\"Pacino's Italian Restaurant\")", "description": "", "name": "thisrest"}, "sv": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[if(ind=6,\"Rubbish\",\" \"),\"Below Average\",\"Average\",\"Above Average\",\n if(ind=3,\" \",\"Good\"),if(ind>4,\"Fantastic\",\" \")]", "description": "", "name": "sv"}, "hascond": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(q=1,cond,\n p=0,\"has not got a drive-thru window\",\n p=1,\"is not open late (after 11pm)\",\n p=2,\"is not located on a public transport route\",\n \"is not located in a town centre\")", "description": "", "name": "hascond"}, "se": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.03..0.15#0.0001)", "description": "", "name": "se"}, "si": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(sval^2,3)", "description": "", "name": "si"}, "place": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(\"South Shields\", \"North Shields\",\"Newcastle\",\"Sunderland\",\"Alnwick\",\"Durham\",\"Metro Centre\")", "description": "", "name": "place"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1,2,3)", "description": "", "name": "p"}, "ansd": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(2.53*sed,3)", "description": "", "name": "ansd"}, "q": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "name": "q"}, "sea": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5.5#0.001)", "description": "", "name": "sea"}, "ansa": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(3.887*sea,3)", "description": "", "name": "ansa"}, "sed": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..2.20#0.001)", "description": "", "name": "sed"}, "cb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-0.5..-0.1#0.0001)", "description": "", "name": "cb"}, "cond": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(p=0,\"has got a drive-thru window\" ,if(p=1,\"is open late (after 11pm)\",if(p=2,\"is located on a public transport route\",\"is located in a town centre\")))", "description": "", "name": "cond"}, "ansc": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(3.82*se,3)", "description": "", "name": "ansc"}, "ansb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(cb)/2.64,3)", "description": "", "name": "ansb"}, "ind": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3,4,5,6)", "description": "", "name": "ind"}, "sval": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.2..0.9#0.000001)", "description": "", "name": "sval"}}, "ungrouped_variables": ["cb", "sed", "cond", "sea", "ind", "pred", "sval", "tol", "thisrest", "thismany", "ansa", "ansb", "ansc", "ansd", "thatmany", "sv", "hascond", "q", "p", "si", "place", "se"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["$X_1$
", "$X_2$
", "$X_3$
"], "displayColumns": 0, "distractors": ["", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [0, 0, 1], "marks": 0}], "type": "gapfill", "prompt": "Which of these variables is an indicator variable?
\n[[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansa+tol", "minValue": "ansa-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansb+tol", "minValue": "ansb-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansc+tol", "minValue": "ansc-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansd+tol", "minValue": "ansd-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansa+tol", "minValue": "ansa-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansc+tol", "minValue": "ansc-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansd+tol", "minValue": "ansd-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "si+tol", "minValue": "si-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Based on a recent survey of some of the restaurants in the North-East, the following (edited) Minitab output was obtained:
\nRegression Analysis: $y$ versus $x_1,\\;x_2,\\;x_3$
\nThe regression equation is: y = ********
\n\n| \n Predictor \n | \n Coef \n | \n SE Coef \n | \n T \n | \n P \n |
|---|---|---|---|---|
| \n Constant \n | \n\n $A$ \n | \n\n {sea} \n | \n\n 3.887 \n | \n\n 0.002$ \n | \n
| \n $x_1$ \n | \n\n {cb} \n | \n\n $B$ \n | \n\n -2.64 \n | \n\n 0.021 \n | \n
| \n $x_2$ \n | \n\n $C$ \n | \n\n {se} \n | \n\n 3.82 \n | \n\n 0.002 \n | \n
| \n $x_3$ \n | \n\n $D$ \n | \n\n {sed} \n | \n\n 2.53 \n | \n\n 0.024 \n | \n
s={sval} R-sq= 92.1% R-Sq(adj)=93.9%
\n(i) Find the values of $A,\\;B,\\;C$ and $D$ to 3 decimal places.
\n$A=\\;$[[0]], $B=\\;$[[1]]
\n$C=\\;$[[2]], $D=\\;$[[3]]
\n(ii) Write down the full fitted regression equation:
\n\\[Y = \\beta_0+ \\beta_1 X_1 + \\beta_2 X_2 + \\beta_3X_3 + \\epsilon,\\;\\;\\epsilon \\sim N(0,\\sigma^2)\\]
\n$Y=\\;$[[4]]-$\\var{abs(cb)}X_1$+[[5]]$X_2$+[[6]]$X_3+\\epsilon$
\nNote that you are given the coefficient of $X_1$ from the Minitab table.
\nAlso find $\\sigma^2=\\;$[[7]] to 3 decimal places where $\\epsilon \\sim N(0,\\sigma^2)$
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "pred+0.1", "minValue": "pred-0.1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Predict sales for a restaurant with {thatmany} competitors, a population of {thismany} within 1 kilometre and that {hascond}.
\nEnter the predicted value to one decimal place:
\n$Y=\\;$[[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ind-1", "minValue": "ind-1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "It is thought that a fourth variable - customer satisfaction based on a recent survey - could also be a predictor of sales.
\nEach of the restaurants was given an overall satisfaction rating by choosing one of the following in the survey:
\n{sv[0]} {sv[1]} {sv[2]} {sv[3]} {sv[4]} {sv[5]}
\nHow many indicator variables would need to be used here?: [[0]]
", "showCorrectAnswer": true, "marks": 0}], "statement": "The management at {thisrest} propose the following model to predict sales Y at their {place} branch.
\n\\[Y = \\beta_0+ \\beta_1 X_1 + \\beta_2 X_2 + \\beta_3X_3 + \\epsilon\\]
\nwhere
\n$X_1=\\;$number of competitors within one kilometre.
\n$X_2=\\;$population within one kilometre (in 1000s).
\n$X_3=\\;$ 1 if {cond}, 0 otherwise.
", "tags": ["ACE2013", "checked2015", "indicator variables", "minitab output", "multiple regression", "regression", "regression equation", "statistics"], "rulesets": {}, "preamble": {"css": ".minitab {\nfont-family: 'Courier', monospace;\n}", "js": ""}, "type": "question", "metadata": {"notes": "09/02/2014:
\nFirst draft. Based on an i-assess question for ACE.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Asking users to interpret a minitab output to give the coefficients of a multiple regression together with a prediction based on the subsequent equation.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a) $X_3$ is the indicator variable.
\nb)
\n\n(i) The values of A, B, C and D are given by:
\nA = 3.887 x SE Coef (A) = 3.887 x {sea} = {ansa} to 3 decimal places.
\nB = Coef (B)/-2.64 = {cb}/-2.64 = {ansb} to 3 decimal places.
\nC = 3.82 x SE Coef (C) = 3.82 x {se} = {ansc} to 3 decimal places.
\nD = 2.53 x SE Coef (D) = 2.53 x {sed} = {ansd} to 3 decimal places.
\nii) The fitted regression equation is:
\n\\[Y=\\var{ansa}-\\var{abs(cb)}X_1+\\var{ansc}X_2+\\var{ansd}X_3+\\epsilon\\] with $\\sigma^2=\\var{sval}^2=\\var{si}$ all to 3 decimal places.
\nc)
\n\nUsing the above fitted model where $X_1=\\var{thatmany}$ and $X_2= \\frac{\\var{thismany}}{1000}=\\var{thismany/1000}$ and since $X_3=\\var{q}$ as the restaurant {hascond} we find :
\n\\[Y=\\var{ansa}-\\var{abs(cb)}\\times \\var{thatmany}+\\var{ansc}\\times \\var{thismany/1000}+\\var{ansd}\\times \\var{q}=\\var{pred}\\] to one decimal place.
\nd)
\nThere are {ind} categories in the survey and so the number of indicator variables is {ind}-1={ind-1}.
"}, {"name": "Linear regression - find line of best fit given summary statistics, ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"tol": {"group": "Ungrouped variables", "templateType": "anything", "definition": "0.001", "name": "tol", "description": ""}, "r1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "repeat(round(normalsample(67,8)),n)", "name": "r1", "description": ""}, "spxy": {"group": "Ungrouped variables", "templateType": "anything", "definition": "sxy-t[0]*t[1]/n", "name": "spxy", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(1/n*(t[1]-spxy/ss[0]*t[0]),3)", "name": "a", "description": ""}, "ssq": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[sum(map(x^2,x,r1)),sum(map(x^2,x,r2))]", "name": "ssq", "description": ""}, "sc": {"group": "Ungrouped variables", "templateType": "anything", "definition": "r1[ch]", "name": "sc", "description": ""}, "ch": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..7)", "name": "ch", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0.25..0.45#0.05)", "name": "b1", "description": ""}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[sum(r1),sum(r2)]", "name": "t", "description": ""}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(10..20)", "name": "a1", "description": ""}, "sxy": {"group": "Ungrouped variables", "templateType": "anything", "definition": "sum(map(r1[x]*r2[x],x,0..n-1))", "name": "sxy", "description": ""}, "ss": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[ssq[0]-t[0]^2/n,ssq[1]-t[1]^2/n]", "name": "ss", "description": ""}, "n": {"group": "Ungrouped variables", "templateType": "anything", "definition": "8", "name": "n", "description": ""}, "obj": {"group": "Ungrouped variables", "templateType": "anything", "definition": "['A','B','C','D','E','F','G','H']", "name": "obj", "description": ""}, "r2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "map(round(a1+b1*x+random(-9..9)),x,r1)", "name": "r2", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(spxy/ss[0],3)", "name": "b", "description": ""}, "ls": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(a+b*sc,2)", "name": "ls", "description": ""}, "res": {"group": "Ungrouped variables", "templateType": "anything", "definition": "map(precround(r2[x]-(a+b*r1[x]),2),x,0..n-1)", "name": "res", "description": ""}, "tsqovern": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[t[0]^2/n,t[1]^2/n]", "name": "tsqovern", "description": ""}}, "ungrouped_variables": ["tsqovern", "a", "b", "obj", "r1", "r2", "ss", "res", "ssq", "ls", "n", "a1", "ch", "spxy", "t", "tol", "sc", "sxy", "b1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "showQuestionGroupNames": false, "functions": {"pstdev": {"type": "number", "language": "jme", "definition": "sqrt(abs(l)/(abs(l)-1))*stdev(l)", "parameters": [["l", "list"]]}}, "parts": [{"marks": 0, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "b-tol", "maxValue": "b+tol", "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "a-tol", "maxValue": "a+tol", "marks": 1}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "prompt": "To find $\\beta_0$ and $\\beta_1$ you first find $\\displaystyle \\beta_1 = \\frac{SPXY}{SSX}$ where:
\n$\\displaystyle SPXY=\\sum xy - \\frac{(\\sum x)\\times (\\sum y)}{\\var{n}}$
\n$\\displaystyle SSX=\\sum x^2 - \\frac{(\\sum x)^2}{\\var{n}}$
\nThen $\\displaystyle \\beta_0= \\frac{1}{\\var{n}}\\left[\\sum y-\\beta_1 \\sum x\\right]$
\nNow go back and fill in the values for $\\beta_0$ and $\\beta_1$.
\n", "scripts": {}, "marks": 0}], "prompt": "
Calculate the equation of the best fitting regression line:
\n\\[Y = \\beta_0 + \\beta_1 X.\\] Find $\\beta_0$ and $\\beta_1$ to 5 decimal places, then input them below to 3 decimal places. You will use these approximate values in the rest of the question.
\n$\\beta_1=\\;$[[0]], $\\beta_0=\\;$[[1]] (both to 3 decimal places.)
\nYou are given the following information:
\n| First Test$(X)$ | \n$\\sum x=\\;\\var{t[0]}$ | \n$\\sum x^2=\\;\\var{ssq[0]}$ | \n
|---|---|---|
| Later Score$(Y)$ | \n$\\sum y=\\;\\var{t[1]}$ | \n$\\sum y^2=\\;\\var{ssq[1]}$ | \n
Also you are given $\\sum xy = \\var{sxy}$.
\nClick on Show steps if you want more information on calculating $\\beta_0$ and $\\beta_1$. You will not lose any marks by doing so.
\n", "stepsPenalty": 0}, {"showCorrectAnswer": true, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "prompt": "
What is the predicted Later score for employee $\\var{obj[ch]}$ in the First test?
Use the values of $\\beta_0$ and $\\beta_1$ you input above.
\nEnter the predicted Later score here: (to 2 decimal places)
", "minValue": "ls-0.01", "correctAnswerFraction": false, "marks": 1, "maxValue": "ls+0.01"}, {"marks": 0, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "res[ch]-0.01", "maxValue": "res[ch]+0.01", "marks": 1}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "prompt": "\nThe residual value is given by:
\nRESIDUAL = OBSERVED - FITTED.
\nIn this case the observed value for $\\var{obj[ch]}$ is $\\var{r2[ch]}$ and you get the fitted value by feeding the First test value $\\var{r1[ch]}$ into the regression equation.
\n\n", "scripts": {}, "marks": 0}], "prompt": "
Use the result above to calculate the residual value for employee $\\var{obj[ch]}$.
\nClick on Show steps to see what is meant by the residual value if you have forgotten. You will not lose any marks by doing so.
\nResidual value = (to 2 decimal places).[[0]]
", "stepsPenalty": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\nTo monitor its staff appraisal methods, a personnel department compares the results of the tests carried out on employees at their first appraisal with an assessment score of the same individuals two years later. The resulting data are as follows:
\n| Employee | $\\var{obj[0]}$ | $\\var{obj[1]}$ | $\\var{obj[2]}$ | $\\var{obj[3]}$ | $\\var{obj[4]}$ | $\\var{obj[5]}$ | $\\var{obj[6]}$ | $\\var{obj[7]}$ |
|---|---|---|---|---|---|---|---|---|
| First Test $(X)$ | \n$\\var{r1[0]}$ | \n$\\var{r1[1]}$ | \n$\\var{r1[2]}$ | \n$\\var{r1[3]}$ | \n$\\var{r1[4]}$ | \n$\\var{r1[5]}$ | \n$\\var{r1[6]}$ | \n$\\var{r1[7]}$ | \n
| Later Score $(Y)$ | \n$\\var{r2[0]}$ | \n$\\var{r2[1]}$ | \n$\\var{r2[2]}$ | \n$\\var{r2[3]}$ | \n$\\var{r2[4]}$ | \n$\\var{r2[5]}$ | \n$\\var{r2[6]}$ | \n$\\var{r2[7]}$ | \n
21/12/2012:
\nChecked rounding, OK. Added tag cr1.
\nPossible use of scenarios, so added tag sc.
\nThe array r2 generates the regression data which has an inbuilt noise via r1[x]+random(-9..9).
\n", "licence": "Creative Commons Attribution 4.0 International", "description": "
Find a regression equation.
"}, "advice": ""}, {"name": "Multiple linear regression - decide which variable to exclude", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"p2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.01..0.6#0.001 except [p0,p1])", "description": "", "name": "p2"}, "p4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.005..0.2#0.001 except [p0,p1,p2,p3])", "description": "", "name": "p4"}, "data": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[[\"$\\\\beta_0$\",random(8..20#0.001),p0],[\"$\\\\beta_1$\",random(-8..-1#0.001),p1],\n [\"$\\\\beta_2$\",random(4..9#0.001),p2],[\"$\\\\beta_3$\",random(2..10#0.001),p3],\n [\"$\\\\beta_4$\",random(1..15#0.001),p4]]", "description": "", "name": "data"}, "p0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.005..0.03#0.001)", "description": "", "name": "p0"}, "p1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.005..0.3#0.001 except p0)", "description": "", "name": "p1"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "max(p)", "description": "", "name": "m"}, "p3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.005..0.2#0.001 except [p0,p1,p2])", "description": "", "name": "p3"}, "mm": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(if(p[j]=m,1,0),j,0..3)", "description": "", "name": "mm"}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "findinlist(p,m)+1", "description": "", "name": "v"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[p1,p2,p3,p4]", "description": "", "name": "p"}}, "ungrouped_variables": ["p2", "p3", "p0", "p1", "p4", "mm", "m", "p", "v", "data"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {"findinlist": {"type": "number", "language": "javascript", "definition": "var r=0;\nfor(j=0;j < l.length;j++)\n {if(l[j]==m){r=j;}\n }\nreturn r;", "parameters": [["l", "list"], ["m", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"displayType": "radiogroup", "choices": ["$X_1$
", "$X_2$
", "$X_3$
", "$X_4$
"], "displayColumns": 0, "prompt": "Which predictor variable would you exclude from the model before re-fitting in Minitab?
", "distractors": ["", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "mm", "marks": 0}], "statement": "A multiple linear regression model of the form:
\n\\[Y=\\beta_0+\\beta_1X_1+ \\beta_2X_2+\\beta_3X_3+\\beta_4X_4+\\epsilon \\]
\nis fitted to some data in Minitab. The following table shows estimates of the parameters with associated $p$-values.
\n09/02/2014:
\nFirst draft finished.
", "licence": "Creative Commons Attribution 4.0 International", "description": "A multiple linear regression model of the form:
\n\\[Y=\\beta_0+\\beta_1X_1+ \\beta_2X_2+\\beta_3X_3+\\beta_4X_4+\\epsilon \\]
\nis fitted to some data in Minitab which generates a table showing estimates of the parameters with associated $p$-values. Determine which variable to exclude first.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "You would choose to exclude the predictor variable which had the largest p-value.
\nIn this example we see that $X_{\\var{v}}$ has the largest $p$-value $\\var{m}$ and and we would exclude it as $\\var{m}>0.05$.
\n"}, {"name": "Calculate centred moving averages", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "tol"}, "tothere3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(ch=3,random(\"Mesa Airways.\",\"KiwiJet.\",\"Frontier Airlines.\",\"EasyAir.\",\"RyanJet.\",\"HorizonAir.\",\"SunCity Airlines.\",\"Air Caribe.\"),\"\")", "description": "", "name": "tothere3"}, "tothere": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(ch=1,random(\"Souter Lighthouse\", \"Washington Old Hall\",\"Gibside\",\"Cherryburn 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true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "y8+tol", "minValue": "y8-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "y9+tol", "minValue": "y9-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "y10+tol", "minValue": "y10-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Find the series of (centred) moving averages for these data and complete the table below:
\n| Jan-Mar | April-June | July-Sep | Oct-Dec | |
|---|---|---|---|---|
| 2007 | \n* | \n* | \n[[0]] | \n[[1]] | \n
| 2008 | \n[[2]] | \n[[3]] | \n[[4]] | \n[[5]] | \n
| 2009 | \n[[6]] | \n[[7]] | \n* | \n* | \n
The following tables shows by quarter over the three years 2007, 2008, 2009 {This} {tothere} {nat} {tothere2} {tothere3} {tothere4}
\n{table(rows,[\" \",\"Jan-Mar\",\"Apr-June\",\"July-Sep\",\"Oct-Dec\"])}
", "tags": ["ACE2013", "centred moving averages", "checked2015", "moving average", "statistics", "time series"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "22/03/2014:
\n
Created from ACE i-assess question.
Centred moving averages.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "| Jan-Mar | April-June | July-Sep | Oct-Dec | |
|---|---|---|---|---|
| 2007 | \n* | \n* | \n$\\var{y3}$ | \n$\\var{y4}$ | \n
| 2008 | \n$\\var{y5}$ | \n$\\var{y6}$ | \n$\\var{y7}$ | \n$\\var{y8}$ | \n
| 2009 | \n$\\var{y9}$ | \n$\\var{y10}$ | \n* | \n* | \n
Let $Y_t$ denote the data table values so $Y_1=\\var{ja7},\\;Y_2=\\var{ap7} ,\\ldots, Y_{12}=\\var{j9}$
\nUsing the centred moving average formula for quarters:
\n\\[Y^*_t=\\frac{Y_{t-2}+2(Y_{t-1}+Y_t+Y_{t+1})+Y_{t+2}}{8}\\] we obtain the values in the table above.
\nThus the moving average for Oct-Dec 2007 is:
\n\\[\\frac{Y_{2}+2(Y_{3}+Y_4+Y_{5})+Y_{6}}{8}=\\frac{\\var{ap7} + 2 (\\var{m7} + \\var{j7} + \\var{ja8}) + \\var{ap8}}{8} = \\var{y4}\\]
\n"}, {"name": "Calculate moving averages, perform linear regression and give seasonal adjustments, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"sdt2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(t2-mt2,1)", "description": "", "name": "sdt2"}, "ms1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(th1+f1+s1+m2+t2)/5", "description": "", "name": "ms1"}, "mf2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(t2+th2+f2+s2+m3)/5", "description": "", "name": 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The first one has been done for you. Input all moving averages to one decimal place.
\n\n| Mon | Tues | Thur | Fri | Sat | |
|---|---|---|---|---|---|
| 7/4/08 | \n* | \n* | \n$\\var{mth1}$ | \n[[0]] | \n[[1]] | \n
| 14/4/08 | \n[[2]] | \n[[3]] | \n[[4]] | \n[[5]] | \n[[6]] | \n
| 21/4/08 | \n[[7]] | \n[[8]] | \n[[9]] | \n* | \n* | \n
Fit the linear regression model $Y=\\alpha+\\beta T+\\epsilon$ to this set $Y$ of moving averages, where $T$ represents time. ($T=1$ for 7/4/08, $T=2$ for 8/4/08, $T=3$ for 10/4/08 etc.)
\nYou may use the following summaries:
\n$\\sum t=88,\\;\\;\\sum y=\\var{summa},\\;\\;\\sum ty=\\var{sumty},\\;\\; \\sum t^2=814,\\;\\;\\sum y^2=\\var{sumysquared}$.
\nEstimate for $\\beta= \\;$[[0]] (estimate to 2 decimal places).
\nEstimate for $\\alpha= \\;$[[1]] (estimate to 2 decimal places). Use the estimate for $\\beta$ to 2 decimal places to estimate $\\alpha$.
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\n| Mon | Tues | Thur | Fri | Sat | |
|---|---|---|---|---|---|
| 7/4/08 | \n* | \n* | \n[[0]] | \n[[1]] | \n[[2]] | \n
| 14/4/08 | \n[[3]] | \n[[4]] | \n[[5]] | \n[[6]] | \n[[7]] | \n
| 21/4/08 | \n[[8]] | \n[[9]] | \n[[10]] | \n* | \n* | \n
| Seasonal Mean | \n[[11]] | \n[[12]] | \n[[13]] | \n[[14]] | \n[[15]] | \n
Input all entries to 2 decimal places.
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"customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sat", "maxValue": "sat", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sath-tol", "maxValue": "sath+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "saf", "maxValue": "saf", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sas", "maxValue": "sas", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.25, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, 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\nSeasonal effect for Monday: $S_M=\\;$ [[0]]
\nSeasonal effect for Tuesday: $S_T=\\;$ [[1]]
\nSeasonal effect for Thursday: $S_{Thu}=\\;$ [[2]]
\nSeasonal effect for Friday: $S_F=\\;$ [[3]]
\nSeasonal effect for Saturday: $S_{Sa}=\\;$ [[4]]
\n", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sm-tol", "maxValue": "sm+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "st-tol", "maxValue": "st+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sth-tol", "maxValue": "sth+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "sf-tol", "maxValue": "sf+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ss-tol", "maxValue": "ss+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "Calculate the adusted seasonal effects for each day of the week: (input all your answers to 2 decimal places).
\nAdjusted seasonal effect for Monday: $S_M=\\;$ [[0]]
\nAdjusted seasonal effect for Tuesday: $S_T=\\;$ [[1]]
\nAdjusted seasonal effect for Thursday: $S_{Thu}=\\;$ [[2]]
\nAdjusted seasonal effect for Friday: $S_F=\\;$ [[3]]
\nAdjusted seasonal effect for Saturday: $S_{Sa}=\\;$ [[4]]
\n", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "asm-tol", "maxValue": "asm+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ast-tol", "maxValue": "ast+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "asth-tol", "maxValue": "asth+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "asf-tol", "maxValue": "asf+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ass-tol", "maxValue": "ass+tol", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "Use the regression equation in part 2 and the adjusted seasonal effects in part 5 to forecast the number of orders on {thisdate}
\nEstimated orders on {thisdate} = ?[[0]] (input to the nearest whole number)
", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "estord-tol1", "maxValue": "estord+tol1", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 2, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "{This} has recently opened for business and is open 5 days a week, Monday, Tuesday, Thursday, Friday and Saturday. The number of orders each day during the first three weeks of business is shown below:
\n{table([[\"7/4/08\",{m1},{t1},{th1},{f1},{s1}],[\"14/4/08\",{m2},{t2},{th2},{f2},{s2}],[\"21/4/08\",{m3},{t3},{th3},{f3},{s3}]],[\" \",\"Mon\",\"Tues\",\"Thur\",\"Fri\",\"Sat\"])}
\n", "tags": ["checked2015", "forecasting", "moving averages", "regression", "seasonal adjustments", "seasonality", "statistics", "time series"], "rulesets": {}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Moving averages, regression and seasonal adjustments.
"}, "advice": "a)
\nThe completed moving average table is as follows:
\n| Mon | Tues | Thur | Fri | Sat | |
|---|---|---|---|---|---|
| 7/4/08 | \n* | \n* | \n$\\var{mth1}$ | \n$\\var{mf1}$ | \n$\\var{ms1}$ | \n
| 14/4/08 | \n$\\var{mm2}$ | \n$\\var{mt2}$ | \n$\\var{mth2}$ | \n$\\var{mf2}$ | \n$\\var{ms2}$ | \n
| 21/4/08 | \n$\\var{mm3}$ | \n$\\var{mt3}$ | \n$\\var{mth3}$ | \n* | \n* | \n
b)
\nThe mean value of $T$ is $\\overline{t}=\\frac{3+4+\\cdots+13}{11}=\\frac{88}{11}=8$.
\nThe mean value of $Y$ is $\\overline{y}=\\frac{\\var{summa}}{11}=\\var{meany}$.
\nWe have:
\n\\[\\begin{align}S_{TY}&=\\sum ty-11\\overline{t}\\overline{y}=\\var{sumty}-11\\times 8 \\times\\var{ meany}\\\\
S_{YY}&=\\sum y^2-11\\overline{y}^2=\\var{sumysquared}-11\\var{meany}^2=\\var{syy}\\\\
S_{TT}&=\\sum t^2-11\\overline{t}^2=814-11\\times 64=110
\\end{align}\\]
Estimate for $\\beta$ is $\\frac{S_{TY}}{S_{TT}}=\\var{testbe}=\\var{estbe}$ to 2 decimal places.
\nEstimate for $\\alpha$ is $\\overline{y}-\\beta\\overline{t}=\\var{meany}-\\var{estbe}\\times\\var{meant}=\\var{testal}=\\var{estal}$ to 2 decimal places.
\nSo the linear regression equation is $Y=\\simplify{{estal}+{estbe}T}+\\epsilon$ coefficients to 2 decimal places.
\nc)
\nThe following table shows the calculated seasonal deviations and seasonal means for each day of the week. These are obtained by taking the moving average data away from the original data on orders. The seasonal means for the days are obtained by taking the means in each column.
\n\n| Mon | Tues | Thur | Fri | Sat | |
|---|---|---|---|---|---|
| 7/4/08 | \n* | \n* | \n$\\var{sdth1}$ | \n$\\var{sdf1}$ | \n$\\var{sds1}$ | \n
| 14/4/08 | \n$\\var{sdm2}$ | \n$\\var{sdt2}$ | \n$\\var{sdth2}$ | \n$\\var{sdf2}$ | \n$\\var{sds2}$ | \n
| 21/4/08 | \n$\\var{sdm3}$ | \n$\\var{sdt3}$ | \n$\\var{sdth3}$ | \n* | \n* | \n
| Seasonal Mean | \n$\\var{sam}$ | \n$\\var{sat}$ | \n$\\var{sath}$ | \n$\\var{saf}$ | \n$\\var{sas}$ | \n
d) The seasonal effects for each day of the week are calculated by first finding the means of all the seasonal deviations found in the last table (not including the seasonal means in the last row). Then you take this away from the seasonal mean for each day.
\nWe find that the mean of the seasonal deviations is:
\n\\[\\simplify[all,!collectNumbers]{({sdth1} + {sdf1} + {sds1} + {sdm2} + {sdt2} + {sdth2} + {sdf2} + {sds2} + {sdm3} + {sdt3} + {sdth3}) / 11} = \\var{om}\\]to 3 decimal places.
\n{comm}
\nSeasonal effect for Monday: $S_M=\\;\\simplify[all,!collectNumbers]{{sam}-{om}}=\\var{sm}$ to 2 decimal places.
\nSeasonal effect for Tuesday: $S_T=\\simplify[all,!collectNumbers]{{sat}-{om}}=\\var{st}$ to 2 decimal places.
\nSeasonal effect for Thursday: $S_{Thu}=\\simplify[all,!collectNumbers]{{sath}-{om}}=\\var{sth}$ to 2 decimal places.
\nSeasonal effect for Friday: $S_F=\\simplify[all,!collectNumbers]{{saf}-{om}}=\\var{sf}$ to 2 decimal places.
\nSeasonal effect for Saturday: $S_{Sa}=\\simplify[all,!collectNumbers]{{sas}-{om}}=\\var{ss}$ to 2 decimal places.
\ne)
\nWe further adjust the seasonal effects for each day by finding the mean of the seasonal effects we have just found and then taking this away from each of the seasonal effects.
\nWe find that the mean of the seasonal deviations is:
\n\\[\\simplify[all,!collectNumbers]{({sm} + {st} + {sth} + {sf} + {ss} ) / 5} = \\var{ms}\\] to 3 decimal places.
\n{comm1}
\nAdjusted seasonal effect for Monday $\\;=\\simplify[all,!collectNumbers]{{sm}-{ms}}=\\var{asm}$ to 2 decimal places.
\nAdjusted seasonal effect for Tuesday$\\;=\\simplify[all,!collectNumbers]{{st}-{ms}}=\\var{ast}$ to 2 decimal places.
\nAdjusted seasonal effect for Thursday$\\;=\\simplify[all,!collectNumbers]{{sth}-{ms}}=\\var{asth}$ to 2 decimal places.
\nAdjusted seasonal effect for Friday$\\;=\\simplify[all,!collectNumbers]{{sf}-{ms}}=\\var{asf}$ to 2 decimal places.
\nAdjusted seasonal effect for Saturday $\\;=\\simplify[all,!collectNumbers]{{ss}-{ms}}=\\var{ass}$ to 2 decimal places.
\nf)
\nWe use the regression equation found above $Y=\\simplify[all,!collectNumbers]{{estal}+{estbe}T}+\\epsilon$ to estimate the number of orders on {thisdate}.
\n\nBut we have to adjust for seasonality using the adjusted seasonal effect found above by adding on the adjusted seasonal effect for {thisday} i.e. $\\var{adj}$.
\nNote that $T=1$ corresponds to 7/4/08 and hence $T=\\var{ti}$ for {thisdate}.
\nSo putting $T=\\var{ti}$ gives $Y=\\simplify[all,!collectNumbers]{{estal}+{estbe}*{ti}+{adj}}=\\var{estal+estbe*ti+adj}=\\var{estord}$ orders to the nearest whole number.
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"correct": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1, \"AR(1)\", \"AR(2)\")", "description": "", "name": "correct"}, "me2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "me1+random(-20,-10,10,20)", "description": "", "name": "me2"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,est11,est12)", "description": "", "name": "p"}, "correct21": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1, \"strong evidence against\", \"moderate evidence against\")", "description": "", "name": "correct21"}, "flse211": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"slight evidence against\"", "description": "", "name": "flse211"}, "flse111": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"no evidence against\"", "description": "", "name": "flse111"}, "flse322": {"templateType": "anything", "group": "Ungrouped variables", "definition": "If(t=1,if(w=1,\"moderate evidence against\", \"slight evidence against\"),\"no evidence against\")", "description": "", "name": "flse322"}, "flse311": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(v=2, \"strong evidence against\",\"moderate evidence against\")", "description": "", "name": "flse311"}, "correctr22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,\"retain this hypothesis\",\"reject this hypothesis\")", "description": "", "name": "correctr22"}, "q3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(v=1,0.01,0.05)", "description": "", "name": "q3"}, "y": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,0,est22)", "description": "", "name": "y"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,\"\",\"s\")", "description": "", "name": "s"}, "q2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,if(w=1,0.05,0.1),0.05)", "description": "", "name": "q2"}, "flse122": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"strong evidence against\"", "description": "", "name": "flse122"}, "flse121": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1,\"moderate evidence against\",\"strong evidence against\")", "description": "", "name": "flse121"}, "forc": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,forc1,precround((me2+est12*(thismany-me2)+est22*(thatmany-me2)),2))", "description": "", "name": "forc"}}, "ungrouped_variables": ["est12", "est11", "flse122", "p22", "an", "me1", "me2", "ine", "flse", "flse121", "q1", "q3", "q2", "correct11", "flse111", "flse221", "flse222", "flse321", "flse322", "tol", "correct", "est22", "that", "forc", "correctr22", "thisno", "forc1", "p11", "p12", "me", "thismany", "flse311", "thatmany", "flser22", "correct21", "flse211", "correct22", "p", "s", "u", "t", "w", "v", "y"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me1", "minValue": "me1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "est11", "minValue": "est11", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me1", "minValue": "me1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me2", "minValue": "me2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "est12", "minValue": "est12", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me2", "minValue": "me2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "est22", "minValue": "est22", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me2", "minValue": "me2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Write down the full estimated AR(1) and AR(2) models:
\n$\\text{AR(1)}:\\;\\;Y_t=a_1+b_1(Y_{t-1}-c_1)+\\epsilon_t$
\nFind $a_1=\\;$[[0]] $b_1=\\;$[[1]] $c_1=\\;$[[2]]
\n$\\text{AR(2)}:\\;\\;Y_t=a_2+b_2(Y_{t-1}-c_2)+d_2(Y_{t-2}-e_2)+\\epsilon_t$
\nFind $a_2=\\;$[[3]], $b_2=\\;$[[4]], $c_2=\\;$[[5]], $d_2=\\;$[[6]], $e_2=\\;$[[7]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["{Correct21}
", "{Flse121}
", "{Flse221}
", "{Flse321}
"], "displayColumns": 0, "distractors": ["", "", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0, 0], "marks": 0}, {"displayType": "radiogroup", "choices": ["retain this hypothesis
", "reject this hypothesis
"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [0, 1], "marks": 0}, {"displayType": "radiogroup", "choices": ["{Correct22}
", "{Flse122}
", "{Flse222}
", "{Flse322}
"], "displayColumns": 0, "distractors": ["", "", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0, 0], "marks": 0}, {"displayType": "radiogroup", "choices": ["{Correctr22}
", "{Flser22}
"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0], "marks": 0}], "type": "gapfill", "prompt": "For the AR(2) fit, choose one of the following:
\n1) For the null hypothesis $\\text{H}_0:\\;\\alpha_1=0$ there is:
\n[[0]]
\nSo we:
\n[[1]]
\n2) For the null hypothesis $\\text{H}_0:\\;\\alpha_2=0$ there is:
\n[[2]]
\nSo we:
\n[[3]]
\n\n", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["{Correct11}
", "{Flse111}
", "{Flse211}
", "{Flse311}
"], "displayColumns": 0, "distractors": ["", "", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0, 0], "marks": 0}, {"displayType": "radiogroup", "choices": ["retain this hypothesis
", "reject this hypothesis
"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [0, 1], "marks": 0}], "type": "gapfill", "prompt": "For the AR(1) fit, choose one of the following:
\nFor the null hypothesis $\\text{H}_0:\\;\\alpha_1=0$ there is:
\n[[0]]
\nSo we:
\n[[1]]
\n", "showCorrectAnswer": true, "marks": 0}, {"displayType": "radiogroup", "choices": ["{Correct}
", "{Flse}
"], "displayColumns": 0, "prompt": "Which of the two models seems most appropriate here?
", "distractors": ["", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0], "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "forc+tol", "minValue": "forc-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Using the most appropriate model as identified in the last part, forecast the next value in the series if the last value was $\\var{thismany}$ {an} $\\var{that}$.
\nForecast= [[0]] (enter to 2 decimal places.)
", "showCorrectAnswer": true, "marks": 0}], "statement": "Minitab was used to fit both an AR(1) model and an AR(2) to a stationary series. The following table summarises the results obtained:
\n\n\n| \n | AR(1) Model | AR(2) Model | ||
|---|---|---|---|---|
| Parameter | \nEstimate | \np-Value | \nEstimate | \np-Value | \n
| $\\alpha_1$ | \n{est11} | \n{p11} | \n{est12} | \n{p12} | \n
| $\\alpha_2$ | \n$\\ldots$ | \n$\\ldots$ | \n{est22} | \n{p22} | \n
| $\\mu$ | \n{me1} | \n$\\ldots$ | \n{me2} | \n$\\ldots$ | \n
23/03/2014:
\nCreated.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Minitab was used to fit both an AR(1) model and an AR(2) to a stationary series. A table is given summarising the results obtained from Minitab. Choose the most appropriate model and make a forecast based on that model.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a) Using the data given we have the models are given by:
\nAR(1):
\n\\[Y_t=a_1+b_1(Y_{t-1}-c_1)+\\epsilon_t\\]
\nwhere $a_1=\\var{me1},\\;\\;b_1=\\alpha_1=\\var{est11},\\;\\;c_1=\\var{me1}$
\nAR(2):
\n\\[Y_t=a_2+b_2(Y_{t-1}-c_2)+d_2(Y_{t-2}-e_2)+\\epsilon_t\\]
\nwhere $a_2=\\var{me2},\\;\\;b_2=\\alpha_1=\\var{est12},\\;\\;c_2=\\var{me2},\\;\\;d_2=\\alpha_2=\\var{est22},\\;\\;e_2=\\var{me2}$
\nb)
\nUsing the above diagram for making decisions on retaining or rejecting the hypotheses on the coefficients we see that:
\nFor the AR(2) fit:
\n$\\var{est12}$ has a p-value of $\\var{p12}$ and so there is {Correct21} the hypothesis that $\\text{H}_0:\\;\\alpha_1=0$ . So we reject this hypothesis.
\n$\\var{est22}$ has a p-value of $\\var{p22}$ and so there is {Correct22} the hypothesis that $\\text{H}_0:\\;\\alpha_2=0$ . So we {Correctr22}.
\nc)
\nFor the AR(1) fit:
\n$\\alpha_1=\\var{est11}$ has a p-value of $\\var{p11} \\lt \\var{q3}$ and so there is {Correct11} the hypothesis that $\\text{H}_0:\\;\\alpha_1=0$ .
\nSo we reject this hypothesis.
\nd)
\nWe see that since in either case we reject the hypothesis that $\\alpha_1=0$ , but that for the AR(2) model we {Correctr22} that $\\alpha_2=0$ , so we choose the {Correct} model as the most appropriate one.
\ne)
\nUsing the {Correct} model we have the forecast for the next period:
\nForecast =$ \\simplify[all,!collectNumbers]{{me}+{p}({thismany}-{me})+{y}({thatmany}-{me})}=\\var{forc}$ to 2 decimal places.
\n\n\n"}, {"name": "Interpret Minitab output of AR(1) model", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"fo": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(fo1=me, fo1+random(40..60),fo1)", "description": "", "name": "fo"}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "tol"}, "fo1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(u*(me-2*mese)+(100-u)*(me+2*mese))/100", "description": "", "name": "fo1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(2.18*sea,2)", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(-0.12*sed,2)", "description": "", "name": "c"}, "sea": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.05..0.45#0.05)", "description": "", "name": "sea"}, "fo3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(afo3,2)", "description": "", "name": "fo3"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..100)", "description": "", "name": "u"}, "afo4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "me+a*(fo3-me)", "description": "", "name": "afo4"}, "sed": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..150#5)", "description": "", "name": "sed"}, "per": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(25..70)", "description": "", "name": "per"}, "afo3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "me+a*(fo2-me)", "description": "", "name": "afo3"}, "me": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(50..150#10)", "description": "", "name": "me"}, "fo2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(afo2,2)", "description": "", "name": "fo2"}, "fo4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(afo4,2)", "description": "", "name": "fo4"}, "afo2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "me+a*(fo -me)", "description": "", "name": "afo2"}, "mese": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(100..300#10)", "description": "", "name": "mese"}}, "ungrouped_variables": ["me", "a", "c", "fo1", "afo2", "fo2", "fo3", "per", "afo4", "sed", "u", "mese", "tol", "afo3", "fo4", "sea", "fo"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "a+tol", "minValue": "a-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "c+tol", "minValue": "c-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Input the values of X and Y:
\nX= [[0]] (to 2 decimal places).
\nY= [[1]] (to 2 decimal places).
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me", "minValue": "me", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "a", "minValue": "a", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me", "minValue": "me", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "The estimated model has the following form:
\n\\[Y_t=a+b(Y_{t-1}-c)+\\epsilon_t\\]
\nFind $a=\\;$[[0]] $b=\\;$[[1]] $c=\\;$[[2]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "fo2", "minValue": "fo2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "fo3", "minValue": "fo3", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "fo4", "minValue": "fo4", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Use your model above to complete the following (edited ) output from Minitab:
\n(Calculate each forecast to 2 decimal places, then use this value to 2 decimal places to calculate the next forecast.)
\n
Forecast from period {per}:
| Period | \nForecast | \n
| {per+1} | \n{fo} | \n
| {per+2} | \n[[0]] | \n
| {per+3} | \n[[1]] | \n
| {per+4} | \n[[2]] | \n
Minitab was used to fit an AR(1) model to a stationary series. The following output was obtained:
\n\nFINAL ESTIMATE OF PARAMETERS
\n| Type | \nCoef | \nSE Coef | \nT | \nP | \n
| AR 1 | \nX | \n{sea} | \n2.18 | \n0.034 | \n
| Constant | \nY | \n{sed} | \n-0.12 | \n0.902 | \n
| Mean | \n{me} | \n{mese} | \n\n | \n |
22/03/2014:
\nCreated.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Minitab was used to fit an AR(1) model to a stationary time series. Given the output answer the following questions about the model and use the model to make forecasts.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a)
\nFrom the table we have $\\displaystyle \\frac{X}{\\var{sea}}=2.18$.
\nHence we find $X=\\var{sea}\\times 2.18=\\var{a}$ to 2 decimal places.
\nSimilarly we have $\\displaystyle \\frac{Y}{\\var{sed}}=-0.12$
\nHence we find $Y=\\var{sed}\\times -0.12=\\var{c}$ to 2 decimal places.
\nb)
\nThe model is
\n\\[Y_t= \\var{me}+\\var{a}(Y_{t-1}-\\var{me})+\\epsilon_t.\\]
\nSo $a=\\var{me},\\;\\;b=\\var{a},\\;\\;c=\\var{me}$.
\nc)
\nPutting $Y_{\\var{per+1}}=\\var{fo1}$ we find:
\n\\[Y_{\\var {per + 2}} = \\var{me} + \\var{a} (Y_{\\var{per + 1}} -\\var{me}) = \\var{me} + \\var{a} \\times (\\var{fo1} -\\var{me}) = \\var{fo2}\\] to 2 decimal places.
\nUsing this value to 2dps we have:
\n\\[Y_{\\var {per + 3}} = \\var{me} + \\var{a} (Y_{\\var{per + 2}} -\\var{me}) = \\var{me} + \\var{a} \\times (\\var{fo2} -\\var{me}) = \\var{fo3}\\] to 2 decimal places.
\nFinally using this last value we have:
\n\\[Y_{\\var {per + 4}} = \\var{me} + \\var{a} (Y_{\\var{per + 3}} -\\var{me}) = \\var{me} + \\var{a} \\times (\\var{fo3} -\\var{me}) = \\var{fo4}\\] to 2 decimal places.
\nIn summary we have.
\nForecast from period {per}:
\n\n| Period | \nForecast | \n
| {per+1} | \n{fo1} | \n
| {per+2} | \n{fo2} | \n
| {per+3} | \n{fo3} | \n
| {per+4} | \n{fo4} | \n