// Numbas version: finer_feedback_settings {"name": "STAT7009 Tutorial 2 Sampling and Confidence Intervals", "metadata": {"description": "

Financial Mathematics

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Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

The following table gives the examination marks in {exam1} and in {exam2} and their total for a sample of $n=\\var{n}$ students.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

{exam1}	$\\var{r0[0]}$	$\\var{r0[1]}$	$\\var{r0[2]}$	$\\var{r0[3]}$	$\\var{r0[4]}$	$\\var{r0[5]}$	$\\var{r0[6]}$	$\\var{r0[7]}$	$\\var{r0[8]}$	$\\var{r0[9]}$	Mean = $\\var{mean1}$
{exam2}	$\\var{r1[0]}$	$\\var{r1[1]}$	$\\var{r1[2]}$	$\\var{r1[3]}$	$\\var{r1[4]}$	$\\var{r1[5]}$	$\\var{r1[6]}$	$\\var{r1[7]}$	$\\var{r1[8]}$	$\\var{r1[9]}$	Mean = $\\var{mean2}$
{total}	$\\var{sscores[0]}$	$\\var{sscores[1]}$	$\\var{sscores[2]}$	$\\var{sscores[3]}$	$\\var{sscores[4]}$	$\\var{sscores[5]}$	$\\var{sscores[6]}$	$\\var{sscores[7]}$	$\\var{sscores[8]}$	$\\var{sscores[9]}$	Mean = $\\var{overallmean}$

Find the sample standard deviation for each of {exam1}, {exam2} and Total Score.

", "advice": "

The solution to the first part is here – the other parts can be done in the same way. Note that in practice we use statistical software such as Minitab.

For {exam1} we have the mean is:

\\[\\simplify[]{({r0[0]} + {r0[1]} + {r0[2]} + {r0[3]} + {r0[4]} + {r0[5]} + {r0[6]} + {r0[7]} + {r0[8]} + {r0[9]}) / {n} = {mean1}}\\]

The sample variance is given by the formula:

\\[\\textrm{Sample Variance} = \\frac{1}{n-1}\\left(\\sum_{j=1}^{n}x_j^2 -n\\mu^2\\right)\\]

where the $x_j$ are the exam scores for {exam1}, $n=\\var{n}$ the number of students and $\\mu=\\var{mean1}$ the sample mean.

We find that
\\[\\begin{eqnarray*}\\sum_{j=1}^{n}x_j^2 &=& \\simplify[]{({r0[0]}^2 + {r0[1]}^2 + {r0[2]}^2 + {r0[3]}^2 + {r0[4]}^2 + {r0[5]}^2 + {r0[6]}^2 + {r0[7]}^2 + {r0[8]}^2 + {r0[9]}^2)}\\\\ &=& \\var{ssq1}\\\\ \\\\ \\\\ n\\mu^2 &=&\\var{n} \\times\\var{mean1}^2\\\\ &=& \\var{n*mean1^2} \\end{eqnarray*} \\]
Hence substituting these values into the formula we find that:

\\[\\begin{eqnarray*} \\textrm{Sample Variance} &=& \\frac{1}{\\var{n-1}}\\left(\\var{ssq1}-\\var{n*mean1^2}\\right)\\\\ &=& \\var{var1} \\end{eqnarray*} \\] to 3 decimal places.

The Sample Standard Deviation is then the square root of the Sample Variance i.e.

Sample Standard Deviation = $\\sqrt{\\var{var1}} = \\var{stdev1}$ to one decimal place.

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{exam1}

\n \n \n \n

Sample Standard Deviation = [[0]] (to one decimal place)

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{exam2}

\n \n \n \n

Sample Standard Deviation = [[0]] (to one decimal place)

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{total}

\n \n \n \n

Sample Standard Deviation = [[0]] (to one decimal place)

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Compute a 95% confidence interval for the population mean given a small sample, and compare it with a confidence interval for a different population.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

The amount of sugar in a particular brand of breakfast cereal is to be determined. $\\var {num_monday}$ replicate samples weighing 10 grams were selected at random and the amount of sugar (grams) measured for each replicate is shown in the table below.

{table([['Sugar (g)']+mon],[])}

", "advice": "

a)

The mean of the sample is:

\\[ \\bar{x} = \\frac{1}{n}(\\sum x_i) = \\var{precround(monday_mean,2)} \\text{ (to 2 d.p.)} \\]

The standard deviation is:

\\[ s = \\sqrt{\\frac{1}{n-1}\\sum \\left(x_i-\\bar{x}\\right)^2} = \\var{precround(monday_sd,2)} \\text{ (to 2 d.p.)} \\]

The standard error is :

$\\frac{s}{\\sqrt {n}}=\\frac{\\var{precround(monday_sd,2)}}{\\sqrt{\\var{num_monday}}}=\\var{precround(monday_se,2)}$

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Data for Monday

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Calculate the mean and standard deviation of the sample.

Enter your answers to 2 decimal places.

Mean: [[0]]

Standard Deviation: [[1]]

Standard Error: [[2]]

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Finding the confidence interval at either 90%, 95% or 99% for the mean given the mean of a sample. The population variance is given and so the z values are used. Various scenarios are included.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

A company {sc[s]} {dothis[s]} $\\var{sd[s]}$ $\\var{units}^2$.

A random sample of $\\var{n}$ {t[s]} gives a mean of $\\var{m[s]}$ {units}.

", "advice": "

We use the standard normal distribution (rather than the t-distribution) to find the confidence interval as we know the population variance.

We now calculate the $\\var{confl}$% confidence interval.

Note that $z_{\\var{con}}=\\var{zval}$ and the confidence interval is given by:

\\[ \\var{m[s]} \\pm z_{\\var{con}}\\sqrt{\\frac{\\var{sd2}}{\\var{n}}}\\]

Hence:

Lower value of the confidence interval $=\\;\\displaystyle \\var{m[s]} -\\var{zval} \\sqrt{\\frac{\\var{sd2}} {\\var{n}}} = \\var{lci}${units} to 2 decimal places.

Upper value of the confidence interval $=\\;\\displaystyle \\var{m[s]} +\\var{zval} \\sqrt{\\frac{\\var{sd2}} {\\var{n}}} = \\var{uci}${units} to 2 decimal places.

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Is the process satisfactory?\",\n \"The vending machines are supposed to fill 100ml cups. Is the machine working satisfactorily?\"\n ]\n ", "description": "", "templateType": "anything", "can_override": false}, "mm": {"name": "mm", "group": "Ungrouped variables", "definition": "[1-test,test]", "description": "", "templateType": "anything", "can_override": false}, "tuci": {"name": "tuci", "group": "Ungrouped variables", "definition": "m[s]+zval*standard_error", "description": "", "templateType": "anything", "can_override": false}, "tlci": {"name": "tlci", "group": "Ungrouped variables", "definition": "m[s]-zval*standard_error", "description": "", "templateType": "anything", "can_override": false}, "sd": {"name": "sd", "group": "Ungrouped variables", "definition": "\n [random(800..1400#20),\n random(1200..1800#20),\n random(300..600#20),\n random(100..200#0.1)]\n \n ", "description": "", "templateType": "anything", "can_override": false}, "con": {"name": "con", "group": "Ungrouped variables", "definition": "(100+confl)/200", "description": "", "templateType": "anything", "can_override": false}, "doornot": {"name": "doornot", "group": "Ungrouped variables", "definition": "if(test=0, \" \",\"does not\")", "description": "", "templateType": "anything", "can_override": false}, "sc4ch": {"name": "sc4ch", "group": "Ungrouped variables", "definition": "random(\"supermarkets\",\"clothing retailers\",\"department stores\",\"fast food outlets\")", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "\n [\"bags \",\n sc2ch,\n \"filled cups \"\n]\n \n \n ", "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 \n \n ", "description": "", "templateType": "anything", "can_override": false}, "sd1": {"name": "sd1", "group": "Ungrouped variables", "definition": "if(s=3,sd[s],sqrt(sd[s]))", "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}, "var2": {"name": "var2", "group": "Ungrouped variables", "definition": "random(\"process variance \",\"population variance \")", "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\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}, "var3": {"name": "var3", "group": "Ungrouped variables", "definition": "random(\"The variance of the filling process \",\"The process variance \")", "description": "", "templateType": "anything", "can_override": false}, "sc2ch": {"name": "sc2ch", "group": "Ungrouped variables", "definition": "random(\"bolts\",\"screws\")", "description": "", "templateType": "anything", "can_override": false}, "confl": {"name": "confl", "group": "Ungrouped variables", "definition": "random(90,95,99)", "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}, "standard_error": {"name": "standard_error", "group": "Ungrouped variables", "definition": "sqrt(sd1^2/n)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["sd1", "sd2", "howwell", "n", "doornot", "uci", "test", "confl", "spec", "var1", "var3", "var2", "sc2ch", "units", "zval", "sc1ch", "lci", "tuci", "lies", "mm", "dothis", "sc4ch", "m", "correct", "aim", "sc3ch", "s", "tlci", "t", "sc", "con", "sd", "standard_error"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Does the sample mean follow a Z-distribution or a t-distribution?

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["Z-distribution", "t-distribution"], "matrix": ["1", 0], "distractors": ["", ""]}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

The standard error is [[0]]

Calculate a $\\var{confl}$% confidence interval $(a,b)$ for the population mean:

$a=\\;$[[0]]{units} $b=\\;$[[1]]{units}

Enter both to 2 decimal places.

Given a random variable $X$ normally distributed as $\\operatorname{N}(m,\\sigma^2)$ find probabilities $P(X \\gt a),\\; a \\gt m;\\;\\;P(X \\lt b),\\;b \\lt m$.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Use the t-distribution to answer the questions below. You must give your answers correct to 3 decimal places.

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What are the values of t_low and t_high which between them contain the central 95% of values in the t- distribution, with:

(i) $\\var{df1}$ degrees of freedom

t_{low =[[0]]}

t_{high =[[1]]}

(ii) $\\var{df2}$ degrees of freedom

t_{low =[[2]]}

t_{high =[[3]]}

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What are the values of t_low and t_high which between them contain the central 99% of values in the t- distribution, with

(i) $\\var{df3}$ degrees of freedom

t_{low =[[0]]}

t_{high =[[1]]}

(ii) $\\var{df4}$ degrees of freedom

t_{low =[[2]]}

t_{high =[[3]]}

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i) What is the critical t-value associated with a sample of size $\\var{n1}$ and a significance level of 0.05 for a two-tailed distribution?

t_crit =[[0]]

ii) What is the critical t-value associated with a sample of size $\\var{n2}$ and a significance level of 0.01 for a one-tailed distribution?

t_crit =[[1]]

i) What is the value of $t_{{\\frac{\\alpha}{2},}{n-1}}$ where $n = \\var{n3}$ and ${\\alpha} = 0.05$?

$t_{{\\frac{\\alpha}{2},}{n-1}}$ = [[0]]

ii) What is the value of $t_{{\\frac{\\alpha}{2},}{n-1}}$ where $n = \\var{n4}$ and ${\\alpha} = 0.01$?

$t_{{\\frac{\\alpha}{2},}{n-1}}$ = [[1]]

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]}.

A 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": "

The population variance is unknown. So we have to use the t tables to find the confidence interval.

We now calculate the $\\var{confl}$ confidence interval:

As we have $\\var{n}-1=\\var{n-1}$ degrees of freedom, the interval is given by:

\\[ \\var{m[s]} \\pm t_{\\var{n-1}}\\sqrt{\\frac{\\var{sd[s]}^2}{\\var{n}}}\\]

Looking up the t tables for $\\var{confl}$% we see that $t_{\\var{n-1}}=\\var{invt}$ to 3 decimal places.

Hence:

Lower 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.

Upper 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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"description": "", "templateType": "anything", "can_override": false}, "sc2ch": {"name": "sc2ch", "group": "Ungrouped variables", "definition": "random(\"local\",\"national\")", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(10..30)", "description": "", "templateType": "anything", "can_override": false}, "sc1ch": {"name": "sc1ch", "group": "Ungrouped variables", "definition": "random(\"hotels\",\"motels\", \"Bed and Breakfasts\",\"budget hotels\")", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "random(0..abs(sc)-1)", "description": "", "templateType": "anything", "can_override": false}, "sc6ch": {"name": "sc6ch", "group": "Ungrouped variables", "definition": "random(\"Cornish pasties\",\"sausage rolls\",\"chicken pies\",\"minced beef pasties\")", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "\n [\"vacated rooms was inspected by the management and this\",\n \"clothing outlets \",\n \"workers\",\n \"tickets \",\n \"chocolate bars \",\n sc6ch+\" \"]\n \n ", "description": "", "templateType": "anything", "can_override": false}, "sc": {"name": "sc", "group": "Ungrouped variables", "definition": "\n [\"a national chain of \"+ sc1ch,\n \"a \"+sc2ch + \" chain of clothing shops \",\n \" a large factory \",\n \" a regional passenger airline in \"+sc4ch,\n \"Choclastic!, a company producing a variety of chocolate bars \",\n \" a large bakery \"]\n ", "description": "", "templateType": "anything", "can_override": false}, "sc5ch": {"name": "sc5ch", "group": "Ungrouped variables", "definition": "random(\"caramel\",\"Turkish delight\",\"honeycomb\",\"nuts\")", "description": "", "templateType": "anything", "can_override": false}, "units": {"name": "units", "group": "Ungrouped variables", "definition": "switch(s=2,\"hours\",s=4,\"g\",s=5,\"g per 100g\",\" \")", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "\n [random(30..100#0.01),\n random(40000..80000#500),\n random(34..48#0.01),\n random(100..300#0.01),\n random(10..20#0.01),\n random(3.5..6#0.01)]\n \n ", "description": "", "templateType": "anything", "can_override": false}, "lci": {"name": "lci", "group": "Ungrouped variables", "definition": "precround(tlci,2)", "description": "", "templateType": "anything", "can_override": false}, "tuci": {"name": "tuci", "group": "Ungrouped variables", "definition": "m[s]+invt*standard_error", "description": "", "templateType": "anything", "can_override": false}, "tinvt": {"name": "tinvt", "group": "Ungrouped variables", "definition": "studenttinv((confl+100)/200,n-1)", "description": "", "templateType": "anything", "can_override": false}, "confl": {"name": "confl", "group": "Ungrouped variables", "definition": "random(90,95,99)", "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}, "sc4ch": {"name": "sc4ch", "group": "Ungrouped variables", "definition": "random(\"the Caribbean\",\"the Mediterranean\",\"North East Ireland\",\"South West Ireand\",\"California\")", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "switch(s=0 or s=1 or s=3,'\u00a3',' ')", "description": "", "templateType": "anything", "can_override": false}, "uci": {"name": "uci", "group": "Ungrouped variables", "definition": "precround(tuci,2)", "description": "", "templateType": "anything", "can_override": false}, "sd": {"name": "sd", "group": "Ungrouped variables", "definition": "\n [random(3..10#0.01),\n random(500..4000#0.5),\n random(2..5#0.01),\n random(10..40#0.01),\n random(1..3#0.01),\n random(0.5..1#0.01)]\n ", "description": "", "templateType": "anything", "can_override": false}, "standard_error": {"name": "standard_error", "group": "Ungrouped variables", "definition": "precround(sqrt(sd[s]^2/n),3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["m", "uci", "invt", "units", "lci", "spec", "sc2ch", "sc1ch", "tinvt", "confl", "tuci", "dothis", "sc4ch", "sc6ch", "n", "p", "s", "tlci", "t", "sc", "sc5ch", "sd", "standard_error"], "variable_groups": [], "functions": {"pounds": {"parameters": [["n", "number"]], "type": "number", "language": "javascript", "definition": "return Numbas.util.currency(n,'\u00a3','p');"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", 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Is the population variance known or unknown?

[[0]]

Which distribution does the sample mean follow?

[[1]]

The standard error of the mean is estimated to be:

[[4]]

Enter to 3 decimal places.

For a $\\var{confl}$% confidence interval, the critical t- value is

[[5]]

Enter to 3 decimal places.

Calculate a $\\var{confl}$% confidence interval $(a,b)$ for the population mean:

$a=\\;${p}[[2]]{units} $b=\\;${p}[[3]]{units}

Enter both to 2 decimal places.

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Calculate confidence interval for the mean, sample variance adaptive marking

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

{scenario1[Index]} {sample_size} {scenario2[Index]} {m[Index]} {units1[Index]} with a standard deviation of {s[Index]} {units1[Index]}.

", "advice": "

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"sample_size": {"name": "sample_size", "group": "Unnamed group", "definition": "random(18..30#1)", "description": "", "templateType": "anything", "can_override": false}, "stuff": {"name": "stuff", "group": "Ungrouped variables", "definition": "[\"assembly time\", \"amount spent on books\"]", "description": "", "templateType": "anything", "can_override": false}, "upper": {"name": "upper", "group": "Ungrouped variables", "definition": "m[Index]+Z[Index]*(s[Index]/sqrt(sample_size))", "description": "", "templateType": "anything", "can_override": false}, "percentage": {"name": "percentage", "group": "Ungrouped variables", "definition": "[\"95\",\"99\"]", "description": "", "templateType": "anything", "can_override": false}, "amount": {"name": "amount", "group": "Ungrouped variables", "definition": "[\"number of grams\", \"volume (in ml)\"]", "description": "", "templateType": "anything", "can_override": false}, "scenario1": {"name": "scenario1", "group": "Unnamed group", "definition": "[\"The operations manager of a large production plant would like to estimate the mean\namount of time a worker takes to assemble a new electronic component. After observing\",\"Suppose we would like to estimate the mean amount of money spent on books by CIT students in a semester. A sample of\"]\n\n", "description": "", "templateType": "anything", "can_override": false}, "scenario3": {"name": "scenario3", "group": "Unnamed group", "definition": "[\"minutes with a standard deviation of\", \"\u20ac with a standard deviation of\"]", "description": "", "templateType": "anything", "can_override": false}, "Index": {"name": "Index", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "scenario2": {"name": "scenario2", "group": "Unnamed group", "definition": "[\"workers assembling similar devices, the manager noticed that their average time was\", \"randomly chosen students was found to spend an average of\"]", "description": "", "templateType": "anything", "can_override": false}, "units1": {"name": "units1", "group": "Ungrouped variables", "definition": "[\"minutes\", \"\u20ac\"]", "description": "", "templateType": "anything", "can_override": false}, "tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.01", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "[random(14..18#0.2),random(80..200#5)]", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "[random(3.2..4#0.1), random(15..20#1)]", "description": "", "templateType": "anything", "can_override": false}, "Z": {"name": "Z", "group": "Ungrouped variables", "definition": "[precround(normalinv(0.975,0,1),2),precround(normalinv(0.995,0,1),2)]", "description": "", "templateType": "anything", "can_override": false}, "standard_error": {"name": "standard_error", "group": "Ungrouped variables", "definition": "precround(s[Index]/sqrt(sample_size),3)", "description": "", "templateType": "anything", "can_override": false}, "lower_t": {"name": "lower_t", "group": "Unnamed group", "definition": "precround(m[Index]-t_used*standard_error,2)", "description": "", "templateType": "anything", "can_override": false}, "upper_t": {"name": "upper_t", "group": "Unnamed group", "definition": "precround(m[Index]+t_used*standard_error,2)", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Unnamed group", "definition": "[precround(studenttinv(0.975,sample_size-1),3),precround(studenttinv(0.995,sample_size-1),3)]", "description": "", "templateType": "anything", "can_override": false}, "lower": {"name": "lower", "group": "Unnamed group", "definition": "m[Index]-Z[Index]*(s[Index]/sqrt(sample_size))", "description": "", "templateType": "anything", "can_override": false}, "t_used": {"name": "t_used", "group": "Unnamed group", "definition": "t[Index]", "description": "

", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["units1", "upper", "m", "s", "amount", "tol", "stuff", "Index", "percentage", "Z", "standard_error"], "variable_groups": [{"name": "Unnamed group", "variables": ["scenario1", "scenario2", "sample_size", "scenario3", "lower_t", "upper_t", "t", "lower", "t_used"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Is the population standard deviation $\\sigma$ known or unknown?

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Which distribution does the sample mean follow?

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The standard error of the mean is estimated to be [[0]]

Enter to 3 decimal places.

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For a {percentage[Index]}% confidence interval, the critical t- value is [[0]]

Enter to 3 decimal places.

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Construct a {percentage[Index]}% confidence interval for the mean {stuff[Index]}.

Lower limit = [[0]]

Upper limit = [[1]]

Enter to 2 decimal places.

Compute a 95% confidence interval for the population mean given a small sample, and compare it with a confidence interval for a different population.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

The ages of a random sample of patients coming into a doctor's surgery during the school holidays are noted down on each Monday and Friday thoughout August.

{table([['Monday']+mon,['Friday']+fri],[])}

", "advice": "

a)

The mean of the sample of patients visiting on Mondays is

\\[ \\bar{x} = \\frac{1}{n}(\\sum x_i) = \\var{precround(monday_mean,2)} \\text{ (to 2 d.p.)} \\]

The standard deviation is

\\[ s = \\sqrt{\\frac{1}{n-1}\\sum \\left(x_i-\\bar{x}\\right)^2} = \\var{precround(monday_sd,2)} \\text{ (to 2 d.p.)} \\]

b)

The sample mean $\\mu_m$ follows a $t$-distribution with $n-1 = \\var{num_monday-1}$ degrees of freedom centred on the population mean, with a standard error of $\\frac{s}{\\sqrt{n}} = \\simplify{{precround(monday_sd,2)}/sqrt({num_monday})} = \\var{precround(monday_se,2)}$.

That means that the 95% confidence interval for the population mean is as follows:

\\begin{array}{lrcl}
&\\left[ \\bar{x} - s \\times t_{\\var{num_monday-1},0.975} \\right. \\kern{-0.5em}&,&\\kern{-0.5em} \\left. \\bar{x} + s \\times t_{\\var{num_monday-1},0.975} \\right] \\\\
= & \\left[ \\simplify[]{{precround(monday_mean,2)} - {precround(monday_se,2)}*{precround(monday_t,2)}} \\right. \\kern{-0.5em}&,&\\kern{-0.5em} \\left. \\simplify[]{{precround(monday_mean,2)} + {precround(monday_se,2)}*{precround(monday_t,2)}} \\right] \\\\
= & \\left[ \\var{precround(monday_low,2)} \\right. \\kern{-0.5em}&,&\\kern{-0.5em} \\left. \\var{precround(monday_high,2)} \\right]
\\end{array}

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Data for Friday

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Data for Monday

", "templateType": "anything", "can_override": false}, "friday_t": {"name": "friday_t", "group": "Ungrouped variables", "definition": "studenttinv(0.975,num_friday-1)", "description": "", "templateType": "anything", "can_override": false}, "monday_mean": {"name": "monday_mean", "group": "Ungrouped variables", "definition": "mean(mon)", "description": "", "templateType": "anything", "can_override": false}, "no_overlap": {"name": "no_overlap", "group": "Ungrouped variables", "definition": "friday_low > monday_high or friday_high < monday_low", "description": "", "templateType": "anything", "can_override": false}, "friday_se": {"name": "friday_se", "group": "Ungrouped variables", "definition": "stdev(fri,true)/sqrt(num_friday)", "description": "", "templateType": "anything", "can_override": false}, "num_friday": {"name": "num_friday", "group": "Ungrouped variables", "definition": "random(12..16)", "description": "", "templateType": "anything", "can_override": false}, "friday_pop_mean": {"name": "friday_pop_mean", "group": "Ungrouped variables", "definition": "random(20..70)", "description": "", "templateType": "anything", "can_override": false}, "num_monday": {"name": "num_monday", "group": "Ungrouped variables", "definition": "12", "description": "", "templateType": "anything", "can_override": false}, "monday_pop_range": {"name": "monday_pop_range", "group": "Ungrouped variables", "definition": "20", "description": "", "templateType": "anything", "can_override": false}, "friday_pop_range": {"name": "friday_pop_range", "group": "Ungrouped variables", "definition": "20", "description": "", "templateType": "anything", "can_override": false}, "monday_sd": {"name": "monday_sd", "group": "Ungrouped variables", "definition": "stdev(mon, true)", "description": "", "templateType": "anything", "can_override": false}, "monday_pop_mean": {"name": "monday_pop_mean", "group": "Ungrouped variables", "definition": "random(20..70)", "description": "", "templateType": "anything", "can_override": false}, "friday_high": {"name": "friday_high", "group": "Ungrouped variables", "definition": "friday_mean+friday_t*friday_se", "description": "", "templateType": "anything", "can_override": false}, "monday_t": {"name": "monday_t", "group": "Ungrouped variables", "definition": "studenttinv(0.975,num_monday-1)", "description": "", "templateType": "anything", "can_override": false}, "friday_low": {"name": "friday_low", "group": "Ungrouped variables", "definition": "friday_mean-friday_t*friday_se", "description": "", "templateType": "anything", "can_override": false}, "friday_mean": {"name": "friday_mean", "group": "Ungrouped variables", "definition": "mean(fri)", "description": "", "templateType": "anything", "can_override": false}, "monday_high": {"name": "monday_high", "group": "Ungrouped variables", "definition": "monday_mean+monday_t*monday_se", "description": "", "templateType": "anything", "can_override": false}, "monday_low": {"name": "monday_low", "group": "Ungrouped variables", "definition": "monday_mean-monday_t*monday_se", "description": "

Lower end of the confidence interval for Monday's mean

", "templateType": "anything", "can_override": false}, "monday_se": {"name": "monday_se", "group": "Ungrouped variables", "definition": "monday_sd/sqrt(num_monday)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["num_monday", "monday_pop_mean", "monday_pop_range", "mon", "num_friday", "friday_pop_mean", "friday_pop_range", "fri", "monday_mean", "monday_sd", "monday_se", "monday_t", "monday_low", "monday_high", "t", "friday_mean", "friday_se", "friday_t", "friday_low", "friday_high", "no_overlap"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Calculate the mean and standard deviation of the sample of patients visiting on Mondays.

Enter your answers to 2 decimal places.

Mean: [[0]]

Standard Deviation: [[1]]

Find the 95% confidence interval for the population mean $\\mu_m$ of the age of patients visiting on Monday, using the $t$-distribution with $\\var{num_monday-1}$ degrees of freedom.

Enter your answers to 2 decimal places.

Lower limit: [[0]]

Upper limit: [[1]]

Calculate confidence interval for the mean, sample variance n>30

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

{scenario1[Index]} {n1[Index]} {scenario2[Index]} {prop1[Index]} {scenario3[Index]} {percentage}% {scenario4[Index]}.

", "advice": "

The estimate for the population proportion is $\\hat{p} =\\frac{{\\var{prop1[Index]}}}{\\var{n1[Index]}}=\\var{precround(p_hat,3)}$

The estimate for the standard error associated with $\\hat{p}$ is $=\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}=\\sqrt{\\frac{\\var{precround(p_hat,3)}\\times (1-\\var{precround(p_hat,3)})}{\\var{n1[Index]}}}=\\var{se}$

To construct a $\\var{percentage}$% confidence interval, $Z_\\frac{\\alpha}{2}=Z_\\frac{\\var{alpha2}}{2} = \\var{Z}$

Putting this information together, the lower bound of the $\\var{percentage}$% confidence interval is:

$\\hat{p} -Z_\\frac{\\alpha}{2}\\times\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{\\var{n1[Index]}}}=\\var{precround(p_hat,3)} -\\var{Z}\\times \\var{se}=\\var{precround(lower,2)}$

The upper bound of the $\\var{percentage}$% confidence interval is:

$\\hat{p} +Z_\\frac{\\alpha}{2}\\times\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{\\var{n1[Index]}}}=\\var{precround(p_hat,3)} +\\var{Z}\\times \\var{se}=\\var{precround(upper,2)}$

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p_hat": {"name": "p_hat", "group": "Unnamed group", "definition": "prop1[Index]/n1[Index]", "description": "", "templateType": "anything", "can_override": false}, "scenario2": {"name": "scenario2", "group": "Unnamed group", "definition": "[\"newly employed graduates from CIT found that\", \"machines is taken and\"]\n", "description": "", "templateType": "anything", "can_override": false}, "n1": {"name": "n1", "group": "Unnamed group", "definition": "[random(100..120#10),random(180..250#10)]", "description": "", "templateType": "anything", "can_override": false}, "upper": {"name": "upper", "group": "Ungrouped variables", "definition": "p_hat+Z*sqrt((p_hat*(1-p_hat))/n1[Index])", "description": "", "templateType": "anything", "can_override": false}, "alpha2": {"name": "alpha2", "group": "Unnamed group", "definition": "alpha*2", "description": "", "templateType": "anything", "can_override": false}, "percentage": {"name": "percentage", "group": "Ungrouped variables", "definition": "random([95,99])", "description": "", "templateType": "anything", "can_override": false}, "lower": {"name": "lower", "group": "Ungrouped variables", "definition": "p_hat-Z*sqrt((p_hat*(1-p_hat))/n1[Index])", "description": "", "templateType": "anything", "can_override": false}, "se": {"name": "se", "group": "Ungrouped variables", "definition": "precround(sqrt((p_hat*(1-p_hat))/n1[Index]),4)", "description": "", "templateType": "anything", "can_override": false}, "prop1": {"name": "prop1", "group": "Unnamed group", "definition": "[random(75..80#1),random(30..50#2)]", "description": "", "templateType": "anything", "can_override": false}, "scenario1": {"name": "scenario1", "group": "Unnamed group", "definition": "[\"A survey of\",\"A manufacturer would like to estimate the proportion of X-ray machines that malfunction and produce excess radiation. A random sample of\"]\n\n", "description": "", "templateType": "anything", "can_override": false}, "Z": {"name": "Z", "group": "Ungrouped variables", "definition": "precround(normalinv(1-alpha,0,1),2)", "description": "", "templateType": "anything", "can_override": false}, "scenario3": {"name": "scenario3", "group": "Unnamed group", "definition": "[\"of the graduates surveyed used statistics in their new roles. Calculate a\" , \"of the machines malfunction. Calculate a\"]", "description": "", "templateType": "anything", "can_override": false}, "Index": {"name": "Index", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "alpha": {"name": "alpha", "group": "Ungrouped variables", "definition": "(100-percentage)/200", "description": "", "templateType": "anything", "can_override": false}, "scenario4": {"name": "scenario4", "group": "Unnamed group", "definition": "[\"confidence interval for the proportion of CIT graduates who use statistics in their place of work.\", \"confidence interval for the proportion of X-ray machines that malfunction in the population.\"]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["upper", "Index", "percentage", "Z", "alpha", "lower", "se"], "variable_groups": [{"name": "Unnamed group", "variables": ["scenario1", "scenario2", "prop1", "scenario3", "p_hat", "n1", "scenario4", "alpha2"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Lower limit = [[0]]

Upper limit = [[1]]

Calculate confidence interval for the proportion

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

A company manufacturing a diagnostic kit sent {n1} samples of the kit to an independent laboratory to be analysed. The number of diagnostic kits that met industry standards was {prop1}.

", "advice": "

The estimate for the population proportion is $\\hat{p} =\\frac{{\\var{prop1}}}{\\var{n1}}=\\var{precround(p_hat,3)}$

To construct a $\\var{percentage}$% confidence interval, $Z_\\frac{\\alpha}{2}=Z_\\frac{\\var{alpha2}}{2} = \\var{Z}$

Putting this information together, the lower bound of the $\\var{percentage}$% confidence interval is:

$\\hat{p} -Z_\\frac{\\alpha}{2}\\times\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{\\var{n1}}}=\\var{precround(p_hat,3)} -\\var{Z}\\times \\var{se}=\\var{precround(lower,2)}$

The upper bound of the $\\var{percentage}$% confidence interval is:

$\\hat{p} +Z_\\frac{\\alpha}{2}\\times\\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{\\var{n1}}}=\\var{precround(p_hat,3)} +\\var{Z}\\times \\var{se}=\\var{precround(upper,2)}$

The sample proportion $\\hat{p}$ is [[0]]

The standard error of the sample proportion is [[0]]

Calculate a {percentage}% confidence interval for the proportion of diagnostic kits that meet idustry standards.

Lower limit = [[0]]

Upper limit = [[1]]

Given a random variable $X$ normally distributed as $\\operatorname{N}(m,\\sigma^2)$ find probabilities $P(X \\gt a),\\; a \\gt m;\\;\\;P(X \\lt b),\\;b \\lt m$.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Use the $\\chi^2$-distribution to answer the questions below. You must give your answers correct to 2 decimal places.

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What are the values of $\\chi^2$_low and $\\chi^2$_high which between them contain the central 95% of values in the $\\chi^2$- distribution, with:

(i) $\\var{df1}$ degrees of freedom

$\\chi^2$_{low =[[0]]}

$\\chi^2$_{high =[[1]]}

(ii) $\\var{df2}$ degrees of freedom

$\\chi^2$_{low =[[2]]}

$\\chi^2$_{high =[[3]]}

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What are the values of $\\chi^2$_low and $\\chi^2$_high which between them contain the central 99% of values in the $\\chi^2$- distribution, with $\\var{df3}$ degrees of freedom.

$\\chi^2$_{low =[[0]]}

$\\chi^2$_{high =[[1]]}

(i) What are the values of $\\chi^2_{{\\frac{\\alpha}{2},}{n-1}}$ and $\\chi^2_{1-{\\frac{\\alpha}{2},}{n-1}}$ where $n = \\var{n3}$ and ${\\alpha} = 0.05$?

$\\chi^2_{{\\frac{\\alpha}{2},}{n-1}}$ = [[0]]

$\\chi^2_{1-{\\frac{\\alpha}{2},}{n-1}}$ = [[1]]

(ii) What is the value of $\\chi^2_{{\\frac{\\alpha}{2},}{n-1}}$ and $\\chi^2_{1-{\\frac{\\alpha}{2},}{n-1}}$ where $n = \\var{n4}$ and ${\\alpha} = 0.01$?

$\\chi^2_{{\\frac{\\alpha}{2},}{n-1}}$ =[[2]]

$\\chi^2_{1-{\\frac{\\alpha}{2},}{n-1}}$ = [[3]]

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The folic acid content ( g per 100g) of 8 samples of a particular brand of breakfast cereal are shown below:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n

{d1}

{d2}

{d3}

{d4}

{d5}

{d6}

{d7}

{d8}

The variance of the folic acid content samples was calculated to be $\\var{sample_var}$.

Calculate a 95% confidence interval for the variance of the folic acid content.

", "advice": "

The $95$% confidence interval for the population variance $\\sigma^2$ is:

$a=\\frac{(n-1)\\times s^2}{\\chi^2_{upper,n-1}}=\\frac{(8-1)\\times \\var{s}^2}{\\chi^2_{0.975,7}}=\\var{lower}$

$b=\\frac{(n-1)\\times s^2}{\\chi^2_{lower,n-1}}=\\frac{(8-1)\\times \\var{s}^2}{\\chi^2_{0.025,7}}=\\var{upper}$

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random(160..179#1)

", "templateType": "anything", "can_override": false}, "d7": {"name": "d7", "group": "Ungrouped variables", "definition": "random(160..179#1)", "description": "", "templateType": "anything", "can_override": false}, "d8": {"name": "d8", "group": "Ungrouped variables", "definition": "random(160..179#1)", "description": "", "templateType": "anything", "can_override": false}, "sample_var": {"name": "sample_var", "group": "Ungrouped variables", "definition": "precround(variance([d1,d2,d3,d4,d5,d6,d7,d8],true),3)", "description": "", "templateType": "anything", "can_override": false}, "chi_lower": {"name": "chi_lower", "group": "Ungrouped variables", "definition": " precround(chisquareinv(0.025,7),3)", "description": "", "templateType": "anything", "can_override": false}, "chi_upper": {"name": "chi_upper", "group": "Ungrouped variables", "definition": "precround(chisquareinv(0.975,7),3)", "description": "", "templateType": "anything", "can_override": false}, "lower": {"name": "lower", "group": "Ungrouped variables", "definition": "precround(sample_var*7/chi_upper,2)", "description": "", "templateType": "anything", "can_override": false}, "upper": {"name": "upper", "group": "Ungrouped variables", "definition": "precround(sample_var*7/chi_lower,2)", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "precround(sqrt(sample_var),4)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["d1", "d2", "d3", "d4", "d5", "d6", "d7", "d8", "sample_var", "chi_lower", "chi_upper", "lower", "upper", "s"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "d1", "part": "p1", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Please give your answers to two decimal places.

What are the values of $\\chi^2_{{\\frac{\\alpha}{2},}{n-1}}$ and $\\chi^2_{1-{\\frac{\\alpha}{2},}{n-1}}$?

$\\chi^2_{{\\frac{\\alpha}{2},}{n-1}}$ = [[0]]

$\\chi^2_{1-{\\frac{\\alpha}{2},}{n-1}}$ = [[1]]

Please give your answer to one decimal place.

The lower limit is [[0]]

The upper limit is [[1]]

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To determine the average sodium concentration of a solution, {n} replicate samples were taken and the sodium content of each sample (mg/L) was recorded:

{data}

The variance of the sodium concentration in the samples was calculated to be $\\var{sample_var}$.

Produce a 99% confidence interval for the standard deviation of the sodium concentration.

", "advice": "

The 99% confidence interval for the population variance $\\sigma^2$ is:

$lower=\\frac{(n-1)\\times s^2}{\\chi^2_{upper,n-1}}=\\frac{(\\var{n}-1)\\times \\var{s}^2}{\\chi^2_{0.995,\\var{n-1}}}=\\var{lower}$

$upper=\\frac{(n-1)\\times s^2}{\\chi^2_{lower,n-1}}=\\frac{(\\var{n}-1)\\times \\var{s}^2}{\\chi^2_{0.005,\\var{n-1}}}=\\var{upper}$

The 99% confidence interval for the population standard deviation $\\sigma$ is:

$lower=\\sqrt{\\var{lower}}=\\var{lower_sd}$

$upper=\\sqrt{\\var{upper}}=\\var{upper_sd}$

Please give your answer to two decimal places.

What are the values of $\\chi^2_{{\\frac{\\alpha}{2},}{n-1}}$ and $\\chi^2_{1-{\\frac{\\alpha}{2},}{n-1}}$?

$\\chi^2_{{\\frac{\\alpha}{2},}{n-1}}$ = [[0]]

$\\chi^2_{1-{\\frac{\\alpha}{2},}{n-1}}$ = [[1]]

Please give your answer to three decimal places.

The lower limit is [[0]]

The upper limit is [[1]]

The question asked for a confidence interval for the standard deviation which is the square root of the variance.

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The question asked for a confidence interval for the standard deviation which is the square root of the variance.

", "useAlternativeFeedback": false, "minValue": "precround(upper,3)-0.001", "maxValue": "precround(upper,3)+0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "upper_sd-0.001", "maxValue": "upper_sd+0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Confidence intervals for the mean and standard deviation of a protein in infant formula adaptive marking", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

The amount of protein in a particular brand of infant formula is to be determined. {n} replicate samples weighing 100 grams were selected at random and the amount of protein (grams) measured for each replicate is shown below.

{data}

(i) Produce a {percentage}% confidence interval for the mean amount of protein per 100 grams

(ii) Produce a {percentage}% confidence interval for the variance of the amount of protein per 100 grams.

", "advice": "

The sample mean $\\bar{x}=\\var{mean}$

The sample standard deviation $s=\\var{s}$

The sample variance is $s^2=\\var{sample_var}$

The sample mean $\\mu_m$ follows a $t$-distribution with $n-1 = \\var{n-1}$ degrees of freedom centred on the population mean, with a standard error of $\\frac{s}{\\sqrt{n}} = \\simplify{{precround(s,2)}/sqrt({n})} = \\var{precround(standard_error,2)}$

That means that the {percentage}% confidence interval for the population mean is as follows:

$a=\\bar{x}-t_{\\frac{\\alpha}{2},n-1}\\times\\frac{s}{\\sqrt{n}}$

$a=\\var{mean}-t_{\\frac{\\var{alpha}}{2},\\var{n-1}}\\times\\frac{\\var{s}}{\\sqrt{\\var{n}}}=\\var{a}$

$a=\\bar{x}+t_{\\frac{\\alpha}{2},n-1}\\times\\frac{s}{\\sqrt{n}}$

$b=\\var{mean}+t_{\\frac{\\var{alpha}}{2},\\var{n-1}}\\times\\frac{\\var{s}}{\\sqrt{\\var{n}}}=\\var{b}$

The {percentage}% confidence interval for the population variance $\\sigma^2$ is:

$a=\\frac{(n-1)\\times s^2}{\\chi^2_{upper,n-1}}=\\frac{(\\var{n}-1)\\times \\var{s}^2}{\\chi^2_{\\simplify{1-{alpha/2}},\\var{n-1}}}=\\var{lower}$

$b=\\frac{(n-1)\\times s^2}{\\chi^2_{lower,n-1}}=\\frac{(\\var{n}-1)\\times \\var{s}^2}{\\chi^2_{\\simplify{{alpha/2}},\\var{n-1}}}=\\var{upper}$

The {percentage}% confidence interval for the population standard deviation $\\sigma$ is:

$a=\\sqrt{\\var{lower}}=\\var{lower_sd}$

$a=\\sqrt{\\var{upper}}=\\var{upper_sd}$

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Please give your answers to three decimal places.

The sample mean is [[0]]

The sample standard deviation is [[1]]

The standard error is [[2]]

Please give your answers correct to 3 decimal places.

What is the critical t-value associated with a sample of size $\\var{n}$ and $\\frac{\\alpha}{2}=\\var{alpha}$?

t_crit =[[0]]

Calculate a $\\var{percentage}$% confidence interval $(a,b)$ for the population mean:

$a=$ [[1]]

$b=$ [[2]]

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Please give your answers correct to two decimal places.

What is the critical $\\chi^2$-value associated with a sample of size $\\var{n}$ and $1-\\frac{\\alpha}{2}=\\var{alpha_comp}$?

[[0]]

What is the critical $\\chi^2$-value associated with a sample of size $\\var{n}$ and $\\frac{\\alpha}{2}=\\var{alpha}$?

[[1]]

Calculate a $\\var{percentage}$% confidence interval $(a,b)$ for the population variance:

$a=\\;$ [[2]]

$b=\\;$ [[3]]

Calculate a $\\var{percentage}$% confidence interval $(a,b)$ for the population standard deviation:

$a=\\;$ [[4]]

$b=\\;$ [[5]]

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