// Numbas version: exam_results_page_options {"showQuestionGroupNames": false, "type": "question", "name": "Etain's copy of Nigel's copy of Mathematics and statistics for bioinformatics", "metadata": {"description": "Questions used in a university course titled \"Mathematics and statistics for bioinformatics\"", "licence": "Creative Commons Attribution 4.0 International"}, "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [{"name": "Find mean and standard deviation of differences between samples", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu1,sig1)),5)", "description": "", "name": "r1"}, "thismany": {"templateType": "anything", "group": "Ungrouped variables", "definition": "5", "description": "", "name": "thismany"}, "sig1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1.5..2.5#0.5)", "description": "", "name": "sig1"}, "performing": {"templateType": "anything", "group": "Ungrouped variables", "definition": " 'working at $\\\\var{100}$ watts on an exercise machine' ", "description": "", "name": "performing"}, "r2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu2,sig2)),5)", "description": "", "name": "r2"}, "attempt": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'attempt'", "description": "", "name": "attempt"}, "meandiff": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mean(d)", "description": "", "name": "meandiff"}, "objects": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'people'", "description": "", "name": "objects"}, "mu1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(16..20#0.5)", "description": "", "name": "mu1"}, "sig2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sig1+random(-0.5..-0.2#0.1)", "description": "", "name": "sig2"}, "mu2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mu1+random(1..3#0.1)", "description": "", "name": "mu2"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "list(vector(r2)-vector(r1))", "description": "", "name": "d"}, "stdiff": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(stdev(d,true),3)", "description": "", "name": "stdiff"}, "object": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'person'", "description": "", "name": "object"}, "something": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'Oxygen uptake values (mL/kg.min)'", "description": "", "name": "something"}}, "ungrouped_variables": ["meandiff", "performing", "attempt", "r1", "objects", "mu2", "object", "sig1", "thismany", "stdiff", "sig2", "something", "r2", "mu1", "d"], "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

Find the mean and standard deviation of the difference between first and second {attempt}s.

\n

Calculate differences for second {attempt} – first {attempt}.

\n

Mean of difference = [[0]] (input as an exact decimal)

\n

Standard deviation of difference = [[1]] (input to 3 decimal places)

", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "{meandiff}", "maxValue": "{meandiff}", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "mustBeReducedPC": 0}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "{stdiff}", "maxValue": "{stdiff}", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0.5, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "

{Something} for $\\var{thismany}$ {objects} {performing} were:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
{capitalise(object)}12345
First {attempt}$\\var{r1[0]}$$\\var{r1[1]}$$\\var{r1[2]}$$\\var{r1[3]}$$\\var{r1[4]}$
Second {attempt}$\\var{r2[0]}$$\\var{r2[1]}$$\\var{r2[2]}$$\\var{r2[3]}$$\\var{r2[4]}$
", "tags": ["checked2015", "cr1", "data analysis", "differences", "elementary statistics", "mean", "mean of differences", "standard deviation", "standard deviation of differences", "statistics", "stats", "tested1", "variance"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

An experiment is performed twice, each with $5$ outcomes

\n

$x_i,\\;y_i,\\;i=1,\\dots 5$ . Find mean and s.d. of their differences $y_i-x_i,\\;i=1,\\dots 5$.

"}, "type": "question", "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

The table of differences is given by:

\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
{capitalise(object)}12345
First {attempt}$\\var{r1[0]}$$\\var{r1[1]}$$\\var{r1[2]}$$\\var{r1[3]}$$\\var{r1[4]}$
Second {attempt}$\\var{r2[0]}$$\\var{r2[1]}$$\\var{r2[2]}$$\\var{r2[3]}$$\\var{r2[4]}$
Differences$\\var{d[0]}$$\\var{d[1]}$$\\var{d[2]}$$\\var{d[3]}$$\\var{d[4]}$
\n

The mean of the differences is $\\var{meandiff}$.

\n

The variance $V$ of the differences is

\n

\\begin{align}
V &= \\frac{1}{4}\\left(\\simplify[]{({d[0]}^2+{d[1]}^2+{d[2]}^2+{d[3]}^2+{d[4]}^2)}-5\\times \\var{meandiff}^2\\right) \\\\
&= \\var{variance(d,true)}
\\end{align}

\n

Hence the standard deviation is $\\sqrt{V}=\\var{stdiff}$ to 3 decimal places.

Obtain the $5$ number summary MQMQM and input their values below as exact decimals:

\n \n \n \n \n \n \n \n \n \n
MinimumLower QuartileMedianUpper QuartileMaximum
[[0]][[1]][[2]][[3]][[4]]
\n \n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{uquartile-lquartile}", "minValue": "{uquartile-lquartile}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n \n \n

Enter the interquartile range: [[0]]

\n \n \n \n

Input as an exact decimal.

\n \n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["

{guess1}

", "

{guess2}

", "

{guess3}

", "

{guess4}

"], "displayColumns": 4, "distractors": ["", "", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0, 0], "marks": 0}], "type": "gapfill", "prompt": "

Without doing any further calculations, which of the following numbers do you think is likely to be closest to the sample standard deviation?
[[0]]

", "showCorrectAnswer": true, "marks": 0}], "statement": "\n \n \n

Given the following table of data, answer all the following questions:

\n \n \n \n \n \n \n \n \n \n
 $\\var{r0[0]}$ $\\var{r0[1]}$ $\\var{r0[2]}$ $\\var{r0[3]}$ $\\var{r0[4]}$ $\\var{r0[5]}$ $\\var{r0[6]}$ $\\var{r0[7]}$ $\\var{r0[8]}$ $\\var{r0[9]}$ $\\var{r0[10]}$ $\\var{r0[11]}$ $\\var{r0[12]}$ $\\var{r0[13]}$ $\\var{r0[14]}$ $\\var{r0[15]}$ $\\var{r0[16]}$ $\\var{r0[17]}$ $\\var{r0[18]}$ $\\var{r0[19]}$ $\\var{r0[20]}$ $\\var{r0[21]}$ $\\var{r0[22]}$ $\\var{r0[23]}$ $\\var{r0[24]}$ $\\var{r0[25]}$ $\\var{r0[26]}$ $\\var{r0[27]}$ $\\var{r0[28]}$ $\\var{r0[29]}$ $\\var{r0[30]}$ $\\var{r0[31]}$
\n \n \n ", "tags": ["average", "checked2015", "cr1", "data analysis", "interquartile range", "lower quartile", "MAS1604", "MAS8380", "MAS8401", "maximum", "mean", "mean ", "median", "minumum", "MQMQM", "ordered data", "quartile", "query", "sample standard deviation", "standard deviation", "tested1", "upper quartile"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

11/07/2012:

\n

\n

Calculations not tested yet.

\n

23/07/2012:

\n

\n

Checked calculations as stats extension now available. OK.

\n

3/08/2012:

\n

\n

Question appears to be working correctly.

\n

19/12/2012:

\n

Changed to new stats functions and replaced the uniform sample data by a normal sample.

\n

Checked calculations. Note that the quartiles are defined differently from the stats extension definition - so used the Newcastle definition! Added query tag so that can be decided upon.

\n

\n

21/12/2012:

\n

\n

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Given 32 datapoints in a table find their minimum, lower quartile, median, upper quartile, and maximum.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

#### a)

\n

If you sort the data in increasing order you get the following table:

\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
 $\\var{r1[0]}$ $\\var{r1[1]}$ $\\var{r1[2]}$ $\\var{r1[3]}$ $\\var{r1[4]}$ $\\var{r1[5]}$ $\\var{r1[6]}$ $\\var{r1[7]}$ $\\var{r1[8]}$ $\\var{r1[9]}$ $\\var{r1[10]}$ $\\var{r1[11]}$ $\\var{r1[12]}$ $\\var{r1[13]}$ $\\var{r1[14]}$ $\\var{r1[15]}$ $\\var{r1[16]}$ $\\var{r1[17]}$ $\\var{r1[18]}$ $\\var{r1[19]}$ $\\var{r1[20]}$ $\\var{r1[21]}$ $\\var{r1[22]}$ $\\var{r1[23]}$ $\\var{r1[24]}$ $\\var{r1[25]}$ $\\var{r1[26]}$ $\\var{r1[27]}$ $\\var{r1[28]}$ $\\var{r1[29]}$ $\\var{r1[30]}$ $\\var{r1[31]}$
\n

Denote the ordered data by $x_j$, thus $x_{10}=\\var{r1[9]}$ for example.

\n

Minimum value: The minimum value is $x_1=\\var{r1[0]}$.

\n

Lower Quartile: As there is an even number of values, the Lower Quartile will lie between two values. Its position is calculated by finding

\n

\$\\frac{n+1}{4}=\\frac{\\var{n+1}}{4}=8\\frac{1}{4}\$

\n

Hence the Lower Quartile lies between the 8th and 9th entries in the ordered table, so it is:

\n

\$0.75\\times x_8+0.25\\times x_9 = 0.75\\times\\var{r1[7]}+0.25\\times \\var{r1[8]}=\\var{lquartile}\$

\n

Median: The position of the median in the table is given by

\n

\$\\frac{2(n+1)}{4} = \\frac{\\var{2*(n+1)}}{4} = 16 \\frac{1}{2}\$

\n

The median lies between the 16th and 17th entries in the ordered table and is given by:

\n

\$0.5\\times x_{16}+0.5\\times x_{17} = 0.5\\times\\var{r1[15]}+0.5\\times \\var{r1[16]}=\\var{median}\$

\n

Upper Quartile: As there is an even number of values, the Upper Quartile will lie between two values. Its position is calculated by finding

\n

\$\\frac{3(n+1)}{4}=\\frac{\\var{3*(n+1)}}{4}=24\\frac{3}{4}\$

\n

Hence the Upper Quartile lies between the 24th and 25th entries in the ordered table.

\n

We find it is \$0.25\\times x_{24}+0.75\\times x_{25} = 0.25\\times\\var{r1[23]}+0.75\\times \\var{r1[24]}=\\var{uquartile}\$

\n

Maximum value: The maximum value is $x_{32}=\\var{r1[31]}$

\n

#### b)

\n

The interquartile range is defined to be

\n

\$\\text{Upper Quartile} – \\text{Lower Quartile} \$

\n

and so in this case we have:

\n

\$\\text{Interquartile range} = \\var{uquartile}-\\var{lquartile}=\\var{uquartile-lquartile} \$

\n

#### c)

\n

Most of the data should be spanned by $4s$ where $s$ is the sample standard deviation.

\n

The range of values is $\\var{r1[31]}-\\var{r1[0]}=\\var{r1[31]-r1[0]}$ and so $s$ should be approximately

\n

\$\\simplify[std]{({r1[31]}-{r1[0]}) / 4 = {(r1[31] -r1[0]) / 4}} \$

\n

The most likely value for the sample standard deviation of the options presented is $\\var{guess1}$.

\n

(The actual value is $\\var{stdev}$ to 2 decimal places).

"}, {"name": "Find sample mean, standard deviation, median and interquartile range, , ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sort(r0)", "description": "", "name": "r1"}, "things": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'tomatoes '", "description": "", "name": "things"}, "median": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.5*(r1[11]+r1[12])", "description": "", "name": "median"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "24", "description": "", "name": "n"}, "uquartile": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.25*r1[17]+0.75*r1[18]", "description": "", "name": "uquartile"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(20..60)", "description": "", "name": "u"}, "whatever": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'weights'", "description": "", "name": "whatever"}, "interq": {"templateType": "anything", "group": "Ungrouped variables", "definition": "uquartile-lquartile", "description": "", "name": "interq"}, "l": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..19)", "description": "", "name": "l"}, "var": {"templateType": "anything", "group": "Ungrouped variables", "definition": "variance(r0,true)", "description": "", "name": "var"}, "description": {"templateType": "anything", "group": "Ungrouped variables", "definition": "' from a new kind of tomato plant.'", "description": "", "name": "description"}, "lquartile": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.75*r1[5]+0.25*r1[6]", "description": "", "name": "lquartile"}, "mean": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mean(r0)", "description": "", "name": "mean"}, "r0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(random(l..u),24)", "description": "", "name": "r0"}, "units": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'grams'", "description": "", "name": "units"}, "stdev": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(stdev(r0,true),2)", "description": "", "name": "stdev"}}, "ungrouped_variables": ["uquartile", "r0", "description", "things", "median", "interq", "whatever", "l", "var", "lquartile", "u", "mean", "stdev", "units", "n", "r1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{precround(mean(r0),1)}", "minValue": "{precround(mean(r0),1)}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{siground(stdev(r0,true),3)}", "minValue": "{siground(stdev(r0,true),3)}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{median}", "minValue": "{median}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{interq}", "minValue": "{interq}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n\n\n\n\n
Sample Mean (1 dp)Sample Standard Deviation (3 sig figs)Median (exact value)Interquartile Range (exact value)
[[0]][[1]][[2]][[3]]
", "showCorrectAnswer": true, "marks": 0}], "statement": "

The following data are the {whatever}, in {units}, of $\\var{n}$ {things} {description}

\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
 $\\var{r0[0]}$ $\\var{r0[1]}$ $\\var{r0[2]}$ $\\var{r0[3]}$ $\\var{r0[4]}$ $\\var{r0[5]}$ $\\var{r0[6]}$ $\\var{r0[7]}$ $\\var{r0[8]}$ $\\var{r0[9]}$ $\\var{r0[10]}$ $\\var{r0[11]}$ $\\var{r0[12]}$ $\\var{r0[13]}$ $\\var{r0[14]}$ $\\var{r0[15]}$ $\\var{r0[16]}$ $\\var{r0[17]}$ $\\var{r0[18]}$ $\\var{r0[19]}$ $\\var{r0[20]}$ $\\var{r0[21]}$ $\\var{r0[22]}$ $\\var{r0[23]}$
", "tags": ["checked2015", "cr1", "data analysis", "elementary statistics", "interquartile range", "IQR", "lower quartile", "LQ", "MAS1604", "MAS8380", "MAS8401", "mean", "mean ", "median", "quartile", "sample mean", "sample standard deviation", "sc", "statistics", "tested1", "upper quartile", "UQ"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

11/07/2012:

\n

\n

Calculation not yet checked.

\n

23/07/2012:

\n

\n

Checked calculation, OK.

\n

Two minor typos changed.

\n

3/08/2012:

\n

\n

Question appears to be working correctly.

\n

19/12/2012:

\n

Changed to new stats extension functions for variance and stdev. Still using the uniform distribution. Checked calculations again.

\n

\n

21/12/2012:

\n

Checked rounding, OK. Added cr1 tag.

\n

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Sample of size $24$ is given in a table. Find sample mean, sample standard deviation, sample median and the interquartile range.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

Sample mean: The sample mean is $\\frac{\\var{sum(r0)}}{\\var{len(r0)}} = \\var{precround(mean(r0),1)}$ to 1 decimal place.

\n

Sample standard deviation: The sample standard deviation is $\\var{stdev(r0,true)}=\\var{siground(stdev(r0,true),3)}$ to 3 significant figures.

\n

If you order the data in increasing order you get the following table:

\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
 $\\var{r1[0]}$ $\\var{r1[1]}$ $\\var{r1[2]}$ $\\var{r1[3]}$ $\\var{r1[4]}$ $\\var{r1[5]}$ $\\var{r1[6]}$ $\\var{r1[7]}$ $\\var{r1[8]}$ $\\var{r1[9]}$ $\\var{r1[10]}$ $\\var{r1[11]}$ $\\var{r1[12]}$ $\\var{r1[13]}$ $\\var{r1[14]}$ $\\var{r1[15]}$ $\\var{r1[16]}$ $\\var{r1[17]}$ $\\var{r1[18]}$ $\\var{r1[19]}$ $\\var{r1[20]}$ $\\var{r1[21]}$ $\\var{r1[22]}$ $\\var{r1[23]}$
\n

Denote the ordered data by $x_j$, thus $x_{10}=\\var{r1[9]}$ for example.

\n

Median: The median lies between the 12th and 13th entries in the ordered table and is given by:

\n

\$0.5\\times x_{12}+0.5\\times x_{13} = 0.5\\times\\var{r1[11]}+0.5\\times \\var{r1[12]}=\\var{median}\$

\n

Interquartile range: As there is an even number of values, the Lower Quartile will lie between two values. Its position is calculated by finding

\n

\$\\frac{n+1}{4}=\\frac{\\var{n+1}}{4}=6\\frac{1}{4}\$

\n

Hence the Lower Quartile lies between the 6th and 7th entries in the ordered table.

\n

It is \$0.75\\times x_6+0.25\\times x_7 = 0.75\\times\\var{r1[5]}+0.25\\times \\var{r1[6]}=\\var{lquartile}\$

\n

Once again as there is an even number of values, the Upper Quartile will lie between two values and its position is calculated by finding

\n

\$\\frac{3(n+1)}{4}=\\frac{\\var{3*(n+1)}}{4}=18\\frac{3}{4}\$

\n

Hence the Upper Quartile lies between the 18th and 19th entries in the ordered table.

\n

We find it is \$0.25\\times x_{18}+0.75\\times x_{19} = 0.25\\times\\var{r1[17]}+0.75\\times \\var{r1[18]}=\\var{uquartile}\$

\n

The interquartile range is defined to be

\n

\$\\text{Upper Quartile} – \\text{Lower Quartile} \$

\n

and so in this case we have:

\n

\$\\text{Interquartile range} = \\var{uquartile}-\\var{lquartile}=\\var{interq} \$

"}, {"name": "Sample standard deviation", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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"}, "r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu,sig1)),n)", "description": "", "name": "r1"}, "mean2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mean(r1)", "description": "", "name": "mean2"}, "sig1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(9..15)", "description": "", "name": "sig1"}, "mean1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mean(r0)", "description": "", "name": "mean1"}, "sig0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(6..12)", "description": "", "name": "sig0"}, "stdev1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(stdev(r0,true),1)", "description": "", "name": "stdev1"}, "stdev2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(stdev(r1,true),1)", "description": "", "name": "stdev2"}, "exam2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'Cell Biology'", "description": "", "name": "exam2"}, "stdevoverall": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(stdev(sscores,true),1)", "description": "", "name": "stdevoverall"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "10", "description": "", "name": "n"}, "mu": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(55..65)", "description": "", "name": "mu"}, "total": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'Total Score'", "description": "", "name": "total"}, "ssq2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(x^2,x,r1))", "description": "", "name": "ssq2"}, "exam1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'Anatomy'", "description": "", "name": "exam1"}, "var2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(variance(r1,true),3)", "description": "", "name": "var2"}, "sscores": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(r0[x]+r1[x],x,0..n-1)", "description": "", "name": "sscores"}, "overallvar": {"templateType": "anything", "group": "Ungrouped variables", "definition": "variance(sscores,true)", "description": "", "name": "overallvar"}, "ssq1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(x^2,x,r0))", "description": "", "name": "ssq1"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "2", "description": "", "name": "s"}, "overallmean": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mean(sscores)", "description": "", "name": "overallmean"}, "r0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(round(normalSample(mu,sig0)),n)", "description": "", "name": "r0"}, "var1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(variance(r0,true),3)", "description": "", "name": "var1"}}, "ungrouped_variables": ["overallmean", "mean1", "mean2", "overallvar", "ssq1", "ssq2", "total", "exam2", "tol", "exam1", "stdev1", "stdev2", "var1", "var2", "sig1", "sig0", "stdevoverall", "r0", "r1", "n", "mu", "s", "sscores"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{stdev1+tol}", "minValue": "{stdev1-tol}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n \n \n

#### {exam1}

\n \n \n \n

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

\n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{stdev2+tol}", "minValue": "{stdev2-tol}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n \n \n

#### {exam2}

\n \n \n \n

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

\n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{stdevoverall+tol}", "minValue": "{stdevoverall-tol}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n \n \n

#### {total}

\n \n \n \n

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

\n \n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\n \n \n

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

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

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

\n \n ", "tags": ["checked2015", "cr1", "data analysis", "elementary statistics", "mean ", "sample", "sample mean", "sample standard deviation", "sample variance", "standard deviation", "statistics", "stats", "tested1", "unused", "variance"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

11/07/2012:

\n

\n

Set new variable tol=0 for all numeric input so that answers have to be accurate to 1 decimal place.

\n

Testing calculation not yet possible due to stats extension unavailability.

\n

23/07/2012:

\n

Corrected error in calculation of variance of Total Score. The variable scores was not used and so mean and variance were not correct.

\n

Checked calculations. OK.

\n

\n

1/08/2012:

\n

\n

Question appears to be working correctly.

\n

19/12/2012:

\n

Changed stats functions to the ones from the new stats extension.

\n

Checked calculations.

\n

\n

21/12/2012:

\n

Checked rounding, OK. Added cr1 tag.

", "licence": "Creative Commons Attribution 4.0 International", "description": "

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

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n \n \n

The solution to the first part is here – the other parts can be done in the same way.

\n \n \n \n

For {exam1} we have the mean is:

\n \n \n \n

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

\n \n \n \n

The sample variance is given by the formula:

\n \n \n \n

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

\n \n \n \n

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

\n \n \n \n

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)}\\\\ \n \n &=& \\var{ssq1}\\\\\n \n \\\\\n \n \\\\\n \n n\\mu^2 &=&\\var{n} \\times\\var{mean1}^2\\\\\n \n &=& \\var{n*mean1^2}\n \n \\end{eqnarray*}\n \n \$
Hence substituting these values into the formula we find that:

\n \n \n \n

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

\n \n \n \n

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

\n \n \n \n

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

\n \n "}, {"name": "Calculate probability of combinations of events happening or not, , ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{intersect-tol}", "maxValue": "{intersect+tol}", "marks": 1}], "type": "gapfill", "prompt": "\n \n \n

$P(A\\cap B)=\\;\\;$[[0]]

\n \n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{intcom-tol}", "maxValue": "{intcom+tol}", "marks": 1}], "type": "gapfill", "prompt": "\n \n \n

$P(A^c\\cap B^c)=\\;\\;$[[0]]

\n \n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{unioncom-tol}", "maxValue": "{unioncom+tol}", "marks": 1}], "type": "gapfill", "prompt": "\n \n \n

$P(A^c\\cup B^c)=\\;\\;$[[0]]

\n \n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{prob4-tol}", "maxValue": "{prob4+tol}", "marks": 1}], "type": "gapfill", "prompt": "\n \n \n

$P(A^c\\cap B)=\\;\\;$[[0]]

\n \n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{prob5-tol}", "maxValue": "{prob5+tol}", "marks": 1}], "type": "gapfill", "prompt": "\n \n \n

$P(A^c\\cup B)=\\;\\;$[[0]]

\n \n \n ", "showCorrectAnswer": true, "marks": 0}], "variables": {"prob4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(1-prob2-intersect,10)", "name": "prob4", "description": ""}, "intcom": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1-prob3", "name": "intcom", "description": ""}, "intersect": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(prob1+1-prob2-prob3,2)", "name": "intersect", "description": "

P(A and B)

"}, "prob2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.1..0.9#0.05)", "name": "prob2", "description": "

P(not B)

"}, "prob3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround((t*(max(prob1,1-prob2))+(100-t)*min(0.95,prob1+1-prob2))/100,2)", "name": "prob3", "description": "

P(A or B)

"}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "name": "tol", "description": ""}, "unioncom": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1-intersect", "name": "unioncom", "description": ""}, "prob1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.1..0.9#0.05)", "name": "prob1", "description": "

P(A)

"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..100)", "name": "t", "description": ""}, "prob5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1-prob1+1-prob2-prob4", "name": "prob5", "description": ""}}, "ungrouped_variables": ["intcom", "intersect", "prob1", "prob2", "prob3", "prob4", "prob5", "t", "tol", "unioncom"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "variable_groups": [], "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n

Let $A$ and $B$ be events with:

\n

1. $P(A) = \\var{prob1}$

\n

2. $P(A \\cup B)=\\var{prob3}$

\n

3. $P(B^c)=\\var{prob2}$

\n

Find the following probabilities (all answers to 2 decimal places):

\n ", "tags": ["axiom", "axioms of probability", "checked2015", "complement", "complement of an event", "cr1", "elementary probability", "intersection of events", "intersection of sets", "laws of sets", "MAS1604", "MAS8380", "MAS8401", "Probability", "probability", "probability laws", "set laws", "sets", "statistics", "tested1", "union", "union of events", "union of sets"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

7/07/2012:

\n

\n

Set tolerances via new variable tol=0 for all answers.

\n

Checked calculations.

\n

22/07/2012:

\n

\n

Switched on stats extension (not needed, but policy for all stats questions).

\n

31/07/2012:

\n

\n

In the Advice section, moved \\Rightarrow to beginning of the line instead of the end of the previous line.

\n

Question appears to be working correctly.

\n

20/12/2012:

\n

Added tested1 tag after checking again - calculations OK.

\n

21/12/2012:

\n

Checked rounding, OK. Added tag cr1.

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Given  $P(A)$, $P(A\\cup B)$, $P(B^c)$ find $P(A \\cap B)$, $P(A^c \\cap B^c)$, $P(A^c \\cup B^c)$ etc..

#### a)

\n

It follows from the axioms of probability that:

\n

\$P(A \\cup B)=P(A)+P(B)-P(A \\cap B)\$

\n

Hence

\n

\\begin{align}
P(A \\cap B) &= P(A)+P(B)-P(A \\cup B) \\\\
&= \\var{prob1}+1-\\var{prob2}-\\var{prob3} \\\\
&= \\var{intersect}
\\end{align}

\n

Note that we have used $P(B)=1-P(B^c)= 1-\\var{prob2}=\\var{1-prob2}$

\n

#### b)

\n

The laws of sets gives:

\n

\$A^c \\cap B^c=(A \\cup B)^c\$

\n

so

\n

\\begin{align}
P(A^c \\cap B^c) &= P((A \\cup B)^c) \\\\
&= 1-P(A \\cup B) \\\\
&= 1-\\var{prob3} \\\\
&= \\var{1-prob3}
\\end{align}

\n

#### c)

\n

Similarly to b), the laws of sets gives:

\n

\$A^c \\cup B^c=(A \\cap B)^c\$

\n

so

\n

\\begin{align}
P(A^c \\cup B^c) &= P((A \\cap B)^c) \\\\
&= 1-P(A \\cap B) \\\\
&= 1-\\var{intersect} \\\\
&= \\var{1-intersect}
\\end{align}

\n

#### d)

\n

Note that $B$ is the following union of disjoint sets:

\n

\$B = (A^c \\cap B) \\cup (A \\cap B)\$

\n

Hence

\n

\\begin{align}
P(B) &= P(A^c \\cap B) + P(A \\cap B) \\\\
\\implies P(A^c \\cap B) &= P(B)-P(A\\cap B) \\\\
&= 1-\\var{prob2}-\\var{intersect} \\\\
&= \\var{prob4}
\\end{align}

\n

#### e)

\n

Once again using a familiar result we have:

\n

\\begin{align}
P(A^c \\cup B) &= P(A^c)+P(B)-P(A^c \\cap B) \\\\
&= 1-\\var{prob1}+1-\\var{prob2}-\\var{prob4} \\\\
&= \\var{prob5}
\\end{align}

\n

Where we used the result from d) that $P(A^c \\cap B)=\\var{prob4}$

"}, {"name": "Calculate probability of either of two events occurring", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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": {"dothisandthat": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"work on both domestic and European routes\"", "description": "", "name": "dothisandthat"}, "ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(1-prob1,2)", "description": "", "name": "ans2"}, "desc2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"are in training\"", "description": "", "name": "desc2"}, "things": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'stewardesses'", "description": "", "name": "things"}, "dothat1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"works on European routes\"", "description": "", "name": "dothat1"}, "p3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "p-random(85..95)", "description": "", "name": "p3"}, "prob1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround((p-p3)/100,2)", "description": "", "name": "prob1"}, "therest": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"The remainder\"", "description": "", "name": "therest"}, "desc1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"with a small UK-based airline\"", "description": "", "name": "desc1"}, "p2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "p-p1", "description": "", "name": "p2"}, "thing": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"stewardess\"", "description": "", "name": "thing"}, "dothis1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"works on domestic routes\"", "description": "", "name": "dothis1"}, "desc4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"is in training\"", "description": "", "name": "desc4"}, "dothat": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"work on European routes\"", "description": "", "name": "dothat"}, "dothis": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"work on domestic routes\"", "description": "", "name": "dothis"}, "p1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(40..70)", "description": "", "name": "p1"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(105..125)", "description": "", "name": "p"}, "desc3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"working with this airline\"", "description": "", "name": "desc3"}}, "ungrouped_variables": ["dothisandthat", "desc4", "p1", "desc1", "dothat", "desc3", "things", "ans2", "p3", "p", "p2", "dothat1", "dothis", "therest", "desc2", "thing", "dothis1", "prob1"], "functions": {}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "\n

Find the probabilities that a randomly chosen {thing} {desc3}:

\n

a) {dothis1} or {dothat1}.

\n

Probability = [[0]]

\n

b) {desc4}.

\n

Probability = [[1]]

\n

Enter both probabilities to 2 decimal places.

\n ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "prob1", "maxValue": "prob1", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ans2", "maxValue": "ans2", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "\n

$\\var{p1}$% of {things} {desc1} {dothis}, $\\var{p2}$% {dothat} and $\\var{p3}$% {dothisandthat}.

\n

{therest} {desc2}

\n ", "tags": ["checked2015"], "rulesets": {}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Example showing how to calculate the probability of A or B using the law $p(A \\;\\textrm{or}\\; B)=p(A)+p(B)-p(A\\;\\textrm{and}\\;B)$.

\n

Also converting percentages to probabilities.

\n

a) There are $\\var{p1}+\\var{p2}-\\var{p3}=\\var{p-p3}$ % of stewardesses working on one of the routes. The probability that a random stewardess is working on one of these routes is therefore $\\displaystyle \\frac{\\var{p-p3}}{100}=\\var{prob1}$.

\n

b) The rest are in training and the probability that a randomly selected stewardess is in training is $1-\\var{prob1}=\\var{1-prob1}$.

"}, {"name": "Convert gambling odds to probabilities, , ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"wdw": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'A bookmaker lists'", "description": "", "name": "wdw"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..20)", "description": "", "name": "t"}, "player1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'Australia'", "description": "", "name": "player1"}, "odds22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "round(odds12*odds21/odds11)-random(2..5 except round(odds12*odds21/odds11))", "description": "", "name": "odds22"}, "den2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "odds21+odds22", "description": "", "name": "den2"}, "odds21": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(8..20)", "description": "", "name": "odds21"}, "between": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'between'", "description": "", "name": "between"}, "odds12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "round((t*round(odds11+1)/2+(20-t)*(odds11-1))/20)", "description": "", "name": "odds12"}, "odds11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(8..20)", "description": "", "name": "odds11"}, "event": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'the next series of games'", "description": "", "name": "event"}, "player2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'England'", "description": "", "name": "player2"}, "den1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "odds11+odds12", "description": "", "name": "den1"}}, "ungrouped_variables": ["odds12", "odds11", "den2", "den1", "wdw", "player2", "player1", "t", "between", "odds22", "event", "odds21"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{odds11}/{den1}", "musthave": {"message": "

Input as a fraction.

", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std, fractionNumbers", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "

You should take the bet if you think that $\\Pr(\\var{player1}\\text{ win})$ is greater than [[0]]

", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{odds22}/{den2}", "musthave": {"message": "

Input as a fraction.

", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std, fractionNumbers", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "

You should take the bet if you think that $\\Pr(\\var{player2}\\text{ win})$ is greater than [[0]]

", "showCorrectAnswer": true, "marks": 0}], "statement": "

The following odds are given for {event} {between} {player1} and {player2}

\n\n\n\n\n\n\n\n
 {player1} $\\var{odds11}: \\var{odds12}$ on {player2} $\\var{odds21}: \\var{odds22}$ against
\n

Convert these statements about odds into probabilities.

\n

", "tags": ["checked2015", "converting odds", "elementary probability", "MAS1604", "MAS8380", "MAS8401", "odds", "odds into probabilities", "probability", "Probability", "statistics", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

7/07/2012:

\n

\n

\n

Checked calculation.

\n

22/07/2012:

\n

\n

31/07/2012:

\n

\n

Question appears to be working correctly.

\n

20/12/2012:

\n

Checked calculations, OK. Added tested1 tag.

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Converting odds to probabilities.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

#### a)

\n

Odds of $\\var{odds11}: \\var{odds12}$ on means that you should take the bet if you think the probability of {player1} winning is greater than:

\n

\$\\frac{\\var{odds11}}{\\var{odds11}+\\var{odds12}} = \\simplify[std]{{odds11}/{odds11+odds12}}\$

\n

#### b)

\n

Odds of $\\var{odds21}: \\var{odds22}$ against means that you should take the bet if you think that the probability of {player2} losing is less than:

\n

\$\\frac{\\var{odds21}}{\\var{odds21}+\\var{odds22}} = \\simplify[std]{{odds21}/{odds21+odds22}}\$

\n

That is, if you think the probability of {player2} winning is greater than:

\n

\$1- \\simplify[std]{{odds21}/{odds21+odds22}}=\\simplify[std]{{odds22}/{odds21+odds22}}\$

\n

Note that the sum of these probabilities is:

\n

\$\\simplify[std]{{odds11}/{odds11+odds12}}+\\simplify[std]{{odds22}/{odds21+odds22}}=\\simplify[std]{{odds11*odds21+odds22*odds12+2*odds11*odds22}/{odds11*odds21+odds22*odds12+odds11*odds22+odds12*odds21}}\$

\n

which is less than $1$, as otherwise you could bet on both to win and not lose any money!

"}, {"name": "Roll a pair of dice - find probability at least one die shows a given number.", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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/"}], "variable_groups": [], "variables": {"number": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..6)", "description": "", "name": "number"}}, "ungrouped_variables": ["number"], "functions": {}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

What is the probability of at least one die showing a $\\var{number}$?

\n

Probability = [[0]]

\n

", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "11/36", "answerSimplification": "std, fractionNumbers", "notallowed": {"message": "

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "checkingType": "absdiff", "vsetRange": [0, 1], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "expectedVariableNames": [], "variableReplacements": [], "failureRate": 1, "musthave": {"message": "

Input as a fraction.

", "showStrings": false, "partialCredit": 0, "strings": ["/", 11, 36]}, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "unitTests": [], "scripts": {}, "vsetRangePoints": 5, "showPreview": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "

Two fair six-sided dice are rolled.

", "tags": ["checked2015", "dice", "die", "elementary probability", "events", "independence", "independent events", "Probability", "probability", "probability dice", "statistics", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Rolling a pair of dice. Find probability that at least one die shows a given number.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n \n \n

Let $A$ be the event that first dice shows a $\\var{number}$ $\\Rightarrow P(A)=\\frac{1}{6}$.

\n \n \n \n

Let $B$ be the event that second dice shows a $\\var{number}$ $\\Rightarrow P(B)=\\frac{1}{6}$.

\n \n \n \n

$A$ and $B$ are independent events so $P(A\\cap B) = P(A)\\times P(B)$.

\n \n \n \n

We want the probability $P(A \\cup B)$ of either $A$ or $B$ showing $\\var{number}$ and

\n \n \n \n

\$\\begin{eqnarray*}\n \n P(A \\cup B) &=& P(A)+P(B)-P(A \\cap B)\\\\\n \n &=& P(A)+P(B)-P(A)P(B)\\\\\n \n &=&\\frac{1}{6}+ \\frac{1}{6}-\\frac{1}{36}\\\\\n \n &=& \\frac{11}{36}\n \n \\end{eqnarray*}\n \n \$

\n \n \n "}, {"name": "Probability - draw without replacement", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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": "0.0001", "description": "", "name": "tol"}, "is": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(m=1,'is','are')", "description": "", "name": "is"}, "colour2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random('white','yellow','blue')", "description": "", "name": "colour2"}, "prob1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(prob,4)", "description": "", "name": "prob1"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..100)", "description": "", "name": "t"}, "k": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(5..8)", "description": "", "name": "k"}, "nochoices": {"templateType": "anything", "group": "Ungrouped variables", "definition": "comb(n,k)*comb(g-n,m-k)", "description": "", "name": "nochoices"}, "prob0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "comb(g-n,m)/possiblechoices", "description": "", "name": "prob0"}, "atleastone": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(1-prob0,4)", "description": "", "name": "atleastone"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "g-k", "description": "", "name": "n"}, "returned": {"templateType": "anything", "group": "Ungrouped variables", "definition": "''", "description": "", "name": "returned"}, "possiblechoices": {"templateType": "anything", "group": "Ungrouped variables", "definition": "comb(g,m)", "description": "", "name": "possiblechoices"}, "colour1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random('black','red','green')", "description": "", "name": "colour1"}, "things": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'balls'", "description": "", "name": "things"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "name": "m"}, "special": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'coloured '+ colour1", "description": "", "name": "special"}, "caught1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "''", "description": "", "name": "caught1"}, "caught2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'drawn'", "description": "", "name": "caught2"}, "later": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'Next'", "description": "", "name": "later"}, "prob": {"templateType": "anything", "group": "Ungrouped variables", "definition": "nochoices/possiblechoices", "description": "", "name": "prob"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..15)", "description": "", "name": "g"}, "indef": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(container='urn', 'n','')", "description": "", "name": "indef"}, "marked": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'coloured '+ colour1+ ' the rest '+ colour2", "description": "", "name": "marked"}, "then": {"templateType": "anything", "group": "Ungrouped variables", "definition": "''", "description": "", "name": "then"}, "container": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random('urn','box','container')", "description": "", "name": "container"}}, "ungrouped_variables": ["is", "marked", "special", "atleastone", "container", "things", "caught2", "caught1", "tol", "prob", "indef", "then", "returned", "nochoices", "prob0", "prob1", "possiblechoices", "g", "k", "later", "m", "n", "colour2", "colour1", "t"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "{atleastone-tol}", "maxValue": "{atleastone+tol}", "marks": 2}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n

A{indef} {container} contains $\\var{g}$ {things}. {Then} $\\var{n}$ of these are {caught1} {marked} {returned}.

\n

{Later} $\\var{m}$ {things} {is} {caught2} without replacement.

\n

What is the probability that at least one of the $\\var{m}$ is {special}?

\n

Input your answer to $4$ decimal places.

\n

Probability = [[0]]?

\n ", "marks": 0}], "statement": "\n \n \n

\n \n \n ", "tags": ["MAS1604", "Probability", "checked2015", "combinations", "cr1", "elementary probability", "number of selections", "sample space", "selection without replacement", "statistics", "tested1", "urn model", "without replacement"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

7/07/2012:

\n

\n

Included more detail in Advice by showing number of combinations explicitly.

\n

Answer tolerance set as new variable tol=0.0001. Perhaps 3 dps and tol=0?

\n

Checked calculation. OK.

\n

22/07/2012:

\n

\n

Checked stats extension box.

\n

31/07/2012:

\n

Question appears to be working correctly.

\n

20/12/2012:

\n

Checked calculation again, OK. Added tested1 tag.

\n

Note that there is scope for setting questions with different wording to that of urn models.

\n

21/12/2012:

\n

Checked rounding, OK. Added tag cr1.

", "licence": "Creative Commons Attribution 4.0 International", "description": "\n \t\t

A box contains $n$ balls, $m$ of these are red the rest white.

\n \t\t

$r$ are drawn without replacement.

\n \t\t

What is the probability that at least one of the $r$ is red?

\n \t\t"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n

The probability that at least one of the {things} is {special} is the same as 1 – probability that none of them are {special}.

\n

The number of selections without {things} {special} is

\n

\${\\var{g-n} \\choose \\var{m}}=\\var{comb(g-n,m)}\$ as all have to be selected from the $\\var{g-n}$ which are {colour2}.

\n

There are $\\displaystyle {\\var{g} \\choose \\var{m}}=\\var{comb(g,m)}$ ways of drawing $\\var{m}$ {things} from the $\\var{g}$.

\n

Hence the probability that none them are {special} is:

\n

\$\\frac{{\\var{g-n} \\choose \\var{m}}}{{\\var{g} \\choose \\var{m}}}=\\frac{\\var{comb(g-n,m)}}{\\var{possiblechoices}}\$

\n

So the probability we want is

\n

\$1- \\frac{\\var{comb(g-n,m)}}{\\var{possiblechoices}} = \\var{atleastone}\$ to 4 decimal places.

\n "}, {"name": "Probability - sum of two numbers drawn without replacement", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"noeven": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(isint(mi/2),di/2+1,di/2)", "description": "", "name": "noeven"}, "botheven": {"templateType": "anything", "group": "Ungrouped variables", "definition": "comb(noeven,2)", "description": "", "name": "botheven"}, "together": {"templateType": "anything", "group": "Ungrouped variables", "definition": "botheven+bothodd", "description": "", "name": "together"}, "numpar": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(parity='even',noeven,noodd)", "description": "", "name": "numpar"}, "mess": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(gcd(comb(numpar,2),together)=1,'','(after reducing to lowest form as a fraction).')", "description": "", "name": "mess"}, "mi": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "name": "mi"}, "otherparity": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(parity='even','odd','even')", "description": "", "name": "otherparity"}, "bothodd": {"templateType": "anything", "group": "Ungrouped variables", "definition": "comb(noodd,2)", "description": "", "name": "bothodd"}, "ma": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mi+random(8..12#2)", "description": "", "name": "ma"}, "di": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ma-mi", "description": "", "name": "di"}, "parity": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random('odd','even')", "description": "", "name": "parity"}, "noodd": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(isint(mi/2),di/2,di/2+1)", "description": "", "name": "noodd"}, "numotherpar": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(parity='even',noodd,noeven)", "description": "", "name": "numotherpar"}, "bothsame": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(parity='odd',bothodd,botheven)", "description": "", "name": "bothsame"}}, "ungrouped_variables": ["parity", "otherparity", "ma", "di", "mess", "mi", "numotherpar", "noodd", "together", "bothodd", "botheven", "numpar", "bothsame", "noeven"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{bothsame}/{botheven+bothodd}", "musthave": {"showStrings": false, "message": "

", "strings": ["/"], "partialCredit": 0}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "

", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n

Probability that both numbers are {parity}= [[0]]

\n

\n ", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n

Two numbers are drawn at random (and without replacement) from the numbers $\\var{mi}$ to $\\var{ma}$.

\n

Find the probability that both numbers are {parity} given that their sum is even.

\n ", "tags": ["MAS1604", "Probability", "checked2015", "conditional probability", "counting", "drawn without replacement", "events", "sampling space", "select without replacement", "sets", "statistics", "subset", "tested1", "urn model", "without replacement"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

7/07/2012:

\n

\n

Reminded user to input answer as a fraction.

\n

Checked calculation.

\n

22/07/2012:

\n

\n

Checked stats extension box.

\n

31/07/2012:

\n

Question appears to be working correctly.

\n

20/12/2012:

\n

Checked calculation, OK. Added tested1 tag.

", "licence": "Creative Commons Attribution 4.0 International", "description": "\n \t\t

Two numbers are drawn at random without replacement from the numbers m to n.

\n \t\t

Find the probability that both are odd given their sum is even.

\n \t\t"}, "advice": "\n \n \n

As we are sampling without replacement the best sampling space is the space of all unordered pairs.

\n \n \n \n

This means that when we count up the number of pairs we use the number of ways of selecting pairs.

\n \n \n \n

Let $A$ be the event that both numbers are {parity} and $B$ the event that their sum is even.

\n \n \n \n

Note that $A$ is a subset of $B$ hence $P(A \\cap B)=P(A)$.

\n \n \n \n

The probability we want to find is $P(A | B)$.

\n \n \n \n

Using the definition of conditional probability:

\n \n \n \n

\$P(A | B) = \\frac{P(A \\cap B)}{P(B)} = \\frac{P(A)}{P(B)} \$

\n \n \n \n

Now there are $\\var{numpar}$ {parity} numbers between $\\var{mi}$ and $\\var{ma}$.

\n \n \n \n

and as we are sampling without replacement there are

\n \n \n \n

\${\\var{numpar} \\choose 2} = \\frac{\\var{numpar}\\times \\var{numpar-1}}{2} = \\var{comb(numpar,2)}\$

\n \n \n \n

such pairs, both {parity}.

\n \n \n \n

This gives the number of elements in $A$.

\n \n \n \n

Also since there are $\\var{ma-mi+1-numpar}$ {otherparity} numbers in the range, there are:

\n \n \n \n

\${\\var{numotherpar} \\choose 2}=\\var{comb(numotherpar,2)}\$ such pairs, both {otherparity}.

\n \n \n \n

There are $\\var{botheven}+\\var{bothodd}=\\var{together}$ pairs with sum even.

\n \n \n \n

This gives the number of events in $B$.

\n \n \n \n

Hence \$\\frac{P(A)}{P(B)}=\\frac{\\var{comb(numpar,2)}}{\\var{together}}\$

\n \n \n \n

So the probability that both are {parity} given their sum is even is

\n \n \n \n

\$\\simplify[std]{{comb(numpar,2)}/{together}}\$

\n \n \n \n

{mess}

\n \n \n "}, {"name": "Probability of not choosing any from a subset", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"noguilty": {"templateType": "anything", "group": "Ungrouped variables", "definition": "comb(men-guilty,suspects-guilty)", "description": "", "name": "noguilty"}, "p4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(suspects=3,1,ns-3)", "description": "", "name": "p4"}, "ns": {"templateType": "anything", "group": "Ungrouped variables", "definition": "men-suspects", "description": "", "name": "ns"}, "is": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(guilty=1,'is','are')", "description": "", "name": "is"}, "overallnumber": {"templateType": "anything", "group": "Ungrouped variables", "definition": "comb(men,suspects)", "description": "", "name": "overallnumber"}, "is2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(suspects-guilty=1,'is','are')", "description": "", "name": "is2"}, "t3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(suspects=5,1,0)", "description": "", "name": "t3"}, "nguilty": {"templateType": "anything", "group": "Ungrouped variables", "definition": "comb(men-suspects,suspects)", "description": "", "name": "nguilty"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(guilty=1,'man','men')", "description": "", "name": "p"}, "q5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(suspects<5,1,men-4)", "description": "", "name": "q5"}, "guilty": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(suspects>2,suspects-random(1,2),suspects-1)", "description": "", "name": "guilty"}, "ans3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(nguilty/overallnumber,3)", "description": "", "name": "ans3"}, "suspects": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3,4,5)", "description": "", "name": "suspects"}, "q6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(suspects<6,1,men-5)", "description": "", "name": "q6"}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(suspects=4,1,0)", "description": "", "name": "t2"}, "t4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(suspects=6,1,0)", "description": "", "name": "t4"}, "singpl": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(suspects-guilty=1,'man','men')", "description": "", "name": "singpl"}, "test": {"templateType": "anything", "group": "Ungrouped variables", "definition": "nguilty/overallnumber", "description": "", "name": "test"}, "t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(suspects=3,1,0)", "description": "", "name": "t1"}, "q4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(suspects=3,1,men-3)", "description": "", "name": "q4"}, "men": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(12..20)", "description": "", "name": "men"}, "p5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(suspects<5,1,ns-4)", "description": "", "name": "p5"}, "p6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(suspects<6,1,ns-5)", "description": "", "name": "p6"}}, "ungrouped_variables": ["guilty", "is", "ans3", "suspects", "noguilty", "q5", "q4", "q6", "is2", "test", "ns", "men", "singpl", "p6", "p4", "p5", "t4", "nguilty", "t2", "t3", "t1", "p", "overallnumber"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{nguilty}/{overallnumber}", "musthave": {"showStrings": false, "message": "

", "strings": ["/"], "partialCredit": 0}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "

", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n \n \n

What is the probability that none of the suspects are chosen?

\n \n \n \n

Probability = [[0]]?

\n \n \n \n

\n \n ", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

A line-up of $\\var{men}$ men is conducted in order that a witness can identify $\\var{suspects}$ suspects.

\n

Suppose that all $\\var{suspects}$ suspects are in the line-up.

\n

Also suppose that the witness does not recognise any of the suspects but simply chooses $\\var{suspects}$ men at random.

", "tags": ["MAS1604", "Probability", "checked2015", "choosing", "combinations", "counting", "cr1", "query", "sample space", "selecting", "selection", "statistics", "tested1", "ways of choosing"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

7/07/2012:

\n

\n

Added an alternative solution to this question (Method 2).

\n

Checked calculation.

\n

22/07/2012:

\n

\n

Checked the stats extension box.

\n

Perhaps the answer should be a decimal rather than a fraction - looks clumsy.

\n

31/07/2012:

\n

Question appears to be working correctly.

\n

20/12/2012:

\n

Could have a variant of this question by using 'scenario' string variables. Added sc tag for this. Also query the above point about a decimal solution rather than a fraction.

\n

Checked calculation, OK. Added tested1 tag.

\n

Improved display of numbers by texifying them.

\n

21/12/2012:

\n

Checked rounding, OK. Added tag cr1.

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Given subset $T \\subset S$ of $m$ objects in $n$ find the probability of choosing without replacement $r\\lt n-m$ from $S$ and not choosing any element in $T$.

We can work out the probability in two ways:

\n

Method 1.

\n

There are $\\var{men-suspects}$ of the $\\var{men}$ who are not suspects.

\n

The probability of picking the first who is not a suspect is therefore:

\n

\$\\simplify[]{{men-suspects}/{men}}\$

\n

The second choice of a non-suspect will be from $\\var{men-suspects-1}$ non-suspects in $\\var{men-1}$ with probability:

\n

\$\\simplify[]{{men-suspects-1}/{men-1}}\$

\n

Hence the probability of choosing two non-suspects will be .

\n

\$\\simplify[]{{men -suspects} / {men}}\\times \\simplify[]{{men -suspects-1} / {men-1}}\$

\n

Continuing in this way we see that the probability of choosing $\\var{suspects}$ non-suspects is:

\n

\$\\simplify[zeroFactor,unitFactor,zeroTerm]{{t1} * ({men -suspects} / {men}) * ({men -suspects -1} / {men -1}) * ({men -suspects -2} / {men -2}) + {t2} * ({men -suspects} / {men}) * ({men -suspects -1} / {men -1}) * ({men -suspects -2} / {men -2}) * ({ns -3} / {men -3}) + {t3} * ({men -suspects} / {men}) * ({men -suspects -1} / {men -1}) * ({men -suspects -2} / {men -2}) * ({ns -3} / {men -3}) * ({ns -4} / {men -4}) + {t4} * ({men -suspects} / {men}) * ({men -suspects -1} / {men -1}) * ({men -suspects -2} / {men -2}) * ({ns -3} / {men -3}) * ({ns -4} / {men -4}) * ({ns -5} / {men -5})}=\\simplify[std]{{nguilty}/{overallnumber}}\$

\n

on reducing the fraction to its lowest form.

\n

Method 2.

\n

There are $\\var{men-suspects}$ of the $\\var{men}$ who are not suspects.

\n

Hence there are \${\\var{men-suspects}\\choose \\var{suspects}}=\\var{comb(men-suspects,suspects)}\$ ways of choosing $\\var{suspects}$ non-suspects.

\n

In total there are \${\\var{men}\\choose \\var{suspects}}=\\var{comb(men,suspects)}\$ ways of choosing $\\var{suspects}$ from all present.

\n

Hence the probability is \$\\frac{\\var{comb(men-suspects,suspects)}}{\\var{comb(men,suspects)}}= \\simplify[std]{{nguilty}/{overallnumber}} \$

"}, {"name": "Calculate expectation and a probability from a frequency table, , , ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [{"variables": ["idef", "thing", "episodes", "period", "activity"], "name": "Strings"}, {"variables": ["p0", "p1", "p2", "p3", "p4", "p5", "p6", "p7", "p8", "probabilities", "values"], "name": "Probabilities"}, {"variables": ["r", "s", "t", "t1", "t2", "u1", "u2", "u3", "d"], "name": "Stuff to generate probabilities"}], "variables": {"p4": {"templateType": "anything", "group": "Probabilities", "definition": "t-p8-p7-p6-p5", "description": "", "name": "p4"}, "expected_number": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(x*y,[x,y],zip(probabilities,values)))", "description": "", "name": "expected_number"}, "u2": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "u1", "description": "", "name": "u2"}, "p1": {"templateType": "anything", "group": "Probabilities", "definition": "p0+t1", "description": "", "name": "p1"}, "p3": {"templateType": "anything", "group": "Probabilities", "definition": "r-p0-p1-p2", "description": "", "name": "p3"}, "t": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "100-r", "description": "", "name": "t"}, "probexceed": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(if(j>expected_number,probabilities[j],0),j,0..8))", "description": "", "name": "probexceed"}, "values": {"templateType": "anything", "group": "Probabilities", "definition": "list(0..8)", "description": "", "name": "values"}, "thing": {"templateType": "string", "group": "Strings", "definition": "\"airline\"", "description": "", "name": "thing"}, "u3": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "u1", "description": "", "name": "u3"}, "u1": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "round(d*random(70..100)/100)", "description": "", "name": "u1"}, "expect_int": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(expected_number)", "description": "", "name": "expect_int"}, "activity": {"templateType": "string", "group": "Strings", "definition": "\"luggage handling\"", "description": "", "name": "activity"}, "probabilities": {"templateType": "anything", "group": "Probabilities", "definition": "map(x/100,x,[p0,p1,p2,p3,p4,p5,p6,p7,p8])", "description": "

Probability of there being $i$ episodes

", "name": "probabilities"}, "d": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "round(t/15)", "description": "", "name": "d"}, "episodes": {"templateType": "string", "group": "Strings", "definition": "\"complaints\"", "description": "", "name": "episodes"}, "t2": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "t1", "description": "", "name": "t2"}, "p8": {"templateType": "anything", "group": "Probabilities", "definition": "d", "description": "", "name": "p8"}, "p7": {"templateType": "anything", "group": "Probabilities", "definition": "p8+u1", "description": "", "name": "p7"}, "p5": {"templateType": "anything", "group": "Probabilities", "definition": "p6+u3", "description": "", "name": "p5"}, "idef": {"templateType": "string", "group": "Strings", "definition": "\"an\"", "description": "", "name": "idef"}, "p2": {"templateType": "anything", "group": "Probabilities", "definition": "p1+t2", "description": "", "name": "p2"}, "t1": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "round(s*random(70..100)/100)", "description": "", "name": "t1"}, "r": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "random(45..65)", "description": "", "name": "r"}, "s": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "round(r/10)", "description": "", "name": "s"}, "p0": {"templateType": "anything", "group": "Probabilities", "definition": "s", "description": "", "name": "p0"}, "p6": {"templateType": "anything", "group": "Probabilities", "definition": "p7+u2", "description": "", "name": "p6"}, "period": {"templateType": "string", "group": "Strings", "definition": "\"day\"", "description": "", "name": "period"}}, "ungrouped_variables": ["expected_number", "expect_int", "probexceed"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "expected_number", "maxValue": "expected_number", "precision": "2", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}], "type": "gapfill", "prompt": "

Find the expected number of {episodes} per {period}.

\n

Expected number = [[0]]

", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "probexceed", "maxValue": "probexceed", "precision": "2", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}], "type": "gapfill", "prompt": "

What is the probability that the number of {episodes} will exceed the expected number?

\n

Probability = [[0]]

", "showCorrectAnswer": true, "marks": 0}], "statement": "

The probabilities that {idef} {thing} will receive {episodes} per {period} about its {activity} are given by the following table:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Complaints{values[0]}{values[1]}{values[2]}{values[3]}{values[4]}{values[5]}{values[6]}{values[7]}{values[8]}
Probability{probabilities[0]}{probabilities[1]}{probabilities[2]}{probabilities[3]}{probabilities[4]}{probabilities[5]}{probabilities[6]}{probabilities[7]}{probabilities[8]}
\n

Answer the following two parts, giving your answers to $2$ decimal places.

", "tags": ["checked2015", "discrete distribution", "expectation", "expected value", "MAS1604", "MAS2304", "MAS8380", "MAS8401", "mass function", "pmf", "PMF", "Probability", "probability", "probability mass function", "query", "sc", "statistics", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

7/07/2012:

\n

\n

Checked calculation.

\n

22/07/2012:

\n

\n

Ticked stats extension box.

\n

31/07/2012:

\n

\n

Question appears to be working correctly.

\n

20/12/2012:

\n

Could increase the number of scenarios by using random string variables. Query tag added for that.

\n

Also very cumbersome use of variables. But no change proposed for now.

\n

Checked calculation, OK. Added tested1 tag.

\n

21/12/2012:

\n

Although asks for solution to 2 dps, there is no rounding as the raw values are to 2 dps. Added sc tag for possible scenarios.

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Given a probability mass function $P(X=i)$ with outcomes $i \\in \\{0,1,2,\\ldots 8\\}$, find the expectation $E$ and $P(X \\gt E)$.

"}, "variablesTest": {"condition": "", "maxRuns": "100"}, "advice": "

#### a)

\n

The expected number of {episodes} is given by:

\n

\$\\simplify[]{{probabilities[0]}*{values[0]} + {probabilities[1]}*{values[1]} + {probabilities[2]}*{values[2]} + {probabilities[3]}*{values[3]} + {probabilities[4]}*{values[4]} + {probabilities[5]}*{values[5]} + {probabilities[6]}*{values[6]} + {probabilities[7]}*{values[7]} + {probabilities[8]}*{values[8]}} = \\var{expected_number} \$

\n

#### b)

\n

We want the probability that the number of {episodes} exceeds $\\var{expected_number}$.

\n

Since the number of {episodes} is a whole number, this is the same as the probability that the number is $\\var{expect_int+1}$ or more and is

\n

\$\\sum_{i=\\var{expect_int+1}}^{i=8} \\left( \\text{Probability}(\\var{episodes} = i ) \\right)= \\simplify[zeroTerm]{ {if(expect_int<1,probabilities[1],0)} + {if(expect_int<2,probabilities[2],0)} + {if(expect_int<3,probabilities[3],0)} + {if(expect_int<4,probabilities[4],0)} + {if(expect_int<5,probabilities[5],0)} + {if(expect_int<6,probabilities[6],0)} + {if(expect_int<7,probabilities[7],0)} + {if(expect_int<8,probabilities[8],0)}} = \\var{probexceed}\$

"}, {"name": "Find expected profit of gambles, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"profit": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(numberbets*bet-numberbets*bet*(odds1+odds2)/(odds2*number),2)", "description": "", "name": "profit"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..6)", "description": "", "name": "r"}, "number": {"templateType": "anything", "group": "Ungrouped variables", "definition": "37", "description": "", "name": "number"}, "odds2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "description": "", "name": "odds2"}, "bet": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,10,50,100)", "description": "", "name": "bet"}, "odds1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "35", "description": "", "name": "odds1"}, "numberbets": {"templateType": "anything", "group": "Ungrouped variables", "definition": "10^r", "description": "", "name": "numberbets"}}, "ungrouped_variables": ["profit", "numberbets", "number", "r", "bet", "odds2", "odds1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "profit", "minValue": "profit", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n

Expected profit=£[[0]]

\n

Enter to two decimal places.

\n

\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "

A roulette table has $\\var{number}$ numbers and pays at $\\var{odds1}$ to $\\var{odds2}$ if the winning number is chosen.

\n

Find the expected profit to the casino if $\\var{10^{r}}$ bets of £$\\var{bet}$ are placed independently.

", "tags": ["checked2015", "MAS8380", "MAS8401"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

04/11/2013

\n

Fix typo \"fod\" -> \"find\".

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Given a large number of gambles, find the expected profit.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n

The probability of winning is $\\dfrac{1}{\\var{number}}$ and the odds of $\\var{odds1}$ to $\\var{odds2}$ tells us that each winning choice realises \$\\text{£}\\simplify{{odds1+odds2}/{odds2}}\\times \\var{bet}=\\text{£}\\var{(odds1+odds2)*bet/odds2}\$on a bet of £ $\\var{bet}$.

\n

Hence the expected payout on a bet of £$\\var{bet}$ is £$\\frac{\\var{(odds1+odds2)*bet}}{\\var{odds2*number}}$

\n

So the expected payout on $\\var{numberbets}$ bets of £$\\var{bet}$ is $\\var{numberbets}\\times \\frac{\\var{(odds1+odds2)*bet}}{\\var{odds2*number}}$

\n

Hence:

\n

Profit = Income - Payout

\n

$=\\text{£}\\var{numberbets}\\times \\var{bet}-\\var{numberbets}\\times \\frac{\\var{(odds1+odds2)*bet}}{\\var{odds2*number}}= \\text{£}\\var{profit}$ to 2 decimal places.

\n

\n "}, {"name": "Is the given function a probability mass function?, , , ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [{"variables": ["is_pmf", "has_negative_probability", "probabilities_dont_sum", "explain_decision"], "name": "Descriptions"}], "variables": {"d3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "d2+random(2..4)", "description": "", "name": "d3"}, "negerror": {"templateType": "anything", "group": "Ungrouped variables", "definition": "round(-b/a)-random(1,2)", "description": "", "name": "negerror"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*(d1+d2+d3+d4)+4*b + error", "description": "", "name": "c"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "has_negative_probability*negerror+(1-has_negative_probability)*random(3..5)", "description": "", "name": "d1"}, "explain_decision": {"templateType": "anything", "group": "Descriptions", "definition": "if(has_negative_probability,\n if(probabilities_dont_sum,\n \"the probabilities do not sum to $1$ and there is a negative probability\",\n \"there is a negative probability\"\n ),\n if(probabilities_dont_sum,\n \"the probabilities do not sum to $1$\",\n \"the probabilities sum to $1$ and all probabilities are non-negative\"\n )\n)", "description": "", "name": "explain_decision"}, "d2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "d1+random(1..5)", "description": "", "name": "d2"}, "has_negative_probability": {"templateType": "anything", "group": "Descriptions", "definition": "random(0,1)", "description": "", "name": "has_negative_probability"}, "is_pmf": {"templateType": "anything", "group": "Descriptions", "definition": "has_negative_probability=0 and probabilities_dont_sum=0", "description": "", "name": "is_pmf"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "a"}, "probabilities_dont_sum": {"templateType": "anything", "group": "Descriptions", "definition": "random(0,1)", "description": "

0 if probabilities sum to 1

", "name": "probabilities_dont_sum"}, "error": {"templateType": "anything", "group": "Ungrouped variables", "definition": "probabilities_dont_sum*random(1..9)", "description": "", "name": "error"}, "d4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "d3+random(3..5)", "description": "", "name": "d4"}}, "ungrouped_variables": ["a", "negerror", "c", "b", "error", "d3", "d4", "d2", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["

Yes, it is a probability mass function

", "

No, it is not a probability mass function

"], "displayColumns": 2, "distractors": ["", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "if(is_pmf,[1,0],[0,1])", "marks": 0}], "type": "gapfill", "prompt": "

Does the following define a valid probability mass function?

\n

\$P(X=x) = \\simplify{({a}x+{b})/{c}},\\;\\;\\;x \\in S=\\{\\var{d1},\\;\\var{d2},\\;\\var{d3},\\;\\var{d4}\\}\$

\n

[[0]]

", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "checkbox", "choices": ["

Probabilities sum to $1$

", "

Probabilities do not sum to $1$

", "

All probabilities are non-negative

", "

There is a negative probability

"], "matrix": "if(probabilities_dont_sum=0,[1,-2],[-2,1])+if(has_negative_probability=0,[1,-2],[-2,1])", "distractors": ["", "", "", ""], "type": "m_n_2", "maxAnswers": 2, "shuffleChoices": false, "warningType": "none", "scripts": {}, "minMarks": 0, "minAnswers": 0, "maxMarks": "2", "showCorrectAnswer": true, "displayColumns": 1, "marks": 0}], "type": "gapfill", "prompt": "

Tick all boxes which describe this function:

\n

[[0]]

\n

Note that if you choose an incorrect option then you will lose 2 marks.

\n

The minimum number of marks you can obtain is 0.

", "showCorrectAnswer": true, "marks": 0}], "statement": "

Determine whether the following defines a valid probability mass function.

\n

Also choose the options which describe the function.

", "tags": ["checked2015", "discrete distribution", "MAS1604", "MAS2304", "MAS8380", "MAS8401", "mass function", "pmf", "PMF", "Probability", "probability", "probability mass function", "statistics", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus", "!simplifyFractions"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

25/02/2015: see the editing history for changes from now on.

\n

\n

7/07/2012:

\n

\n

\n

22/07/2012:

\n

\n

Ticked stats extension box.

\n

Issue with the multiple response question.The feedback on choosing only one correct answer out of the two says that both marks are awarded. This needs to be modified to the correct number of marks awarded and also in practice mode should give the information that there are other correct responses.

\n

Another linked issue is that there should be an option for forcing a number of choices for multiple response.

\n

31/07/2012:

\n

\n

20/12/2012:

\n

The above issue on multiple response has been resolved. Changed the MR so that lose 2 marks if choose an incorrect response (min mark 0).

\n

Corrected error in setting up negative values for function, but still claiming that it was a PMF.

\n

Checked calculation, OK. Added tested1 tag.

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Determine if the following describes a probability mass function.

\n

$P(X=x) = \\frac{ax+b}{c},\\;\\;x \\in S=\\{n_1,\\;n_2,\\;n_3,\\;n_4\\}\\subset \\mathbb{R}$.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

A probability mass function $f(x)=P(X=x)$ has to satisfy:

\n

1. $f(x) \\ge 0$, $\\forall x \\in S$

\n

2. $\\sum_{x \\in S} f(x) = 1$

\n

To verify this we calculate the function as follows:

\n

\\begin{align}
P(X = \\var{d1}) &= \\simplify[std]{({a} * {d1} + {b}) / {c} = {a * d1 + b} / {c}} \\\\ \\\\
P(X = \\var{d2}) &= \\simplify[std]{({a} * {d2} + {b}) / {c} = {a * d2 + b} / {c}} \\\\ \\\\
P(X = \\var{d3}) &= \\simplify[std]{({a} * {d3} + {b}) / {c} = {a * d3 + b} / {c}} \\\\ \\\\
P(X = \\var{d4}) &= \\simplify[std]{({a} * {d4} + {b}) / {c} = {a * d4 + b} / {c}}
\\end{align}

\n

and

\n

\$\\sum_{x \\in S} f(x) =\\simplify[std]{ {a*d1+b}/{c} + {a*d2+b}/{c} + {a*d3+b}/{c} + {a*d4+b}/{c}} = \\simplify[fractionNumbers]{{c-error}/{c}} = \\simplify[std,simplifyFractions]{{c-error}/{c}} \$

\n

In this case, {if(is_pmf,\"this is a probability mass function\",\"this is not a probability mass function\")} as {explain_decision}.

"}, {"name": "Calculate expectation, variance and CDF of uniform distribution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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", "name": "tol", "description": ""}, "ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround((upper-lower)^2/12,3)", "name": "ans2", "description": ""}, "ans3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround((c-lower)/(upper-lower),3)", "name": "ans3", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(d1=f1,d1-1,min(d1,f1))", "name": "d", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "round((upper-lower-2)*random(0..60)/60+lower+1)", "name": "c", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "max(d1,f1)", "name": "f", "description": ""}, "ans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(upper+lower)/2", "name": "ans1", "description": ""}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "round((upper-lower-2)*random(0..60)/60+lower+1)", "name": "d1", "description": ""}, "ans4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround((f-d)/(upper-lower),3)", "name": "ans4", "description": ""}, "upper": {"templateType": "anything", "group": "Ungrouped variables", "definition": "lower+random(2..20#2)", "name": "upper", "description": ""}, "lower": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-10..10)", "name": "lower", "description": ""}, "f1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "round((upper-lower-2)*random(0..60)/60+lower)", "name": "f1", "description": ""}}, "ungrouped_variables": ["upper", "lower", "f", "d", "f1", "ans1", "ans2", "ans3", "ans4", "c", "tol", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "ans1", "correctAnswerFraction": false, "marks": 1, "maxValue": "ans1"}, {"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "ans2-tol", "correctAnswerFraction": false, "marks": 1, "maxValue": "ans2+tol"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

The expectation $\\operatorname{E}[Y]=\\;$[[0]]  (to 3 decimal places).

\n

The variance $\\operatorname{Var}(Y)=\\;$[[1]]  (to 3 decimal places).

\n

", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "ans3-tol", "correctAnswerFraction": false, "marks": 1, "maxValue": "ans3+tol"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

$P(Y \\le \\var{c})=\\;$[[0]]

\n

(to 3 decimal places).

", "marks": 0}, {"scripts": {}, "gaps": [{"answer": "(y-{lower})/({upper-lower})", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answersimplification": "basic", "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "

Input all numbers as fractions or integers.

", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

Input the CDF $F_Y(y)$ for $y$ in the range $\\var{lower} \\le y \\le \\var{upper}$

\n

$F_Y(y)=\\;$[[0]]

\n

Input all numbers as fractions or integers

", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "ans4-tol", "correctAnswerFraction": false, "marks": 1, "maxValue": "ans4+tol"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

$P(\\var{d} \\lt Y \\lt \\var{f})=\\;$[[0]]

\n

\n

", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

Let $Y$ be a random variable with the uniform distribution

\n

\$Y \\sim \\operatorname{U}(\\var{lower},\\var{upper})\$

", "tags": ["CDF uniform", "checked2015", "continuous distributions", "expectation", "MAS1604", "Probability", "probability", "sc", "statistical distributions", "uniform distribution", "uniformly distributed", "variance"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

25/01/2013:

\n

Copy made of 1403CBA3Q5 and then edited.

\n

\n

To be tested.

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Exercise using a given uniform distribution $Y$, calculating the expectation and variance as well as asking for the CDF. Also finding $P(Y \\le a)$ and $P( b \\lt Y \\lt c)$ for a given values $a,\\;b,\\;c$.

a) For a Uniform distribution \$Y \\sim \\operatorname{U}(\\var{lower},\\var{upper})\$ we have:

\n

$\\displaystyle \\operatorname{E}[Y] = \\simplify[!collectNumbers]{({lower}+{upper})/2}=\\var{ans1}$

\n

$\\displaystyle \\operatorname{Var}(Y) =\\simplify[basic,!collectNumbers,!noleadingminus]{({upper}-{lower})^2/12}=\\frac{\\var{upper-lower}^2}{12}=\\var{ans2}$ to 3 decimal places.

\n

b)

\n

$\\displaystyle P(Y \\le \\var{c})=\\simplify[basic,!collectNumbers,!noleadingminus]{({c} -{lower})/({upper}-{lower})}=\\var{ans3}$ to 3 decimal places.

\n

c) The CDF for a uniform distribution $Y$ on the interval $a \\le y \\le b$ is given by:

\n

\$F_Y(y) = \\begin{cases} 0 & y \\lt a \\\\ \\frac{y-a}{b-a} & a \\le y \\le b, \\\\ 1 & y \\gt b. \\end{cases}\$

\n

Hence in this case we have:

\n

\$F_Y(y) = \\simplify[basic]{(y-{lower})/({upper-lower})}\$ for $\\var{lower}\\le y \\le \\var{upper}$

\n

d) Using the CDF we have:

\n

\$\\begin{eqnarray}P(\\var{d} \\lt Y \\lt \\var{f})&=&F_Y(\\var{f})-F_Y(\\var{d})\\\\&=& \\simplify[basic]{({f}-{lower})/({upper-lower})- ({d}-{lower})/({upper-lower})}\\\\&=&\\var{ans4}\\end{eqnarray}\$

\n

to 3 decimal places.

"}, {"name": "Calculate probabilities from a normal distribution, ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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": "0.0001", "description": "", "name": "tol"}, "amount": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"electricity consumption\"", "description": "", "name": "amount"}, "p1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "normalcdf((upper-m)/s,0,1)", "description": "", "name": "p1"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(750..1250#50)", "description": "", "name": "m"}, "prob1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(normalcdf(lower,m,s),4)", "description": "", "name": "prob1"}, "stuff": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"a frozen foods warehouse each week in the summer months \"", "description": "", "name": "stuff"}, "lower": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(m-1.5*s..m-0.5s#5)", "description": "", "name": "lower"}, "prob2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(1-normalcdf(upper,m,s),4)", "description": "", "name": "prob2"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(60..100#10)", "description": "", "name": "s"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "normalcdf((m-lower)/s,0,1)", "description": "", "name": "p"}, "upper": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(m+0.5s..m+1.5*s#5)", "description": "", "name": "upper"}, "units1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"k Wh\"", "description": "", "name": "units1"}}, "ungrouped_variables": ["units1", "upper", "lower", "p1", "m", "s", "p", "amount", "stuff", "tol", "prob2", "prob1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "prob1+tol", "minValue": "prob1-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "prob2+tol", "minValue": "prob2-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "

Find the probability that in a particular week the {amount} is less than {lower} {units1}:

\n

Probability = [[0]](to 4  decimal places)

\n

Find the probability that in a particular week the {amount} is greater than {upper} {units1}:

\n

Probability = [[1]](to 4  decimal places)

", "showCorrectAnswer": true, "marks": 0}], "statement": "

The {amount}, $X$, of {stuff}  is normally distributed with mean {m}{units1} and standard deviation {s}{units1}.

\n

i.e.   \$X \\sim \\operatorname{N}(\\var{m},\\var{s}^2)\$

\n

", "tags": ["checked2015", "continuous random variable", "MAS8380", "MAS8401", "mean", "mean ", "Normal distribution", "normal distribution", "normal tables", "probabilities", "random variable", "sc", "standard deviation", "statistical distributions", "statistics", "z-scores"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t

1/1/2012:

\n \t\t \t\t

Can be configured to other applications using the string variables suppplied. Included tag sc.

\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

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

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

1. Converting to $\\operatorname{N}(0,1)$

\n

$\\simplify[all,!collectNumbers]{P(X < {lower}) = P(Z < ({lower} -{m}) / {s}) = P(Z < {lower-m}/{s}) = 1 -P(Z < {m-lower}/{s})} = 1 -\\var{p} = \\var{precround(1 -p,4)}$ to 4 decimal places.

\n

Here the probability could have been looked up from normal CDF tables. Alternatively we can simply do the whole

\n

calculation in R by typing $\\operatorname{pnorm}(\\var{lower},\\var{m},\\var{s})$.

\n

2. Converting to $\\operatorname{N}(0,1)$

\n

$\\simplify[all,!collectNumbers]{P(X > {upper}) = P(Z > ({upper} -{m}) / {s}) = P(Z > {upper-m}/{s}) = 1 -P(Z < {upper-m}/{s})} = 1-\\var{p1} = \\var{precround(1 -p1,4)}$ to 4 decimal places.

\n

Here the probability could have been looked up from normal CDF tables. Alternatively we can

\n

simply do the whole calculation in R by typing 1 - pnorm({upper},{m},{s}).

"}, {"name": "Calculate probabilities using the exponential distribution, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.5..0.8#0.01)", "description": "", "name": "r"}, "thismany": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(6..10#0.2)", "description": "", "name": "thismany"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5#0.1)", "description": "", "name": "m"}, "p1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(e^(-thismany/m1),3)", "description": "", "name": "p1"}, "place": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(\n 'London',\n 'Manhattan',\n 'Mexico City',\n 'Beijing',\n 'Los Angeles',\n 'Buenos Aires',\n 'Bangkok'\n )", "description": "", "name": "place"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(e^(-thismany/m),3)", "description": "", "name": "p"}, "m1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(r*m,1)", "description": "", "name": "m1"}, "stuff": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(\n 'carbon monoxide',\n 'Freon-22',\n 'hydrogen sulphide',\n 'mercury',\n 'polynuclear aromatic hydrocarbon',\n 'crystalline silica')", "description": "", "name": "stuff"}}, "ungrouped_variables": ["p1", "thismany", "m", "p", "stuff", "m1", "tol", "place", "r"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "prompt": "

Find the probability that the {stuff} concentration exceeds $\\var{thismany}$ parts per million in a one hour period.

\n

", "minValue": "p-tol", "maxValue": "p+tol", "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "prompt": "

A control strategy reduced the mean to $\\var{m1}$ parts per million.

\n

Now find the probability that a concentration exceeds $\\var{thismany}$ parts per million in a one hour period.

\n

", "minValue": "p1-tol", "maxValue": "p1+tol", "marks": 1, "showPrecisionHint": false}], "statement": "

One hour {stuff} concentrations in samples of air taken at a location in {place} have an approximate exponential distribution with mean $\\var{m}$ parts per million.

", "tags": ["checked2015", "continuous distributions", "distributions", "exponential distribution", "MAS1604", "MAS2304", "Probability", "statistical distributions", "statistics"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

29/01/2013:

\n

First draft completed.

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Calculating simple probabilities using the exponential distribution.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

The random variable $X$ is {stuff} concentration {ppm} and $\\displaystyle X \\sim \\operatorname{Exp}\\left(\\frac{1}{\\var{m}}\\right )$.

\n

Hence the probability that $X \\lt x$ is $P(X \\lt x)=1-e^{-x/\\var{m}}$.

\n

a)

\n

$P(X \\gt \\var{thismany})=1-P(X \\lt \\var{thismany})=1-(1-e^{-\\var{thismany}/\\var{m}})=\\var{p}$ to 3 decimal places.

\n

\n

b) Changing the mean value gives:

\n

$P(X \\gt \\var{thismany})=1-P(X \\lt \\var{thismany})=1-(1-e^{-\\var{thismany}/\\var{m1}})=\\var{p1}$ to 3 decimal places.

\n

"}, {"name": "Expectation and variance of combinations of estimators", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'B'", "name": "b", "description": ""}, "e3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*m", "name": "e3", "description": ""}, "unb2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb1=0,1,random(0,1))", "name": "unb2", "description": ""}, "s7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s7", "description": ""}, "sqsum1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "cw1^2+cx1^2+cy1^2+cz1^2", "name": "sqsum1", "description": ""}, "wrong": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb1*unb2>0,'C',if(unb1=1,'B','A'))", "name": "wrong", "description": ""}, "t3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb3=1,1,random(2..9))", "name": "t3", "description": ""}, "v1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sqsum1*sd^2", "name": "v1", "description": ""}, "su2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "cw2+cx2+cy2+cz2", "name": "su2", "description": ""}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random('W','X','Y','Z')", "name": "p", "description": ""}, "cy1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(cx1=0,random(1..9),random(-9..9))", "name": "cy1", "description": ""}, "sd": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "name": "sd", "description": ""}, "cw1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "cw1", "description": ""}, "t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb1=1,1,random(2..9))", "name": "t1", "description": ""}, "tw": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "name": "tw", "description": ""}, "cy3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "cy3", "description": ""}, "sqsum2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "cw2^2+cx2^2+cy2^2+cz2^2", "name": "sqsum2", "description": ""}, "v3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sqsum3*sd^2", "name": "v3", "description": ""}, "s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "cw1+cx1+cy1+cz1", "name": "s1", "description": ""}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.01", "name": "tol", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'A'", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'C'", "name": "c", "description": ""}, "sqsum3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "cw3^2+cx3^2+cy3^2+cz3^2", "name": "sqsum3", "description": ""}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb2=1,su2,su2+random(1..4))", "name": "s2", "description": ""}, "unb1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "name": "unb1", "description": ""}, "cz3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t3-cw3-cx3-cy3", "name": "cz3", "description": ""}, "cx2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "cx2", "description": ""}, "cw3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "tw*random(10..40)", "name": "cw3", "description": ""}, "sx": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "sx", "description": ""}, "correct2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb1=1,if(unb2=1,'B','C'),'C')", "name": "correct2", "description": ""}, "cw2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "cw2", "description": ""}, "q": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(p='X',random('W','Y','Z'),p='W',random('X','Y','Z'),p='Y',random('W','X','Z'),random('W','X','Y'))", "name": "q", "description": ""}, "e2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(su2*m/S2,2)", "name": "e2", "description": ""}, "e1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*m", "name": "e1", "description": ""}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s7*random(1..10)", "name": "m", "description": ""}, "correct1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb1=1,'A','B')", "name": "correct1", "description": ""}, "cz2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "cz2", "description": ""}, "unb3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb1*unb2>0,0,1)", "name": "unb3", "description": ""}, "cy2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "cy2", "description": ""}, "cx1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "name": "cx1", "description": ""}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "cw3+cx3+cy3+cz3", "name": "s3", "description": ""}, "v2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(sqsum2*sd^2/S2^2,2)", "name": "v2", "description": ""}, "cx3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "tx*sx*random(1..9)", "name": "cx3", "description": ""}, "cz1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t1-cx1-cw1-cy1", "name": "cz1", "description": ""}, "tx": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(tw=0,1,0)", "name": "tx", "description": ""}}, "ungrouped_variables": ["cy3", "cy2", "cy1", "t3", "correct2", "correct1", "t1", "unb3", "unb2", "unb1", "s3", "tw", "s1", "s7", "tol", "cx1", "cx2", "cx3", "sqsum1", "sqsum3", "sqsum2", "cz2", "cz3", "cz1", "v1", "v2", "v3", "e1", "e3", "e2", "a", "c", "b", "tx", "cw1", "cw3", "cw2", "m", "wrong", "sx", "q", "p", "su2", "s2", "sd"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{e1}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{e1}"}, {"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{v1}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{v1}"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

$\\simplify[std]{A = {cw1} * W + {cx1} * X + {cy1} * Y + {cz1} * Z}$

\n

$\\operatorname{E}[A]=\\;\\;$[[0]]$\\;\\;\\;\\;\\operatorname{Var}(A)=\\;\\;\\;$[[1]]

\n

Input both as integers.

", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{e2-tol}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{e2+tol}"}, {"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{v2-tol}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{v2+tol}"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

$\\simplify[std]{B = (1 / {S2}) * ({cw2} * W + {cx2} * X + {cy2} * Y + {cz2} * Z)}$

\n

$\\operatorname{E}[B]=\\;\\;$[[0]]$\\;\\;\\;\\;\\operatorname{Var}(B)=\\;\\;\\;$[[1]]

\n

Input both to 2 decimal places.

", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{e3}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{e3}"}, {"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{v3}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{v3}"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

$\\simplify[std]{C = {cw3} * W + {cx3} * X + {cy3} * Y + {cz3} * Z}$

\n

$\\operatorname{E}[C]=\\;\\;$[[0]]$\\;\\;\\;\\;\\operatorname{Var}(C)=\\;\\;\\;$[[1]]

\n

Input both as integers.

", "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "checkbox", "choices": ["

$\\var{Correct1}$

", "

$\\var{Correct2}$

", "

$\\var{Wrong}$

"], "matrix": [1, 1, -1], "distractors": ["", "", ""], "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "scripts": {}, "maxMarks": 0, "type": "m_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 3, "marks": 0}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n \n \n

Which of the estimators $A,\\;\\;B$ or $C$ above are unbiased for $\\mu$? Select the correct choices.
You will lose a mark for selecting a wrong choice.

\n \n \n \n

[[0]]

\n \n ", "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["

$\\var{B}$

", "

$\\var{A}$

", "

$\\var{C}$

"], "matrix": [1, 0, 0], "distractors": ["", "", ""], "shuffleChoices": true, "scripts": {}, "maxMarks": 0, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 3, "marks": 0}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n \n \n

Which of the estimators $A,\\;\\;B$ or $C$ above is the most efficient?

\n \n \n \n

[[0]]

\n \n ", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{m^2}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{m^2}"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

Find $\\operatorname{E}[\\var{p}\\var{q}]=\\;\\;$[[0]]

", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

Suppose $W,\\;\\;X,\\;\\;Y,\\;\\;$ and $Z$ are i.i.d. variables with mean $\\mu=\\var{m}$, standard deviation $\\sigma=\\var{sd}$

\n

Find the expectation and variance of each of the following estimators of $\\mu$.

", "tags": ["IID", "MAS1604", "biased", "checked2015", "cr1", "efficient estimators", "estimators", "expectation", "i.i.d", "identical independent distributions", "iid", "independent identical distributions", "mean ", "random variables", "standard deviation", "statistics", "tested1", "unbiased", "unbiased estimators", "variance"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

13/07/2012:

\n

\n

Improved and made consistent the display in various content areas.

\n

Set new tolerance variable tol=0 for 2 dps numeric input questions.

\n

Added formula for $\\operatorname{Var}(aR+bS)$.

\n

Checked calculation.

\n

\n

1/08/2012:

\n

\n

Question appears to be working correctly.

\n

21/12/2012:

\n

Checked calculation, OK. Added tested1 tag.

\n

Checked rounding, OK. Added tag cr1.

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Given three linear combinations of four i.i.d. variables, find the expectation and variance of these estimators of the mean $\\mu$. Which are unbiased and efficient?

We use the result throughout these solutions that for independent random variables $R$ and $S$ then $\\operatorname{E}[aR+bS]=a \\operatorname{E}[R]+b\\operatorname{E}[S]$ and $\\operatorname{Var}(aR+bS)=a^2\\operatorname{Var}(R)+b^2\\operatorname{Var}(S)$.

\n

a)
\$\\begin{eqnarray*} \\operatorname{E}[A] &=& \\operatorname{E}[\\simplify[std]{{cw1} * W + {cx1} * X + {cy1} * Y + {cz1} * Z}]\\\\ &=& \\simplify[std]{{cw1} * {m} + {cx1} * {m} + {cy1} * {m} + {cz1} * {m}}\\\\ &=& \\var{e1}\\\\ \\\\ \\\\ \\operatorname{Var}(A) &=& \\operatorname{Var}[\\simplify[std]{{cw1} * W + {cx1} * X + {cy1} * Y + {cz1} * Z}]\\\\ &=& \\simplify[std]{ {cw1 ^ 2} * Var(W) + {cx1 ^ 2} * Var(X) + {cy1 ^ 2} * Var(Y) + {cz1 ^ 2} * Var(Z)}\\\\ &=& \\simplify[std]{{cw1 ^ 2} * {sd ^ 2} + {cx1 ^ 2} * {sd ^ 2} + {cy1 ^ 2} * {sd ^ 2} + {cz1 ^ 2} * {sd ^ 2} }\\\\ &=& \\var{v1} \\end{eqnarray*} \$

\n

b)
\$\\begin{eqnarray*} \\operatorname{E}[B] &=& \\frac{1}{\\var{S2}}\\left(\\operatorname{E}[\\simplify[std]{{cw2} * W + {cx2} * X + {cy2} * Y + {cz2} * Z}]\\right)\\\\ &=& \\frac{1}{\\var{S2}}\\left(\\simplify[std]{{cw2} * {m} + {cx2} * {m} + {cy2} * {m} + {cz2} * {m}}\\right)\\\\ &=& \\var{e2}\\\\ \\\\ \\\\ \\operatorname{Var}(B) &=& \\frac{1}{\\var{S2^2}}\\left(\\operatorname{Var}[\\simplify[std]{{cw2} * W + {cx2} * X + {cy2} * Y + {cz2} * Z}]\\right)\\\\ &=& \\frac{1}{\\var{S2^2}}\\left(\\simplify[std]{ {cw2 ^ 2} * Var(W) + {cx2 ^ 2} * Var(X) + {cy2 ^ 2} * Var(Y) + {cz2 ^ 2} * Var(Z)}\\right)\\\\ &=& \\frac{1}{\\var{S2^2}}\\left(\\simplify[std]{{cw2 ^ 2} * {sd ^ 2} + {cx2 ^ 2} * {sd ^ 2} + {cy2 ^ 2} * {sd ^ 2} + {cz2 ^ 2} * {sd ^ 2} }\\right)\\\\ &=& \\var{v2} \\end{eqnarray*} \$
c)
\$\\begin{eqnarray*} \\operatorname{E}[C] &=& \\operatorname{E}[\\simplify[std,collectNumbers]{{cw3} * W + {cx3} * X + {cy3} * Y + {cz3} * Z}]\\\\ &=& \\simplify[std,collectNumbers]{{cw3} * {m} + {cx3} * {m} + {cy3} * {m} + {cz3} * {m}}\\\\ &=& \\var{e3}\\\\ \\\\ \\\\ \\operatorname{Var}(C) &=& \\operatorname{Var}[\\simplify[std]{{cw3} * W + {cx3} * X + {cy3} * Y + {cz3} * Z}]\\\\ &=& \\simplify[std]{ {cw3 ^ 2} * Var(W) + {cx3 ^ 2} * Var(X) + {cy3 ^ 2} * Var(Y) + {cz3 ^ 2} * Var(Z)}\\\\ &=& \\simplify[std]{{cw3 ^ 2} * {sd ^ 2} + {cx3 ^ 2} * {sd ^ 2} + {cy3 ^ 2} * {sd ^ 2} + {cz3 ^ 2} * {sd ^ 2} }\\\\ &=& \\var{v3} \\end{eqnarray*} \$

\n

d)

\n

We see that $\\var{Correct1},\\;\\;\\var{Correct2}\\;\\;$ are unbiased estimators for $\\mu=\\var{m}$ as their expectations are $\\var{m}$.

\n

e)

\n

The most efficient estimator is $B$ as it has the smallest variance.

\n

f)
Since $\\var{p}$ and $\\var{q}$ are independent we have:

\n

$\\operatorname{E}[\\var{p}\\var{q}]=\\operatorname{E}[\\var{p}]\\operatorname{E}[\\var{q}] = \\var{m}\\times \\var{m} = \\var{m^2}$

"}, {"name": "Find expectation, variance and probability sample mean in range for normal distribution, , ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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": "0.0001", "description": "", "name": "tol"}, "zlowsam": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sqrt(sa)*zlow", "description": "", "name": "zlowsam"}, "pupsam": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(normalCDF(zupsam,0,1),6)", "description": "", "name": "pupsam"}, "tol1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "tol1"}, "sva": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(s^2/sa,2)", "description": "", "name": "sva"}, "pup": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(normalcdf(zup,0,1),6)", "description": "", "name": "pup"}, "plowsam": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(normalCDF(abs(zlowsam),0,1),6)", "description": "", "name": "plowsam"}, "zlow": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(lower-m)/s", "description": "", "name": "zlow"}, "sup": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.1..0.75#0.005)", "description": "", "name": "sup"}, "upper": {"templateType": "anything", "group": "Ungrouped variables", "definition": "m+sup*s", "description": "", "name": "upper"}, "lower": {"templateType": "anything", "group": "Ungrouped variables", "definition": "m-slow*s", "description": "", "name": "lower"}, "plow": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(normalcdf(abs(zlow),0,1),6)", "description": "", "name": "plow"}, "nationality": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random('English','Australian','African','American','Chinese','Mediterranean')", "description": "", "name": "nationality"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(100..300#10)", "description": "", "name": "m"}, "animals": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random('rabbits','goats','mice','cows','rats')", "description": "", "name": "animals"}, "sb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(sa=9,random(16,25,36),if(sa=16,random(25,36),36))", "description": "", "name": "sb"}, "probsam": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(pupsam+plowsam-1,4)", "description": "", "name": "probsam"}, "zup": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(upper-m)/s", "description": "", "name": "zup"}, "stuff": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random('sodium choride','fatty acid','potassium','protein','carbonic anhydrase','fibrinogen')", "description": "", "name": "stuff"}, "slow": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.1..0.75#0.005)", "description": "", "name": "slow"}, "zupsam": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sqrt(sa)*zup", "description": "", "name": "zupsam"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(15..30#5)", "description": "", "name": "s"}, "svb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(s^2/sb,2)", "description": "", "name": "svb"}, "prob": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(pup+plow-1,4)", "description": "", "name": "prob"}, "sa": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(s=25,random(9,16),random(9,16,25))", "description": "", "name": "sa"}}, "ungrouped_variables": ["upper", "zlowsam", "zlow", "m", "plowsam", "plow", "slow", "tol", "sup", "prob", "zup", "probsam", "zupsam", "nationality", "pup", "lower", "animals", "pupsam", "svb", "sva", "s", "stuff", "sb", "sa", "tol1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{prob+tol}", "minValue": "{prob-tol}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n

Find:

\n

$P(\\var{lower} \\lt X \\lt \\var{upper})=\\;\\;$[[0]]

\n

Correct to 4 decimal places.

\n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{m}", "minValue": "{m}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{sva+tol1}", "minValue": "{sva-tol1}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{m}", "minValue": "{m}", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{svb+tol1}", "minValue": "{svb-tol1}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "

Let $\\overline{X}$ be the random variable given by the sample mean.

\n

Find $\\operatorname{E}[ \\overline{X}]$ and $\\operatorname{Var}(\\overline{X})$ in the following cases:

\n

1) A sample of size $\\var{sa}$

\n

$\\operatorname{E}[ \\overline{X}]=\\;\\;$[[0]]$\\;\\;\\;\\operatorname{Var}(\\overline{X})=\\;\\;$[[1]]

\n

2) A sample of size $\\var{sb}$

\n

$\\operatorname{E}[ \\overline{X}]=\\;\\;$[[2]]$\\;\\;\\;\\operatorname{Var}(\\overline{X})=\\;\\;$[[3]]

\n

Enter the variances to 2 decimal places.

", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{probsam+tol}", "minValue": "{probsam-tol}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n \n \n

Assuming that $\\overline{X}$ also follows a normal distribution.

\n \n \n \n

Find $P(\\var{lower} \\lt \\overline{X} \\lt \\var{upper})$ in a sample of size $\\var{sa}$.

\n \n \n \n

$P(\\var{lower} \\lt \\overline{X} \\lt \\var{upper})=\\;\\;$[[0]]

\n \n \n \n

Enter the value correct to 4 decimal places.

\n \n \n ", "showCorrectAnswer": true, "marks": 0}], "statement": "

The total {stuff} content of the blood plasma of {nationality} {animals} ($X$, in mg/100ml) is known to follow a $N(\\var{m},\\var{s^2})$ distribution.

", "tags": ["checked2015", "cr1", "distribution of sample mean", "distributions", "MAS1604", "MAS8380", "MAS8401", "normal distribution", "Normal distribution", "Probability", "probability", "random variables", "sample", "sample distribution", "sample mean", "sc", "statistics", "tested1", "z scores"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

13/07/2012:

\n

\n

Cannot check calculations as yet as cannot access stats extension.

\n

Set new tolerance variable tol=0.0001 for numeric entries to 4 dps.

\n

Set new tolerance variable tol=0 for numeric entries to 2 dps.

\n

21/12/2012:

\n

Checked calculations against standard tables, OK. Added tested1 tag.

\n

Corrected a typos and improved display in Advice.

\n

Checked rounding, OK. Added tag cr1.

\n

This has scenarios - could be extended. Added sc tag.

\n

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Normal distribution $X \\sim N(\\mu,\\sigma^2)$ given. Find $P(a \\lt X \\lt b)$. Find expectation, variance, $P(c \\lt \\overline{X} \\lt d)$ for sample mean $\\overline{X}$.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

a) Converting to $Z$ scores in $N(0,1)$ we have for $X \\sim N(\\var{m},\\var{s^2})$

\n

\$\\begin{eqnarray*} P(\\var{lower} \\lt X \\lt \\var{upper})&=&P(X \\lt \\var{upper})-P(X \\lt \\var{lower})\\\\ &=&P\\left(Z \\lt \\frac{\\var{upper}-\\var{m}}{\\var{s}}\\right)-P\\left(Z \\lt \\frac{\\var{lower}-\\var{m}}{\\var{s}}\\right)\\\\ &=&P(Z \\lt \\var{zup})-P( Z \\lt \\var{zlow})\\\\ &=&\\var{pup}-\\var{1-plow}\\\\ &=&\\var{prob} \\end{eqnarray*} \$ to 4 decimal places.

\n

Here the probabilities could have been looked up from normal CDF tables. Alternatively we can simply do the whole calculation in

\n

R by typing $\\operatorname{pnorm}(\\var{upper},\\var{m},\\var{s})-\\operatorname{pnorm}(\\var{lower},\\var{m},\\var{s})$.

\n

b)
This part reminds you that for the sample mean for samples of size $n$ from a normal distribution $N(\\mu,\\sigma^2)$ has a normal distribution $\\displaystyle N\\left(\\mu,\\frac{\\sigma^2}{n}\\right)$.

\n

Hence for a sample size $\\var{sa}$:

\n

$\\displaystyle \\operatorname{E}[\\overline{X}] = \\mu=\\var{m}$ and $\\displaystyle \\operatorname{Var}(\\overline{X}) = \\frac{\\sigma^2}{n}=\\frac{\\var{s^2}}{\\var{sa}}=\\var{sva}$

\n

Hence for a sample size $\\var{sb}$:

\n

$\\operatorname{E}[\\overline{X}] = \\mu=\\var{m}$ and $\\displaystyle \\operatorname{Var}(\\overline{X}) = \\frac{\\sigma^2}{n}=\\frac{\\var{s^2}}{\\var{sb}}=\\var{svb}$

\n

c)
Since the sample size is $\\var{sa}$ we are dealing with the normal distribution $N(\\var{m},\\simplify[std]{{s^2}/{sa}})$.

\n

Converting to $Z$ scores in $N(0,1)$ we have for $\\overline{X} \\sim N(\\var{m},\\simplify[std]{{s^2}/{sa}})$

\n

\$\\begin{eqnarray*} P(\\var{lower} \\lt \\overline{X} \\lt \\var{upper})&=&P(\\overline{X} \\lt \\var{upper})-P(\\overline{X}\\lt \\var{lower})\\\\ &=&P\\left(Z \\lt \\frac{\\sqrt{\\var{sa}}(\\var{upper}-\\var{m})}{\\var{s}}\\right)-P\\left(Z \\lt \\frac{\\sqrt{\\var{sa}}(\\var{lower}-\\var{m})}{\\var{s}}\\right)\\\\ &=&P(Z \\lt \\var{zupsam})-P( Z \\lt \\var{zlowsam})\\\\ &=&\\var{pupsam}-\\var{1-plowsam}\\\\ &=&\\var{probsam} \\end{eqnarray*} \$ to 4 decimal places.

\n

Here the probabilities could have been looked up from normal CDF tables. Alternatively we can simply do the whole calculation in

\n

R by typing $\\operatorname{pnorm}(\\var{upper},\\var{m},\\frac{\\var{s}}{\\sqrt{\\var{sa}}})-\\operatorname{pnorm}(\\var{lower},\\var{m},\\frac{\\var{s}}{\\sqrt{\\var{sa}}})$ i.e. $\\operatorname{pnorm}(\\var{upper},\\var{m},\\var{s/sqrt(sa)})-\\operatorname{pnorm}(\\var{lower},\\var{m},\\var{s/sqrt(sa)})$

"}], "pickQuestions": 0}], "variable_groups": [], "contributors": [{"name": "Etain Kiely", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1608/"}], "extensions": ["stats"], "custom_part_types": [], "resources": []}