// Numbas version: exam_results_page_options {"name": "Find mean and standard deviation of differences between samples, , , , ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"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 $y_i-x_i,\\;i=1,\\dots 5$.
", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["meandiff", "performing", "attempt", "r1", "objects", "mu2", "object", "sig1", "thismany", "stdiff", "sig2", "something", "r2", "mu1", "d"], "statement": "{Something} for $\\var{thismany}$ {objects} {performing} were:
\n{capitalise(object)} | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
First {attempt} | \n$\\var{r1[0]}$ | \n$\\var{r1[1]}$ | \n$\\var{r1[2]}$ | \n$\\var{r1[3]}$ | \n$\\var{r1[4]}$ | \n
Second {attempt} | \n$\\var{r2[0]}$ | \n$\\var{r2[1]}$ | \n$\\var{r2[2]}$ | \n$\\var{r2[3]}$ | \n$\\var{r2[4]}$ | \n
The table of differences is given by:
\n{capitalise(object)} | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
First {attempt} | \n$\\var{r1[0]}$ | \n$\\var{r1[1]}$ | \n$\\var{r1[2]}$ | \n$\\var{r1[3]}$ | \n$\\var{r1[4]}$ | \n
Second {attempt} | \n$\\var{r2[0]}$ | \n$\\var{r2[1]}$ | \n$\\var{r2[2]}$ | \n$\\var{r2[3]}$ | \n$\\var{r2[4]}$ | \n
Differences | \n$\\var{d[0]}$ | \n$\\var{d[1]}$ | \n$\\var{d[2]}$ | \n$\\var{d[3]}$ | \n$\\var{d[4]}$ | \n
The mean of the differences is $\\var{meandiff}$.
\nThe 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}
Hence the standard deviation is $\\sqrt{V}=\\var{stdiff}$ to 3 decimal places.
", "preamble": {"css": "", "js": ""}, "parts": [{"type": "gapfill", "scripts": {}, "variableReplacements": [], "marks": 0, "gaps": [{"type": "numberentry", "allowFractions": false, "marks": "2", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "correctAnswerFraction": false, "scripts": {}, "variableReplacements": [], "maxValue": "{meandiff}", "notationStyles": ["plain", "en", "si-en"], "minValue": "{meandiff}", "correctAnswerStyle": "plain", "showCorrectAnswer": true}, {"type": "numberentry", "allowFractions": false, "marks": "2", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "correctAnswerFraction": false, "scripts": {}, "variableReplacements": [], "maxValue": "{stdiff}", "notationStyles": ["plain", "en", "si-en"], "minValue": "{stdiff}", "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "variableReplacementStrategy": "originalfirst", "prompt": "Find the mean and standard deviation of the difference between first and second {attempt}s.
\nCalculate differences for second {attempt} – first {attempt}.
\nMean of difference = [[0]] (input as an exact decimal)
\nStandard deviation of difference = [[1]] (input to 3 decimal places)
", "showCorrectAnswer": true, "showFeedbackIcon": true}], "type": "question", "contributors": [{"name": "Nigel Atkins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/275/"}]}]}], "contributors": [{"name": "Nigel Atkins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/275/"}]}