// 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": [{"name": "Find mean and standard deviation of differences between samples", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>An experiment is performed twice, each with $5$ outcomes</p>\n<p>$x_i,\\;y_i,\\;i=1,\\dots 5$&nbsp;. Find mean and s.d. of their differences $y_i-x_i,\\;i=1,\\dots 5$.</p>"}, "advice": "<p>The table of differences is given by:</p>\n<table border=\"1\">\n<tbody>\n<tr><th>{capitalise(object)}</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr>\n<tr><th align=\"left\">First {attempt}</th>\n<td>$\\var{r1[0]}$</td>\n<td>$\\var{r1[1]}$</td>\n<td>$\\var{r1[2]}$</td>\n<td>$\\var{r1[3]}$</td>\n<td>$\\var{r1[4]}$</td>\n</tr>\n<tr><th align=\"left\">Second {attempt}</th>\n<td>$\\var{r2[0]}$</td>\n<td>$\\var{r2[1]}$</td>\n<td>$\\var{r2[2]}$</td>\n<td>$\\var{r2[3]}$</td>\n<td>$\\var{r2[4]}$</td>\n</tr>\n<tr><th align=\"left\">Differences</th>\n<td>$\\var{d[0]}$</td>\n<td>$\\var{d[1]}$</td>\n<td>$\\var{d[2]}$</td>\n<td>$\\var{d[3]}$</td>\n<td>$\\var{d[4]}$</td>\n</tr>\n</tbody>\n</table>\n<p>The mean of the differences is $\\var{meandiff}$.</p>\n<p>The variance $V$ of the differences is</p>\n<p>\\begin{align} <br/>V &amp;= \\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) \\\\ <br/>&amp;= \\var{variance(d,true)} <br/>\\end{align}</p>\n<p>Hence the standard deviation is $\\sqrt{V}=\\var{stdiff}$ to 3 decimal places.</p>", "extensions": ["stats"], "type": "question", "statement": "<p>{Something} for $\\var{thismany}$ {objects} {performing} were:</p>\n<table border=\"1\">\n<tbody>\n<tr><th>{capitalise(object)}</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr>\n<tr><th align=\"left\">First {attempt}</th>\n<td>$\\var{r1[0]}$</td>\n<td>$\\var{r1[1]}$</td>\n<td>$\\var{r1[2]}$</td>\n<td>$\\var{r1[3]}$</td>\n<td>$\\var{r1[4]}$</td>\n</tr>\n<tr><th align=\"left\">Second {attempt}</th>\n<td>$\\var{r2[0]}$</td>\n<td>$\\var{r2[1]}$</td>\n<td>$\\var{r2[2]}$</td>\n<td>$\\var{r2[3]}$</td>\n<td>$\\var{r2[4]}$</td>\n</tr>\n</tbody>\n</table>", "variable_groups": [], "functions": {}, "variables": {"attempt": {"name": "attempt", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "'attempt'"}, "sig2": {"name": "sig2", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "sig1+random(-0.5..-0.2#0.1)"}, "mu2": {"name": "mu2", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "mu1+random(1..3#0.1)"}, "performing": {"name": "performing", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": " 'working at $\\\\var{100}$ watts on an exercise machine' "}, "object": {"name": "object", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "'person'"}, "thismany": {"name": "thismany", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "5"}, "r2": {"name": "r2", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu2,sig2)),5)"}, "objects": {"name": "objects", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "'people'"}, "mu1": {"name": "mu1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(16..20#0.5)"}, "d": {"name": "d", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "list(vector(r2)-vector(r1))"}, "r1": {"name": "r1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu1,sig1)),5)"}, "meandiff": {"name": "meandiff", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "mean(d)"}, "sig1": {"name": "sig1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(1.5..2.5#0.5)"}, "something": {"name": "something", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "'Oxygen uptake values (mL/kg.min)'"}, "stdiff": {"name": "stdiff", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "precround(stdev(d,true),3)"}}, "tags": ["checked2015", "cr1", "data analysis", "differences", "elementary statistics", "mean", "mean of differences", "standard deviation", "standard deviation of differences", "statistics", "stats", "tested1", "variance"], "parts": [{"variableReplacements": [], "type": "gapfill", "gaps": [{"variableReplacements": [], "minValue": "{meandiff}", "type": "numberentry", "maxValue": "{meandiff}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "mustBeReducedPC": 0, "unitTests": [], "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": "", "marks": 0.5}, {"variableReplacements": [], "minValue": "{stdiff}", "type": "numberentry", "maxValue": "{stdiff}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "mustBeReducedPC": 0, "unitTests": [], "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": "", "marks": 0.5}], "variableReplacementStrategy": "originalfirst", "unitTests": [], "sortAnswers": false, "showFeedbackIcon": true, "prompt": "<p>Find the mean and standard deviation of the difference between first and second {attempt}s.</p>\n<p>Calculate differences for second {attempt} &ndash; first {attempt}.</p>\n<p>Mean of difference = [[0]] (input as an exact decimal)</p>\n<p>Standard deviation of difference = [[1]] (input to 3 decimal places)</p>", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": "", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"css": "", "js": ""}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "ungrouped_variables": ["meandiff", "performing", "attempt", "r1", "objects", "mu2", "object", "sig1", "thismany", "stdiff", "sig2", "something", "r2", "mu1", "d"], "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/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "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/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}