// Numbas version: finer_feedback_settings
{"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": [{"rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"mu2": {"group": "Ungrouped variables", "name": "mu2", "description": "", "templateType": "anything", "definition": "mu1+random(1..3#0.1)"}, "object": {"group": "Ungrouped variables", "name": "object", "description": "", "templateType": "anything", "definition": "'person'"}, "mu1": {"group": "Ungrouped variables", "name": "mu1", "description": "", "templateType": "anything", "definition": "random(16..20#0.5)"}, "meandiff": {"group": "Ungrouped variables", "name": "meandiff", "description": "", "templateType": "anything", "definition": "mean(d)"}, "sig1": {"group": "Ungrouped variables", "name": "sig1", "description": "", "templateType": "anything", "definition": "random(1.5..2.5#0.5)"}, "performing": {"group": "Ungrouped variables", "name": "performing", "description": "", "templateType": "anything", "definition": " 'working at $\\\\var{100}$ watts on an exercise machine' "}, "thismany": {"group": "Ungrouped variables", "name": "thismany", "description": "", "templateType": "anything", "definition": "5"}, "stdiff": {"group": "Ungrouped variables", "name": "stdiff", "description": "", "templateType": "anything", "definition": "precround(stdev(d,true),3)"}, "r1": {"group": "Ungrouped variables", "name": "r1", "description": "", "templateType": "anything", "definition": "repeat(round(normalsample(mu1,sig1)),5)"}, "something": {"group": "Ungrouped variables", "name": "something", "description": "", "templateType": "anything", "definition": "'Oxygen uptake values (mL/kg.min)'"}, "objects": {"group": "Ungrouped variables", "name": "objects", "description": "", "templateType": "anything", "definition": "'people'"}, "attempt": {"group": "Ungrouped variables", "name": "attempt", "description": "", "templateType": "anything", "definition": "'attempt'"}, "d": {"group": "Ungrouped variables", "name": "d", "description": "", "templateType": "anything", "definition": "list(vector(r2)-vector(r1))"}, "r2": {"group": "Ungrouped variables", "name": "r2", "description": "", "templateType": "anything", "definition": "repeat(round(normalsample(mu2,sig2)),5)"}, "sig2": {"group": "Ungrouped variables", "name": "sig2", "description": "", "templateType": "anything", "definition": "sig1+random(-0.5..-0.2#0.1)"}}, "extensions": ["stats"], "name": "Find mean and standard deviation of differences between samples", "preamble": {"css": "", "js": ""}, "parts": [{"variableReplacements": [], "showFeedbackIcon": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": 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>", "type": "gapfill", "gaps": [{"variableReplacements": [], "showFeedbackIcon": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReduced": false, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "unitTests": [], "maxValue": "{meandiff}", "correctAnswerFraction": false, "showCorrectAnswer": true, "minValue": "{meandiff}", "marks": 0.5, "mustBeReducedPC": 0}, {"variableReplacements": [], "showFeedbackIcon": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReduced": false, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "unitTests": [], "maxValue": "{stdiff}", "correctAnswerFraction": false, "showCorrectAnswer": true, "minValue": "{stdiff}", "marks": 0.5, "mustBeReducedPC": 0}], "scripts": {}, "variableReplacementStrategy": "originalfirst", "unitTests": [], "sortAnswers": false, "showCorrectAnswer": true, "marks": 0}], "metadata": {"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>", "licence": "Creative Commons Attribution 4.0 International"}, "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>", "tags": [], "ungrouped_variables": ["meandiff", "performing", "attempt", "r1", "objects", "mu2", "object", "sig1", "thismany", "stdiff", "sig2", "something", "r2", "mu1", "d"], "functions": {}, "variable_groups": [], "statement": "<p>{Something} for $\\var{thismany}$ {objects} {performing} were sampled as follows:</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<th>{capitalise(object)}</th>\n<th>1</th>\n<th>2</th>\n<th>3</th>\n<th>4</th>\n<th>5</th>\n</tr>\n<tr>\n<th align=\"left\">First {attempt}</th>\n<td>{r1[0]}</td>\n<td>{r1[1]}</td>\n<td>{r1[2]}</td>\n<td>{r1[3]}</td>\n<td>{r1[4]}</td>\n</tr>\n<tr>\n<th align=\"left\">Second {attempt}</th>\n<td>{r2[0]}</td>\n<td>{r2[1]}</td>\n<td>{r2[2]}</td>\n<td>{r2[3]}</td>\n<td>{r2[4]}</td>\n</tr>\n</tbody>\n</table>", "type": "question", "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Mario Orsi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/427/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Mario Orsi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/427/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}