// Numbas version: finer_feedback_settings
{"name": "Paired sample t-test.", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Paired sample t-test.", "tags": ["average", "data analysis", "differences", "elementary statistics", "hypothesis testing", "mean", "mean of differences", "paired t-test", "standard deviation", "standard deviation of differences", "statistics", "stats", "t-test", "variance"], "metadata": {"description": "<p>Paired t-test to see if there is a difference between times take in a task.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n  <p>The following data was obtained from $12$ individuals. The observations consist of the time taken to complete a dexterity task using their left and right hands.</p>\n  <table border=\"1\">\n  <tbody>\n  <tr><th>{object}</th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th><th>F</th><th>G</th><th>H</th><th>I</th><th>J</th><th>K</th><th>L</th></tr>\n  <tr><th align=\"left\">Right</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  <td>$\\var{r1[5]}$</td>\n  <td>$\\var{r1[6]}$</td>\n  <td>$\\var{r1[7]}$</td>\n  <td>$\\var{r1[8]}$</td>\n  <td>$\\var{r1[9]}$</td>\n  <td>$\\var{r1[10]}$</td>\n  <td>$\\var{r1[11]}$</td>\n  </tr>\n  <tr><th align=\"left\">Left</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  <td>$\\var{r2[5]}$</td>\n  <td>$\\var{r2[6]}$</td>\n  <td>$\\var{r2[7]}$</td>\n  <td>$\\var{r2[8]}$</td>\n  <td>$\\var{r2[9]}$</td>\n  <td>$\\var{r2[10]}$</td>\n  <td>$\\var{r2[11]}$</td>\n  </tr>\n  </tbody>\n  </table>\n  <p>Carry out by hand a paired t-test to test whether there is evidence of a difference in the average times for the left and right hands.</p>\n  ", "advice": "\n  <p>The table of differences is given by:</p>\n    <table border=\"1\">\n    <tbody>\n    <tr><th>{object}</th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th><th>F</th><th>G</th><th>H</th><th>I</th><th>J</th><th>K</th><th>L</th></tr>\n    <tr><th align=\"left\">Right</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    <td>$\\var{r1[5]}$</td>\n    <td>$\\var{r1[6]}$</td>\n    <td>$\\var{r1[7]}$</td>\n    <td>$\\var{r1[8]}$</td>\n    <td>$\\var{r1[9]}$</td>\n    <td>$\\var{r1[10]}$</td>\n    <td>$\\var{r1[11]}$</td>\n    </tr>\n    <tr><th align=\"left\">Left</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    <td>$\\var{r2[5]}$</td>\n    <td>$\\var{r2[6]}$</td>\n    <td>$\\var{r2[7]}$</td>\n    <td>$\\var{r2[8]}$</td>\n    <td>$\\var{r2[9]}$</td>\n    <td>$\\var{r2[10]}$</td>\n    <td>$\\var{r2[11]}$</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    <td>$\\var{d[5]}$</td>\n    <td>$\\var{d[6]}$</td>\n    <td>$\\var{d[7]}$</td>\n    <td>$\\var{d[8]}$</td>\n    <td>$\\var{d[9]}$</td>\n    <td>$\\var{d[10]}$</td>\n    <td>$\\var{d[11]}$</td>\n    </tr>\n    </tbody>\n    </table>\n    <p>We test the following hypothesis:</p>\n    <p style=\"text-align: center;\">$H_0:\\;\\mu_d=0$ versus $H_1:\\;\\mu_d\\neq 0$</p>\n    <p>$n=\\var{n}$ and the mean of the differences is $\\overline{d}=\\var{meandiff}$.</p>\n    <p>The variance $V$ of the differences is calculated to be $\\var{pstdev(d)^2}$<br/><br/>Hence we have the standard deviation $s_d= \\sqrt{V}=\\var{stdiff}$ to 3 decimal places.</p>\n    <p>The paired t-statistic is given by:</p>\n    <p>\\[\\begin{eqnarray*} T&amp;=&amp;\\frac{\\overline{d}-\\mu_d}{\\frac{s_d}{\\sqrt{n}}}\\\\&amp;=&amp;\\frac{\\var{meandiff}-0}{\\frac{\\var{stdiff}}{\\sqrt{\\var{n}}}}\\\\&amp;=&amp;\\var{tvalue}\\end{eqnarray*}\\]</p>\n    <p>(Using the&nbsp;null hypothesis that the means are the same i.e. $\\mu_d=0$.)</p>\n    <p>Hence our test statistic &nbsp;$|T|=\\var{tvalue}$.</p>\n    <p>Looking up this value on the T-distribution table for $t_{11}$</p>\n    <p>\\[\\begin{array}{r|rrrrr}&amp;0.20&amp;0.10&amp;0.05&amp;0.01&amp;0.001\\\\\\hline11&amp;1.363&amp;1.796&amp;2.201&amp;3.106&amp;4.437\\end{array}\\]</p>\n    <p>We see that the t-statistic {msg[t]} and the table tells us that the $p$ value {pmsg[t]}.</p>\n    <p>Hence we conclude that we {cmsg[t]} the null hypothesis. There is {cmsg1[t]} evidence of a difference between the average scores of the two groups.</p>\n    <p>&nbsp;</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"pmsg": {"name": "pmsg", "group": "Ungrouped variables", "definition": "[' is less than $0.001$',' lies between $0.001$ and $0.01$',' lies between $0.01$ and $0.05$',' lies between $0.05$ and $0.10$',' is greater than $0.10$']", "description": "", "templateType": "anything", "can_override": false}, "msg": {"name": "msg", "group": "Ungrouped variables", "definition": "['is greater than $\\\\var{t999}$','lies between $\\\\var{t99}$ and $\\\\var{t999}$','lies between $\\\\var{t95}$ and $\\\\var{t99}$','lies between $\\\\var{t90}$ and $\\\\var{t95}$','is less than $\\\\var{t90}$']", "description": "", "templateType": "anything", "can_override": false}, 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"anything", "can_override": false}, "stdiff": {"name": "stdiff", "group": "Ungrouped variables", "definition": "precround(pstdev(d),3)", "description": "", "templateType": "anything", "can_override": false}, "sig2": {"name": "sig2", "group": "Ungrouped variables", "definition": "random(2..4#0.2)", "description": "", "templateType": "anything", "can_override": false}, "t99": {"name": "t99", "group": "Ungrouped variables", "definition": "3.106", "description": "", "templateType": "anything", "can_override": false}, "t95": {"name": "t95", "group": "Ungrouped variables", "definition": "2.201", "description": "", "templateType": "anything", "can_override": false}, "t90": {"name": "t90", "group": "Ungrouped variables", "definition": "1.796", "description": "", "templateType": "anything", "can_override": false}, "cmsg": {"name": "cmsg", "group": "Ungrouped variables", "definition": "['do reject','do reject','do not reject','do not reject','do not reject']", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "list(vector(r2)-vector(r1))", "description": "", "templateType": "anything", "can_override": false}, "mu1": {"name": "mu1", "group": "Ungrouped variables", "definition": "random(18..25#0.5)", "description": "", "templateType": "anything", "can_override": false}, "r1": {"name": "r1", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu1,sig1)),12)", "description": "", "templateType": "anything", "can_override": false}, "r2": {"name": "r2", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu2,sig2)),12)", "description": "", "templateType": "anything", "can_override": false}, "mu2": {"name": "mu2", "group": "Ungrouped variables", "definition": "mu1+random(2..4#0.1)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "12", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "switch(v[0]=1,0,v[1]=1,1,v[2]=1,2,v[3]=1,3,4)", "description": "", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "if(tvalue>=t999,[1,0,0,0,0],if(tvalue>=t99,[0,1,0,0,0],if(tvalue>=t95,[0,0,1,0,0],if(tvalue>=t90,[0,0,0,1,0],[0,0,0,0,1]))))", "description": "", "templateType": "anything", "can_override": false}, "pvalue": {"name": "pvalue", "group": "Ungrouped variables", "definition": "precround(ttest(0,d,2),3)", "description": "", "templateType": "anything", "can_override": false}, "attempt": {"name": "attempt", "group": "Ungrouped variables", "definition": "", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["pmsg", "msg", "cmsg1", "t999", "meandiff", "object", "sig1", "tvalue", "stdiff", "sig2", "t99", "t95", "t90", "cmsg", "d", "mu1", "r1", "r2", "mu2", "n", "t", "v", "pvalue", "attempt"], "variable_groups": [], "functions": {"pstdev": {"parameters": [["l", "list"]], "type": "number", "language": "jme", "definition": "sqrt(abs(l)/(abs(l)-1))*stdev(l)"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n      <p>Find the mean and standard deviations of the difference between left and right {attempt}s.</p>\n        <p>Calculate differences for <strong>left</strong> {attempt} times &ndash; <strong>right</strong> {attempt} times. Make sure you take the differences this way round.</p>\n        <p>Mean of difference = [[0]] (input&nbsp; to 3 decimal places&nbsp;)</p>\n        <p>Standard deviation of difference = [[1]] (input to 3 decimal places)</p>\n        <p>Now find the t-test statistic $T$ using the values you have just calculated and &nbsp;input the absolute value $|T|$ here: [[2]] (3 decimal places).&nbsp;</p>\n        <p>&nbsp;</p>\n      ", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.5, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{meandiff}", "maxValue": "{meandiff}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 0.5, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{stdiff}", "maxValue": "{stdiff}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{tvalue}", "maxValue": "{tvalue}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Give the value of the t-statistic you have found, choose the range for the $p$ value by looking up the t-statistic tables:</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>$p$ less than $0.1 \\%$</p>", "<p>$p$ lies between  $0.1\\%$ and $1 \\%$</p>", "<p>$p$ lies between  $1 \\%$ and $5\\%$</p>", "<p>$p$ lies between  $5 \\%$ and $10\\%$</p>", "<p>$p$ is greater than $10\\%$</p>"], "matrix": "v"}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Given the $p$-value and the range you have found, what is the strength of evidence against the null hypothesis that there is no difference in the average times for the left and right hands?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayColumns": 0, "showCellAnswerState": true, "choices": ["Very strong", "Strong", "Moderate", "Slight", "None"], "matrix": "v"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}], "resources": []}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}]}