// Numbas version: finer_feedback_settings
{"name": "A comparison of the means of two groups (2).", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {"pstdev": {"definition": "stdev(l)*sqrt(abs(l)/(abs(l)-1))", "type": "number", "parameters": [["l", "list"]], "language": "jme"}}, "ungrouped_variables": ["attempt", "d", "meandiff", "mu1", "mu2", "object", "objects", "r1", "r2", "sig1", "sig2", "stdiff", "t95", "thismany", "tcalc", "tupper", "tlower", "mdupper", "mdlower", "stdupper", "stdlower", "outcome", "sed", "test"], "name": "A comparison of the means of two groups (2).", "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"], "advice": "<p>First, recognise that this is a paired t-test: paired because twins are genetically identical.</p>\n<p>So calculate the differences (placebo&nbsp;$-$ drug) between those two observations:</p>\n<p style=\"padding-left: 30px;\">{d}</p>\n<p><strong>Note:</strong> If you have a graphical calculator, you can enter the two sets of&nbsp;observations into two lists (say L1 and L2), then in the heading (title/label) row for L3 type \"=L1 $-$ L2\" to get the calculator to work out the differences.</p>\n<p>Then calculate the mean difference:</p>\n<p style=\"padding-left: 30px;\">{meandiff}</p>\n<p>Then the standard deviation (sd) of the differences:</p>\n<p style=\"padding-left: 30px;\">{stdiff}</p>\n<p><strong>Note:</strong> You should use your calculator to get both the mean and standard deviation.</p>\n<p>The standard error (se) is calculated as:</p>\n<p>\\[ se = \\frac{ sd }{ \\sqrt{ n }} \\]</p>\n<p>It is:</p>\n<p style=\"padding-left: 30px;\">{sed}</p>\n<p>Finally, calculate the test statisitic, t-calc, as the mean difference divided by the standard error:</p>\n<p><span>\\[ t-calc&nbsp;= \\frac{ mean difference&nbsp;}{ se} \\]</span></p>\n<p style=\"padding-left: 30px;\"><span>t-calc = {tcalc}</span></p>\n<p><span>There are 6&nbsp;subjects, so 5 degrees of freedom, and the test is two-tailed, so the t-table = 2.571.</span></p>\n<p><span>If t-calc is greater than 2.571, the null is rejected. As this is a two-tailed test, the sign of t-calc (positive or negative) is ignored.</span></p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>Find the mean and standard deviations of the difference between the placebo&nbsp;and drug&nbsp;responses</p>\n<p>Calculate differences between placebo&nbsp;and&nbsp;drug&nbsp;responses.</p>\n<p>Mean of difference = [[0]] (input&nbsp; to 3&nbsp;decimal places&nbsp;)</p>\n<p>Standard deviation of difference = [[1]] (input to 3&nbsp;decimal places)</p>\n<p>Now find the paired t-test statistic using the values you have just calculated =[[2]] (input 3&nbsp;decimal places)</p>\n<p>Is the null hypothesis rejected?&nbsp;[[3]] Enter 0 for \"No\" and 1 for \"Yes\".</p>", "marks": 0, "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "scripts": {}, "precision": "3", "maxValue": "{mdupper}", "minValue": "{mdlower}", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "marks": "1", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "scripts": {}, "precision": "3", "maxValue": "{stdupper}", "minValue": "{stdlower}", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": "50", "marks": "1", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "scripts": {}, "precision": "3", "maxValue": "{tupper}", "minValue": "{tlower}", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "scripts": {}, "maxValue": "{outcome}", "minValue": "{outcome}", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>A medical researcher is concerned that a new drug may be affecting a patient&rsquo;s ability to concentrate. She finds six (6) sets of identical twins, randomly gives one a placebo pill (a medically inert and inactive substance) and the other the drug, then measures their reaction times.&nbsp;The data is in the table&nbsp;below:</p>\n<table>\n<tbody>\n<tr>\n<td><strong>{object}</strong></td>\n<td style=\"text-align: center;\"><strong>A</strong></td>\n<td style=\"text-align: center;\"><strong>B</strong></td>\n<td style=\"text-align: center;\"><strong>C</strong></td>\n<td style=\"text-align: center;\"><strong>D</strong></td>\n<td style=\"text-align: center;\"><strong>E</strong></td>\n<td style=\"text-align: center;\"><strong>F</strong></td>\n</tr>\n<tr>\n<td><strong>Placebo</strong></td>\n<td style=\"text-align: center;\"><span>$\\var{r1[0]}$</span></td>\n<td style=\"text-align: center;\"><span>$\\var{r1[1]}$</span></td>\n<td style=\"text-align: center;\"><span>$\\var{r1[2]}$</span></td>\n<td style=\"text-align: center;\"><span>$\\var{r1[3]}$</span></td>\n<td style=\"text-align: center;\"><span>$\\var{r1[4]}$</span></td>\n<td style=\"text-align: center;\"><span>$\\var{r1[5]}$</span></td>\n</tr>\n<tr>\n<td><strong>Drug</strong></td>\n<td style=\"text-align: center;\"><span>$\\var{r2[0]}$</span></td>\n<td style=\"text-align: center;\"><span>$\\var{r2[1]}$</span></td>\n<td style=\"text-align: center;\"><span>$\\var{r2[2]}$</span></td>\n<td style=\"text-align: center;\"><span>$\\var{r2[3]}$</span></td>\n<td style=\"text-align: center;\"><span>$\\var{r2[4]}$</span></td>\n<td style=\"text-align: center;\"><span>$\\var{r2[5]}$</span></td>\n</tr>\n</tbody>\n</table>\n<p><strong>Is there a difference in response time between the placebo and the drug?</strong></p>\n<p>In other words, test the following null hypothesis:</p>\n<p style=\"padding-left: 30px;\">H0: mean time placebo&nbsp;= mean time drug.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"tlower": {"definition": "tcalc-0.025", "templateType": "anything", "group": "Ungrouped variables", "name": "tlower", "description": ""}, "sed": {"definition": "precround(stdiff/sqrt(thismany),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "sed", "description": ""}, "t95": {"definition": "2.571", "templateType": "anything", "group": "Ungrouped variables", "name": "t95", "description": ""}, "mdupper": {"definition": "meandiff+0.025", "templateType": "anything", "group": "Ungrouped variables", "name": "mdupper", "description": ""}, "stdupper": {"definition": "stdiff+0.025", "templateType": "anything", "group": "Ungrouped variables", "name": "stdupper", "description": ""}, "test": {"definition": "stdev(d)*sqrt(thismany/(thismany-1))", "templateType": "anything", "group": "Ungrouped variables", "name": "test", "description": ""}, "meandiff": {"definition": "precround(mean(d),6)", "templateType": "anything", "group": "Ungrouped variables", "name": "meandiff", "description": ""}, "object": {"definition": "'Twin set'", "templateType": "anything", "group": "Ungrouped variables", "name": "object", "description": ""}, "tcalc": {"definition": "precround(meandiff/(stdiff/sqrt(thismany)),6)", "templateType": "anything", "group": "Ungrouped variables", "name": "tcalc", "description": ""}, "sig1": {"definition": "random(0.025..0.05)", "templateType": "anything", "group": "Ungrouped variables", "name": "sig1", "description": ""}, "thismany": {"definition": "6", "templateType": "anything", "group": "Ungrouped variables", "name": "thismany", "description": ""}, "stdiff": {"definition": "precround(pstdev(d),6)", "templateType": "anything", "group": "Ungrouped variables", "name": "stdiff", "description": ""}, "sig2": {"definition": "random(0.025..0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "sig2", "description": ""}, "mdlower": {"definition": "meandiff-0.025", "templateType": "anything", "group": "Ungrouped variables", "name": "mdlower", "description": ""}, "attempt": {"definition": "'hand'", "templateType": "anything", "group": "Ungrouped variables", "name": "attempt", "description": ""}, "stdlower": {"definition": "stdiff-0.025", "templateType": "anything", "group": "Ungrouped variables", "name": "stdlower", "description": ""}, "d": {"definition": "list(vector(r1)-vector(r2))", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "mu1": {"definition": "random(0.2..0.3)", "templateType": "anything", "group": "Ungrouped variables", "name": "mu1", "description": ""}, "r1": {"definition": "repeat(min(precround(normalsample(mu1,sig1),2),25),6)", "templateType": "anything", "group": "Ungrouped variables", "name": "r1", "description": ""}, "r2": {"definition": "repeat(min(precround(normalsample(mu2,sig2),2),25),6)", "templateType": "anything", "group": "Ungrouped variables", "name": "r2", "description": ""}, "mu2": {"definition": "mu1+random(0.05..0.10)", "templateType": "anything", "group": "Ungrouped variables", "name": "mu2", "description": ""}, "objects": {"definition": "'twins'", "templateType": "anything", "group": "Ungrouped variables", "name": "objects", "description": ""}, "tupper": {"definition": "tcalc+0.025", "templateType": "anything", "group": "Ungrouped variables", "name": "tupper", "description": ""}, "outcome": {"definition": "if((abs(tcalc)>t95),1,0)", "templateType": "anything", "group": "Ungrouped variables", "name": "outcome", "description": ""}}, "metadata": {"notes": "", "description": "<p>A comparison of the mean of two groups.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Keith McGuinness", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/523/"}], "resources": []}]}], "contributors": [{"name": "Keith McGuinness", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/523/"}]}