// Numbas version: finer_feedback_settings
{"name": "LSD and Tukey yardsticks, and one-way ANOVA - Mock Test - SES2003", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>LSD and Tukey yardsticks on three treatments. Also one-way Anova test on same set of data.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "name": "LSD and Tukey yardsticks, and one-way ANOVA - Mock Test - SES2003", "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "gaps": [{"correctAnswerStyle": "plain", "scripts": {}, "variableReplacements": [], "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "precision": "2", "precisionMessage": "You have not given your answer to the correct precision.", "mustBeReducedPC": 0, "showFeedbackIcon": true, "strictPrecision": false, "showPrecisionHint": false, "allowFractions": false, "precisionPartialCredit": 0, "marks": "0.5", "notationStyles": ["plain", "en", "si-en"], "minValue": "btss-tol", "showCorrectAnswer": true, "precisionType": "dp", "extendBaseMarkingAlgorithm": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "maxValue": "btss+tol", "type": "numberentry"}, {"maxValue": "dfrs+tol", "minValue": "dfrs-tol", "correctAnswerStyle": "plain", "scripts": {}, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "unitTests": [], "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "mustBeReducedPC": 0, "type": "numberentry", "allowFractions": false, "showCorrectAnswer": true, "marks": "0.5"}, {"maxValue": "n-1", "minValue": "n-1", "correctAnswerStyle": "plain", "scripts": {}, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "unitTests": [], "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "mustBeReducedPC": 0, "type": "numberentry", "allowFractions": false, "showCorrectAnswer": true, "marks": "0.5"}, {"correctAnswerStyle": "plain", "scripts": {}, "variableReplacements": [], "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "precision": "2", "precisionMessage": "You have not given your answer to the correct precision.", "mustBeReducedPC": 0, "showFeedbackIcon": true, "strictPrecision": false, "showPrecisionHint": false, "allowFractions": false, "precisionPartialCredit": 0, "marks": "0.5", "notationStyles": ["plain", "en", "si-en"], "minValue": "tss-tol", "showCorrectAnswer": true, "precisionType": "dp", "extendBaseMarkingAlgorithm": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "maxValue": "tss+tol", "type": "numberentry"}], "variableReplacements": [], "customMarkingAlgorithm": "", "prompt": "<p>Fill in the&nbsp;following ANOVA table for this data, entering your answers to 2 decimal places</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<th align=\"left\">Source</th>\n<th>df</th>\n<th>SS</th>\n<th>MS</th>\n<th>VR</th>\n</tr>\n<tr>\n<th align=\"left\">Between Treatments</th>\n<td>$\\var{dfbt}$</td>\n<td>[[0]]</td>\n<td>$\\var{mbt}$</td>\n<td>$\\var{vr}$</td>\n</tr>\n<tr>\n<th align=\"left\">Residual</th>\n<td>[[1]]</td>\n<td>$\\var{rss}$</td>\n<td>$\\var{mrs}$</td>\n<td>-</td>\n</tr>\n<tr>\n<th align=\"left\">Total</th>\n<td>[[2]]</td>\n<td>[[3]]</td>\n<td>-</td>\n<td>-</td>\n</tr>\n</tbody>\n</table>\n<p>&nbsp;$\\sqrt{RMS} = \\var{sqrms}$,&nbsp;to 2 decimal places.&nbsp;This will be used to calculate the LSD and Tukey yardstick values later.</p>", "unitTests": [], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "type": "gapfill", "showCorrectAnswer": true, "marks": 0}, {"choices": ["$p$ less than $0.1\\%$", "$p$ lies between $0.1\\%$ and $1\\%$", "$p$ lies between $1 \\%$ and $5\\%$", "$p$ lies between $5 \\%$ and $10\\%$", "$p$ is greater than $10\\%$"], "displayType": "radiogroup", "displayColumns": "1", "scripts": {}, "minMarks": 0, "extendBaseMarkingAlgorithm": true, "maxMarks": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "prompt": "<p style=\"text-align: left;\"><strong>Using ANOVA</strong></p>\n<p>Given the value of $VR$ in the table above, find the range for the $p$ value by using the critical values of $F_{2,15}$ (one-sided) below.</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td>$10\\%$</td>\n<td>$5\\%$</td>\n<td>$1\\%$</td>\n<td>$0.1\\%$</td>\n</tr>\n<tr>\n<td>$2.7$</td>\n<td>$3.68$</td>\n<td>$6.36$</td>\n<td>$11.34$</td>\n</tr>\n</tbody>\n</table>\n<p></p>", "showCellAnswerState": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "matrix": "v", "showFeedbackIcon": true, "shuffleChoices": false, "type": "1_n_2", "showCorrectAnswer": true, "marks": 0}, {"choices": ["Very strong", "Strong", "Moderate", "Weak", "None"], "displayType": "radiogroup", "displayColumns": 0, "scripts": {}, "minMarks": 0, "extendBaseMarkingAlgorithm": true, "maxMarks": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "prompt": "<p style=\"text-align: left;\">What is the strength of evidence against the null hypothesis that the mean times taken are the same for the three groups?<strong></strong></p>", "showCellAnswerState": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "matrix": "v", "showFeedbackIcon": true, "shuffleChoices": false, "type": "1_n_2", "showCorrectAnswer": true, "marks": 0}, {"choices": ["We reject the null hypothesis at the $0.1\\%$ level", "We reject the null hypothesis at the $1\\%$ level.", "We reject the null hypothesis at the $5\\%$ level.", "We do not reject the null hypothesis but consider further investigation.", "We do not reject the null hypothesis."], "displayType": "radiogroup", "displayColumns": 1, "scripts": {}, "minMarks": 0, "extendBaseMarkingAlgorithm": true, "maxMarks": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "prompt": "<p>Hence what is your decision based on the above ANOVA analysis?</p>", "showCellAnswerState": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "matrix": "v", "showFeedbackIcon": true, "shuffleChoices": false, "type": "1_n_2", "showCorrectAnswer": true, "marks": 0}, {"sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "gaps": [{"maxValue": "lsd+tol", "minValue": "lsd-tol", "correctAnswerStyle": "plain", "scripts": {}, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "unitTests": [], "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "mustBeReducedPC": 0, "type": "numberentry", "allowFractions": false, "showCorrectAnswer": true, "marks": 1}, {"maxValue": "tukey+tol", "minValue": "tukey-tol", "correctAnswerStyle": "plain", "scripts": {}, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "unitTests": [], "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "mustBeReducedPC": 0, "type": "numberentry", "allowFractions": false, "showCorrectAnswer": true, "marks": 1}, {"displayType": "radiogroup", "answers": ["<strong>Definite Significant Difference</strong>", "<strong>Possible Significant Difference</strong>", "<strong>No Significant Difference</strong>"], "scripts": {}, "minMarks": 0, "minAnswers": 0, "variableReplacements": [], "shuffleAnswers": false, "customMarkingAlgorithm": "", "maxAnswers": 0, "type": "m_n_x", "showCorrectAnswer": true, "marks": 0, "choices": ["Groups $1$ and $2$", "Groups $2$ and $3$", "Groups $1$ and $3$"], "showCellAnswerState": true, "layout": {"type": "all", "expression": ""}, "maxMarks": 0, "extendBaseMarkingAlgorithm": true, "warningType": "none", "unitTests": [], "variableReplacementStrategy": "originalfirst", "matrix": "w", "shuffleChoices": false, "showFeedbackIcon": true}], "variableReplacements": [], "customMarkingAlgorithm": "", "prompt": "<p>The mean values of the groups are</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td>&nbsp;</td>\n<td>$\\overline{x}_i$</td>\n</tr>\n<tr>\n<th align=\"left\">Group&nbsp;1</th>\n<td>{m1}</td>\n</tr>\n<tr>\n<th align=\"left\">Group&nbsp;2</th>\n<td>{m2}</td>\n</tr>\n<tr>\n<th align=\"left\">Group&nbsp;3</th>\n<td>{m3}</td>\n</tr>\n</tbody>\n</table>\n<p><br/>Given $q_{t,\\nu}(\\alpha) =3.67$, $t_{\\nu}(\\alpha) =2.131$ and the value for $\\sqrt{RMS}$ above, find the LSD and Tukey yardsticks from the data.</p>\n<p>LSD= [[0]]&nbsp; (to 2 decimal places)</p>\n<p>Tukey= [[1]]&nbsp; (to 2 decimal places)&nbsp;&nbsp;</p>\n<p>Using these yardsticks fill in the following table indicating if there is a possible or definite significant difference between the pairs of groups mean times in undertaking the tasks:</p>\n<p>[[2]]</p>", "unitTests": [], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "type": "gapfill", "showCorrectAnswer": true, "marks": 0}], "variable_groups": [], "ungrouped_variables": ["lsd", "tukey", "sd1", "sd2", "sd3", "vr", "w3", "w2", "w1", "mbt", "n", "pmsg", "mu3", "stderror", "sqrms", "dfrs", "m1", "m3", "m2", "btss", "stovern", "tss", "dfbt", "tol", "yn", "ssq", "sig1", "v1", "sig3", "sig2", "cmsg", "rss", "tsqovern", "mu1", "g", "r2", "mu2", "ss", "k", "r3", "mrs", "r1", "u", "t", "w", "v", "n1", "n2", "n3", "pvalue"], "extensions": ["stats"], "tags": [], "statement": "<p>The following data arose in a comparison of the effects of hydration on the time taken (in minutes) to complete a set of exercises. There were three groups of subjects; group 1 (fully hydrated), group 2 (partially hydrated) and group 3 (dehydrated).</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<th align=\"left\">Group&nbsp;1 (fully hydrated)</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</tr>\n<tr>\n<th align=\"left\">Group&nbsp;2 (partially hydrated)</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</tr>\n<tr>\n<th align=\"left\">Group&nbsp;3 (dehydrated)</th>\n<td>$\\var{r3[0]}$</td>\n<td>$\\var{r3[1]}$</td>\n<td>$\\var{r3[2]}$</td>\n<td>$\\var{r3[3]}$</td>\n<td>$\\var{r3[4]}$</td>\n<td>$\\var{r3[5]}$</td>\n</tr>\n</tbody>\n</table>\n<p>&nbsp;</p>", "advice": "<p><strong>Using the Yardsticks</strong></p>\n<p>The mean values for each group are:</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td style=\"text-align: left;\">&nbsp;</td>\n<td>$\\overline{x}_i$</td>\n</tr>\n<tr>\n<th align=\"left\">Group 1&nbsp;</th>\n<td>$\\var{m1}$</td>\n</tr>\n<tr>\n<th align=\"left\">Group&nbsp;2</th>\n<td>$\\var{m2}$</td>\n</tr>\n<tr>\n<th style=\"text-align: left;\" align=\"left\">Group&nbsp;3</th>\n<td style=\"text-align: left;\">$\\var{m3}$<span style=\"text-align: center;\">&nbsp;</span><span style=\"text-align: center;\">&nbsp;</span></td>\n</tr>\n</tbody>\n</table>\n<p>The differences between the mean values for the groups are:</p>\n<p style=\"text-align: left;\">Between $1$ and $2=\\;|\\var{m1}-\\var{m2}|=\\var{abs(m1-m2)}$</p>\n<p style=\"text-align: left;\">Between $2$ and $3=\\;|\\var{m2}-\\var{m3}|=\\var{abs(m2-m3)}$</p>\n<p style=\"text-align: left;\">Between $1$ and $3=\\;|\\var{m1}-\\var{m3}|=\\var{abs(m1-m3)}$</p>\n<p style=\"text-align: left;\">We compare these differences with the LSD and Tukey yardsticks:</p>\n<p style=\"text-align: left;\">LSD yardstick = $2.131\\times\\var{sqrms}\\times\\sqrt{2/\\var{n1}}=\\var{lsd}$ to 2 decimal places, where $\\var{sqrms}$ is the value of $\\sqrt{RMS}$ found above.</p>\n<p style=\"text-align: left;\">Tukey yardstick = $3.67\\times\\var{sqrms}\\times\\sqrt{1/\\var{n1}}=\\var{tukey}$ to 2 decimal places.</p>\n<p style=\"text-align: left;\">If the difference of the means:</p>\n<ul style=\"text-align: left;\">\n<li>is greater than the Tukey yardstick we say that there is evidence of a definite significant difference.</li>\n</ul>\n<ul style=\"text-align: left;\">\n<li>less than the Tukey yardstick but greater than the LSD yardstick&nbsp;we say that there is evidence of a possible significant difference.</li>\n</ul>\n<ul style=\"text-align: left;\">\n<li>is less than the LSD yardstick then we say&nbsp;that there is no &nbsp;evidence of a significant difference.</li>\n</ul>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">Hence we have the following for the groups:</p>\n<table style=\"float: left;\" border=\"1\">\n<tbody>\n<tr>\n<th align=\"left\">Pairs of Groups</th>\n<th>Definite Significant Difference</th>\n<th>Possible Significant Difference</th>\n<th>No Significant Difference</th>\n</tr>\n<tr>\n<th align=\"left\">Means of Groups&nbsp;1 and 2</th>\n<td><strong>{yn[0][0]}</strong></td>\n<td><strong>{yn[0][1]}</strong></td>\n<td><strong>{yn[0][2]}</strong></td>\n</tr>\n<tr>\n<th align=\"left\">Means of Groups&nbsp;2 and 3</th>\n<td><strong>{yn[1][0]}</strong></td>\n<td><strong>{yn[1][1]}</strong></td>\n<td><strong>{yn[1][2]}</strong></td>\n</tr>\n<tr>\n<th align=\"left\">Means of Groups&nbsp;1 and 3</th>\n<td><strong>{yn[2][0]}</strong></td>\n<td><strong>{yn[2][1]}</strong></td>\n<td><strong>{yn[2][2]}</strong></td>\n</tr>\n</tbody>\n</table>", "functions": {"yeaornay": {"type": "string", "parameters": [["n", "number"]], "language": "jme", "definition": "if(n=1,'Yes','No')"}, "pstdev": {"type": "number", "parameters": [["l", "list"]], "language": "jme", 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"group": "Ungrouped variables", "name": "r3", "definition": "repeat(round(normalsample(mu3,sig3)),6)", "templateType": "anything"}, "ss": {"description": "", "group": "Ungrouped variables", "name": "ss", "definition": "sum(ssq)", "templateType": "anything"}, "dfrs": {"description": "", "group": "Ungrouped variables", "name": "dfrs", "definition": "N-k", "templateType": "anything"}, "sig3": {"description": "", "group": "Ungrouped variables", "name": "sig3", "definition": "sig1", "templateType": "anything"}, "g": {"description": "", "group": "Ungrouped variables", "name": "g", "definition": "sum(t)", "templateType": "anything"}, "tss": {"description": "", "group": "Ungrouped variables", "name": "tss", "definition": "precround(ss-G^2/N,2)", "templateType": "anything"}, "m1": {"description": "", "group": "Ungrouped variables", "name": "m1", "definition": "precround(mean(r1),2)", "templateType": "anything"}, "pvalue": {"description": "", "group": "Ungrouped variables", "name": "pvalue", 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"n1+n2+n3", "templateType": "anything"}, "t": {"description": "", "group": "Ungrouped variables", "name": "t", "definition": "[sum(r1),sum(r2),sum(r3)]", "templateType": "anything"}, "dfbt": {"description": "", "group": "Ungrouped variables", "name": "dfbt", "definition": "k-1", "templateType": "anything"}, "m3": {"description": "", "group": "Ungrouped variables", "name": "m3", "definition": "precround(mean(r3),2)", "templateType": "anything"}, "sd2": {"description": "", "group": "Ungrouped variables", "name": "sd2", "definition": "precround(pstdev(r2),2)", "templateType": "anything"}, "m2": {"description": "", "group": "Ungrouped variables", "name": "m2", "definition": "precround(mean(r2),2)", "templateType": "anything"}, "n2": {"description": "", "group": "Ungrouped variables", "name": "n2", "definition": "6", "templateType": "anything"}, "mrs": {"description": "", "group": "Ungrouped variables", "name": "mrs", "definition": "precround(RSS/dfrs,2)", "templateType": "anything"}, "w1": 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"precround(pstdev(r3),2)", "templateType": "anything"}, "sd1": {"description": "", "group": "Ungrouped variables", "name": "sd1", "definition": "precround(pstdev(r1),2)", "templateType": "anything"}, "r2": {"description": "", "group": "Ungrouped variables", "name": "r2", "definition": "repeat(round(normalsample(mu2,sig2)),6)", "templateType": "anything"}, "stovern": {"description": "", "group": "Ungrouped variables", "name": "stovern", "definition": "sum(tsqovern)", "templateType": "anything"}, "mu2": {"description": "", "group": "Ungrouped variables", "name": "mu2", "definition": "mu1+random(3..6)", "templateType": "anything"}}, "preamble": {"js": "", "css": ""}, "type": "question", "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/"}]}]}], "contributors": [{"name": "Chris Graham", 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