// Numbas version: finer_feedback_settings
{"name": "LSD and Tukey yardsticks, and one-way ANOVA - SES2003", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "LSD and Tukey yardsticks, and one-way ANOVA - SES2003", "tags": [], "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"}, "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>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"w2": {"name": "w2", "group": "Ungrouped variables", "definition": "switch(abs(m3-m2)>=tukey,[1,0,0],abs(m3-m2)>=lsd,[0,1,0],[0,0,1])", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "3", "description": "", "templateType": "anything", "can_override": false}, "mbt": {"name": "mbt", "group": "Ungrouped variables", "definition": "precround(BTSS/dfbt,2)", "description": "", "templateType": "anything", "can_override": false}, "m3": {"name": "m3", "group": "Ungrouped variables", "definition": "precround(mean(r3),2)", "description": "", "templateType": "anything", "can_override": false}, "btss": {"name": "btss", "group": "Ungrouped variables", "definition": "precround(stovern-G^2/N,2)", "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}, "r1": {"name": "r1", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu1,sig1)),6)", "description": "", "templateType": "anything", "can_override": false}, "sd2": {"name": "sd2", "group": "Ungrouped variables", "definition": "precround(pstdev(r2),2)", "description": "", "templateType": "anything", "can_override": false}, "u": {"name": "u", "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}, "dfrs": {"name": "dfrs", "group": "Ungrouped variables", "definition": "N-k", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "[sum(r1),sum(r2),sum(r3)]", "description": "", "templateType": "anything", "can_override": false}, "n1": {"name": "n1", "group": "Ungrouped variables", "definition": "6", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "n1+n2+n3", "description": "", "templateType": "anything", "can_override": false}, "mu3": {"name": "mu3", "group": "Ungrouped variables", "definition": "mu2+random(4..6)", "description": "", "templateType": "anything", "can_override": false}, "mu1": {"name": "mu1", "group": "Ungrouped variables", "definition": "random(20..25#0.5)", "description": "", "templateType": "anything", "can_override": false}, "n3": {"name": "n3", "group": "Ungrouped variables", "definition": "6", "description": "", "templateType": "anything", "can_override": false}, "vr": {"name": "vr", "group": "Ungrouped variables", "definition": "precround(MBT/MRS,2)", "description": "", "templateType": "anything", "can_override": false}, "mrs": {"name": "mrs", "group": "Ungrouped variables", "definition": "precround(RSS/dfrs,2)", "description": "", "templateType": "anything", "can_override": false}, "sig2": {"name": "sig2", "group": "Ungrouped variables", "definition": "sig1", "description": "", "templateType": "anything", "can_override": false}, "stderror": {"name": "stderror", "group": "Ungrouped variables", "definition": "precround(mrs/sqrt(n1),2)", "description": "", "templateType": "anything", "can_override": false}, "r3": {"name": "r3", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu3,sig3)),6)", "description": "", "templateType": "anything", "can_override": false}, "sd3": {"name": "sd3", "group": "Ungrouped variables", "definition": "precround(pstdev(r3),2)", "description": "", "templateType": "anything", "can_override": false}, "tsqovern": {"name": "tsqovern", "group": "Ungrouped variables", "definition": "[t[0]^2/n1,t[1]^2/n2,t[2]^2/n3]", "description": "", "templateType": "anything", "can_override": false}, "mu2": {"name": "mu2", "group": "Ungrouped variables", "definition": "mu1+random(3..6)", "description": "", "templateType": "anything", "can_override": false}, "sig1": {"name": "sig1", "group": "Ungrouped variables", "definition": "random(3..4#0.2)", "description": "", "templateType": "anything", "can_override": false}, "w3": {"name": "w3", "group": "Ungrouped variables", "definition": "switch(abs(m3-m1)>=tukey,[1,0,0],abs(m3-m1)>=lsd,[0,1,0],[0,0,1])", "description": "", "templateType": "anything", "can_override": false}, "rss": {"name": "rss", "group": "Ungrouped variables", "definition": "precround(ss-stovern,2)", "description": "", "templateType": "anything", "can_override": false}, "m1": {"name": "m1", "group": "Ungrouped variables", "definition": "precround(mean(r1),2)", "description": "", "templateType": "anything", "can_override": false}, "sd1": {"name": "sd1", "group": "Ungrouped variables", "definition": "precround(pstdev(r1),2)", "description": "", "templateType": "anything", "can_override": false}, "yn": {"name": "yn", "group": "Ungrouped variables", "definition": "map(map(yeaornay(x),x,w[z]),z,0..2)", "description": "", "templateType": "anything", "can_override": false}, "tss": {"name": "tss", "group": "Ungrouped variables", "definition": "precround(ss-G^2/N,2)", "description": "", "templateType": "anything", "can_override": false}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "6", "description": "", "templateType": "anything", "can_override": false}, "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}, "stovern": {"name": "stovern", "group": "Ungrouped variables", "definition": "sum(tsqovern)", "description": "", "templateType": "anything", "can_override": false}, "lsd": {"name": "lsd", "group": "Ungrouped variables", "definition": "precround(2.131*sqrms*sqrt(2/n1),2)", "description": "", "templateType": "anything", "can_override": false}, "r2": {"name": "r2", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu2,sig2)),6)", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "sum(t)", "description": "", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "switch(vr>=11.34,[1,0,0,0,0],vr>=6.36,[0,1,0,0,0],vr>=3.68,[0,0,1,0,0],vr>=2.7,[0,0,0,1,0],[0,0,0,0,1])", "description": "", "templateType": "anything", "can_override": false}, "sig3": {"name": "sig3", "group": "Ungrouped variables", "definition": "sig1", "description": "", "templateType": "anything", "can_override": false}, "sqrms": {"name": "sqrms", "group": "Ungrouped variables", "definition": "precround(sqrt(mrs),2)", "description": "", "templateType": "anything", "can_override": false}, "tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.01", "description": "", "templateType": "anything", "can_override": false}, "w": {"name": "w", "group": "Ungrouped variables", "definition": "[w1,w2,w3]", "description": "", "templateType": "anything", "can_override": false}, "pvalue": {"name": "pvalue", "group": "Ungrouped variables", "definition": "precround(ftest(VR,2,15),3)", "description": "", "templateType": "anything", "can_override": false}, "dfbt": {"name": "dfbt", "group": "Ungrouped variables", "definition": "k-1", "description": "", "templateType": "anything", "can_override": false}, "ss": {"name": "ss", "group": "Ungrouped variables", "definition": "sum(ssq)", "description": "", "templateType": "anything", "can_override": false}, "tukey": {"name": "tukey", "group": "Ungrouped variables", "definition": "precround(3.67*sqrms*1/sqrt(n1),2)", "description": "", "templateType": "anything", "can_override": false}, "v1": {"name": "v1", "group": "Ungrouped variables", "definition": "switch(pvalue<=0.001,[1,0,0,0,0],pvalue<=0.01,[0,1,0,0,0],pvalue<=0.05,[0,0,1,0,0],pvalue<=0.1,[0,0,0,1,0],[0,0,0,0,1])", "description": "", "templateType": "anything", "can_override": false}, "ssq": {"name": "ssq", "group": "Ungrouped variables", "definition": "[sum(map(x^2,x,r1)),sum(map(x^2,x,r2)),sum(map(x^2,x,r3))]", "description": "", "templateType": "anything", "can_override": false}, "m2": {"name": "m2", "group": "Ungrouped variables", "definition": "precround(mean(r2),2)", "description": "", "templateType": "anything", "can_override": false}, "w1": {"name": "w1", "group": "Ungrouped variables", "definition": "switch(abs(m1-m2)>=tukey,[1,0,0],abs(m1-m2)>=lsd,[0,1,0],[0,0,1])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "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"], "variable_groups": [], "functions": {"pstdev": {"parameters": [["l", "list"]], "type": "number", "language": "jme", "definition": "sqrt(abs(l)/(abs(l)-1))*stdev(l)"}, "yeaornay": {"parameters": [["n", "number"]], "type": "string", "language": "jme", "definition": "if(n=1,'Yes','No')"}}, "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>You are given the following ANOVA table for this data:</p>\n        <table border=\"1\">\n        <tbody>\n        <tr><th align=\"left\">Source</th><th>df</th><th>SS</th><th>MS</th><th>VR</th></tr>\n        <tr><th align=\"left\">Between Treatments</th>\n        <td>$\\var{dfbt}$</td>\n        <td>$\\var{btss}$</td>\n        <td>$\\var{mbt}$</td>\n        <td>$\\var{vr}$</td>\n        </tr>\n        <tr><th align=\"left\">Residual</th>\n        <td>$\\var{dfrs}$</td>\n        <td>$\\var{rss}$</td>\n        <td>$\\var{mrs}$</td>\n        <td>-</td>\n        </tr>\n        <tr><th align=\"left\">Total</th>\n        <td>$\\var{n-1}$</td>\n        <td>$\\var{tss}$</td>\n        <td>-</td>\n        <td>-</td>\n        </tr>\n        </tbody>\n        </table>\n        <p>&nbsp;</p>\n        <p>&nbsp;Input $\\sqrt{RMS}$ here: [[0]] to 2 decimal places.</p>\n        <p>&nbsp;</p>\n        <p>This will be used to calculate the LSD and Tukey yardstick values later.</p>\n      \n      ", "gaps": [{"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": "sqrms-tol", "maxValue": "sqrms+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "\n      <p style=\"text-align: center;\"><strong>Using ANOVA</strong></p>\n      <p>Using the $VR$ value given in the table and one-way ANOVA, what is the strength of evidence against the null hypothesis that the mean times taken are the same for the three groups?</p>\n      ", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["Very strong", "Strong", "Moderate", "Weak", "None"], "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>Hence what is your decision based on the above ANOVA analysis?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 1, "showCellAnswerState": true, "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."], "matrix": "v"}, {"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": "<p style=\"text-align: center;\"><strong>Using the Yardsticks</strong></p>\n<p>Fill in this table with the appropriate values for the mean values of the groups, all decimals to 2 decimal places:</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>[[0]]</td>\n</tr>\n<tr>\n<th align=\"left\">Group&nbsp;2</th>\n<td>[[1]]</td>\n</tr>\n<tr>\n<th align=\"left\">Group&nbsp;3</th>\n<td>[[2]]</td>\n</tr>\n</tbody>\n</table>\n<p>&nbsp;</p>", "gaps": [{"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": "m1", "maxValue": "m1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "m2", "maxValue": "m2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": "m3", "maxValue": "m3", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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": "<p>Now find the LSD and Tukey yardsticks from the above data. 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