// Numbas version: finer_feedback_settings
{"name": "LSD and Tukey yardsticks, and one-way ANOVA", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"pvalue": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ftest(VR,2,15),3)", "name": "pvalue", "description": ""}, "r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu1,sig1)),6)", "name": "r1", "description": ""}, "ssq": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[sum(map(x^2,x,r1)),sum(map(x^2,x,r2)),sum(map(x^2,x,r3))]", "name": "ssq", "description": ""}, "btss": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(stovern-G^2/N,2)", "name": "btss", "description": ""}, "sd1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(pstdev(r1),2)", "name": "sd1", "description": ""}, "tss": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ss-G^2/N,2)", "name": "tss", "description": ""}, "k": {"templateType": "anything", "group": "Ungrouped variables", "definition": "3", "name": "k", "description": ""}, "dfbt": {"templateType": "anything", "group": "Ungrouped variables", "definition": "k-1", "name": "dfbt", "description": ""}, "sig2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sig1", "name": "sig2", "description": ""}, "v1": {"templateType": "anything", "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])", "name": "v1", "description": ""}, "vr": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(MBT/MRS,2)", "name": "vr", "description": ""}, "w1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(abs(m1-m2)>=tukey,[1,0,0],abs(m1-m2)>=lsd,[0,1,0],[0,0,1])", "name": "w1", "description": ""}, "rss": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ss-stovern,2)", "name": "rss", "description": ""}, "n1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "6", "name": "n1", "description": ""}, "sd3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(pstdev(r3),2)", "name": "sd3", "description": ""}, "ss": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(ssq)", "name": "ss", "description": ""}, "w2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(abs(m3-m2)>=tukey,[1,0,0],abs(m3-m2)>=lsd,[0,1,0],[0,0,1])", "name": "w2", "description": ""}, "sig3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sig1", "name": "sig3", "description": ""}, "mbt": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(BTSS/dfbt,2)", "name": "mbt", "description": ""}, "r2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu2,sig2)),6)", "name": "r2", "description": ""}, "mu2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mu1+random(3..6)", "name": "mu2", "description": ""}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(t)", "name": "g", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[sum(r1),sum(r2),sum(r3)]", "name": "t", "description": ""}, "sd2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(pstdev(r2),2)", "name": "sd2", "description": ""}, "w": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[w1,w2,w3]", "name": "w", "description": ""}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.01", "name": "tol", "description": ""}, "v": {"templateType": "anything", "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])", "name": "v", "description": ""}, "pmsg": {"templateType": "anything", "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 $0.10$']", "name": "pmsg", "description": ""}, "sig1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..4#0.2)", "name": "sig1", "description": ""}, "sqrms": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(sqrt(mrs),2)", "name": "sqrms", "description": ""}, "mrs": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(RSS/dfrs,2)", "name": "mrs", "description": ""}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(v[0]=1,0,v[1]=1,1,v[2]=1,2,v[3]=1,3,4)", "name": "u", "description": ""}, "m3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(mean(r3),2)", "name": "m3", "description": ""}, "n2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "6", "name": "n2", "description": ""}, "cmsg": {"templateType": "anything", "group": "Ungrouped variables", "definition": "['do reject','do reject','do not reject','do not reject','do not reject']", "name": "cmsg", "description": ""}, "mu1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(20..25#0.5)", "name": "mu1", "description": ""}, "stovern": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(tsqovern)", "name": "stovern", "description": ""}, "lsd": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(2.131*sqrms*sqrt(2/n1),2)", "name": "lsd", "description": ""}, "dfrs": {"templateType": "anything", "group": "Ungrouped variables", "definition": "N-k", "name": "dfrs", "description": ""}, "yn": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(map(yeaornay(x),x,w[z]),z,0..2)", "name": "yn", "description": ""}, "tukey": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(3.67*sqrms*1/sqrt(n1),2)", "name": "tukey", "description": ""}, "w3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(abs(m3-m1)>=tukey,[1,0,0],abs(m3-m1)>=lsd,[0,1,0],[0,0,1])", "name": "w3", "description": ""}, "m1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(mean(r1),2)", "name": "m1", "description": ""}, "r3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu3,sig3)),6)", "name": "r3", "description": ""}, "m2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(mean(r2),2)", "name": "m2", "description": ""}, "n3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "6", "name": "n3", "description": ""}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "n1+n2+n3", "name": "n", "description": ""}, "stderror": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(mrs/sqrt(n1),2)", "name": "stderror", "description": ""}, "mu3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mu2+random(4..6)", "name": "mu3", "description": ""}, "tsqovern": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[t[0]^2/n1,t[1]^2/n2,t[2]^2/n3]", "name": "tsqovern", "description": ""}}, "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"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "LSD and Tukey yardsticks, and one-way ANOVA", "functions": {"pstdev": {"type": "number", "language": "jme", "definition": "sqrt(abs(l)/(abs(l)-1))*stdev(l)", "parameters": [["l", "list"]]}, "yeaornay": {"type": "string", "language": "jme", "definition": "if(n=1,'Yes','No')", "parameters": [["n", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "sqrms-tol", "maxValue": "sqrms+tol", "marks": 1}], "type": "gapfill", "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      ", "showCorrectAnswer": true, "marks": 0}, {"displayType": "radiogroup", "choices": ["Very strong", "Strong", "Moderate", "Weak", "None"], "displayColumns": 0, "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      ", "distractors": ["", "", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "v", "marks": 0}, {"displayType": "radiogroup", "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."], "displayColumns": 1, "prompt": "<p>Hence what is your decision based on the above ANOVA analysis?</p>", "distractors": ["", "", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "v", "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "m1", "maxValue": "m1", "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "m2", "maxValue": "m2", "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "m3", "maxValue": "m3", "marks": 1}], "type": "gapfill", "prompt": "\n      <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><th align=\"left\">Group A</th>\n      <td>[[0]]</td>\n      </tr>\n      <tr><th align=\"left\">Group B</th>\n      <td>[[1]]</td>\n      </tr>\n      <tr><th align=\"left\">Group C</th>\n      <td>[[2]]</td>\n      </tr>\n      </tbody>\n      </table>\n      <p>&nbsp;</p>\n      ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "lsd-tol", "maxValue": "lsd+tol", "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "tukey-tol", "maxValue": "tukey+tol", "marks": 1}, {"layout": {"expression": ""}, "choices": ["Groups $A$ and $B$", "Groups $B$ and $C$", "Groups $A$ and $C$"], "matrix": "w", "type": "m_n_x", "maxAnswers": 0, "shuffleChoices": false, "marks": 0, "scripts": {}, "minMarks": 0, "minAnswers": 0, "maxMarks": 0, "shuffleAnswers": false, "showCorrectAnswer": true, "answers": ["<strong>Definite Significant Difference</strong>", "<strong>Possible Significant Difference</strong>", "<strong>No Significant Difference</strong>"]}], "type": "gapfill", "prompt": "<p>Now find the LSD and Tukey yardsticks from the above data. Use the value to 2 decimal places you found for $\\sqrt{RMS}$:</p>\n      <p>&nbsp; &nbsp;LSD= [[0]]</p>\n      <p>Tukey= [[1]]</p>\n      <p>&nbsp;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>", "showCorrectAnswer": true, "marks": 0}], "statement": "\n  <p>The following data arose in a comparison of the effects of alcohol on the time taken to complete a task. There were three groups of subjects; Group A had no alcohol, Group B had two units over 1 hour and Group C had 4 units over 1 hour. The responses are the times (in seconds) taken to complete a word-matching task.</p>\n    <table border=\"1\">\n    <tbody>\n    <tr><th align=\"left\">Group A (0 units)</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><th align=\"left\">Group B (2 units)</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><th align=\"left\">Group C (4 units)</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>\n  \n  ", "tags": ["ANOVA", "average", "checked2015", "data analysis", "definite significant difference", "degrees of freedom", "F-test", "hypothesis testing", "Least significant difference", "lsd", "LSD", "mean ", "possible significant difference", "PSY2010", "standard deviation", "statistics", "stats", "Tukey", "tukey ", "variance", "yardsticks", "Yardsticks"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n                                                    \t\t        \t\t<p><strong>15/11/2012:</strong></p>\n                                                    \t\t        \t\t                <p><strong><br /></strong>This question cones from editing a one-way Anova example</p>\n                                                    \t\t        \t\t                <p>Added tags and description</p>\n                                                    \t\t        \t\t                <p>&nbsp;</p>\n                                                    \t\t        \n                                                    \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>LSD and Tukey yardsticks on three treatments. Also one-way Anova test on same set of data.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p><strong>Using the Yardsticks</strong></p>\n    <p>The mean values for each group are:</p>\n    <p>&nbsp;&nbsp;</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><th align=\"left\">Group A</th>\n    <td>$\\var{m1}$</td>\n    </tr>\n    <tr><th align=\"left\">Group B</th>\n    <td>$\\var{m2}$</td>\n    </tr>\n    <tr><th style=\"text-align: left;\" align=\"left\">Group C</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 $A$ and $B=\\;|\\var{m1}-\\var{m2}|=\\var{abs(m1-m2)}$</p>\n    <p style=\"text-align: left;\">Between $B$ and $C=\\;|\\var{m2}-\\var{m3}|=\\var{abs(m2-m3)}$</p>\n    <p style=\"text-align: left;\">Between $A$ and $C=\\;|\\var{m1}-\\var{m3}|=\\var{abs(m1-m3)}$</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    <p style=\"text-align: left;\">&nbsp;</p>\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    <p style=\"text-align: left;\">&nbsp;</p>\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><th align=\"left\">Pairs of Groups</th><th>Definite Significant Difference</th><th>Possible Significant Difference</th><th>No Significant Difference</th></tr>\n    <tr><th align=\"left\">Means of Groups A and B</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><th align=\"left\">Means of Groups B and C</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><th align=\"left\">Means of Groups A and C</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>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}