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{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Statistics and probability. 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"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}], "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>"}, {"name": "LSD and Tukey yardsticks, and two-way ANOVA", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"btss": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(sum(map(x^2,x,cols))/n-tot^2/(m*n),2)", "name": "btss", "description": ""}, "sig": {"templateType": "anything", "group": "Ungrouped variables", "definition": 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"definition": "precround(btss/(m-1),2)", "name": "msbt", "description": ""}, "w4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(abs(me[1]-me[2])>=tukey,[1,0,0],abs(me[1]-me[2])>=lsd,[0,1,0],[0,0,1])", "name": "w4", "description": ""}, "bbss": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(sum(map(x^2,x,t))/m-tot^2/20,2)", "name": "bbss", "description": ""}, "lsd": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(2.179*sqrms*sqrt(2/5),2)", "name": "lsd", "description": ""}, "w5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(abs(me[1]-me[3])>=tukey,[1,0,0],abs(me[1]-me[3])>=lsd,[0,1,0],[0,0,1])", "name": "w5", "description": ""}, "vr": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(msbt/rs,2)", "name": "vr", "description": ""}, "cols": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(sum(rt[x]),x,0..m-1)", "name": "cols", "description": ""}, "w1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(abs(me[0]-me[1])>=tukey,[1,0,0],abs(me[0]-me[1])>=lsd,[0,1,0],[0,0,1])", "name": "w1", "description": ""}, "pvalue": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ftest(VR,3,12),3)", "name": "pvalue", "description": ""}, "tot": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(cols)", "name": "tot", "description": ""}, "dfr": {"templateType": "anything", "group": "Ungrouped variables", "definition": "m*n-m-n+1", "name": "dfr", "description": ""}, "tukey": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(4.2*sqrms*sqrt(1/5),2)", "name": "tukey", "description": ""}, "w": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[w1,w2,w3,w4,w5,w6]", "name": "w", "description": ""}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "4", "name": "m", "description": ""}, "msbb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(bbss/(n-1),2)", "name": "msbb", "description": ""}, "dfbb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "n-1", "name": "dfbb", "description": ""}, "w3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(abs(me[0]-me[3])>=tukey,[1,0,0],abs(me[0]-me[3])>=lsd,[0,1,0],[0,0,1])", "name": "w3", "description": ""}, "sqrms": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(sqrt(rs),2)", "name": "sqrms", "description": ""}, "t99": {"templateType": "anything", "group": "Ungrouped variables", "definition": "20.09", "name": "t99", "description": ""}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.01", "name": "tol", "description": ""}, "t95": {"templateType": "anything", "group": "Ungrouped variables", "definition": "15.51", "name": "t95", "description": ""}, "w2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(abs(me[0]-me[2])>=tukey,[1,0,0],abs(me[0]-me[2])>=lsd,[0,1,0],[0,0,1])", "name": "w2", "description": ""}, "yn": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(map(yeaornay(x),x,w[z]),z,0..5)", "name": "yn", "description": ""}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "list(transpose(matrix(r1)))", "name": "r", "description": ""}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "5", "name": "n", "description": ""}, "me": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(mean(r1[x]),x,0..3)", "name": "me", "description": ""}, "stderror": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(sqrt(rs/n),2)", "name": "stderror", "description": ""}, "rt": {"templateType": "anything", "group": "Ungrouped variables", "definition": "list(transpose(matrix(r)))", "name": "rt", "description": ""}, "rss": {"templateType": "anything", "group": "Ungrouped variables", "definition": "tss-btss-bbss", "name": "rss", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(sum(r[x]),x,0..n-1)", "name": "t", "description": ""}}, "ungrouped_variables": ["lsd", "tukey", "stderror", "bbss", "cols", "vrbb", "w6", "vr", "w4", "w3", "w2", "w1", "dfr", "rt", "rs", "t90", "tot", "sqrms", "sig", "tol", "btss", "tss", "dfbt", "yn", "msbt", "ssq", "v1", "dfbb", "t99", "t95", "msbb", "rss", "me", "r1", "m", "w5", "n", "mu", "r", "t", "w", "v", "pvalue"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {"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": "<p>Here is the ANOVA table corresponding to this data:</p>\n      <p>&nbsp;</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>$\\var{m-1}$</td>\n      <td>$\\var{btss}$</td>\n      <td>$\\var{msbt}$</td>\n      <td>$\\var{vr}$</td>\n      </tr>\n      <tr><th align=\"left\">Between Blocks</th>\n      <td>$\\var{n-1}$</td>\n      <td>$\\var{bbss}$</td>\n      <td>$\\var{msbb}$</td>\n      <td>$\\var{vrbb}$</td>\n      </tr>\n      <tr><th align=\"left\">Residual</th>\n      <td>$\\var{dfr}$</td>\n      <td>$\\var{rss}$</td>\n      <td>$\\var{rs}$</td>\n      <td>-</td>\n      </tr>\n      <tr><th align=\"left\">Total</th>\n      <td>$\\var{m*n-1}$</td>\n      <td>$\\var{tss}$</td>\n      <td>-</td>\n      <td>-</td>\n      </tr>\n      </tbody>\n      </table>\n      <p>Input $\\sqrt{RMS}$ here: [[0]] to 2 decimal places.</p>\n      <p>This will be used later to calculate the yardsticks.</p>", "showCorrectAnswer": true, "marks": 0}, {"displayType": "radiogroup", "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>"], "displayColumns": 1, "prompt": "<p>Given the value of $VR$ in the table above, find the range for the $p$ value by looking up the critical values of $F_{3,12}$ (one-sided).</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>$2.61$</td>\n      <td>$3.49$</td>\n      <td>$5.95$</td>\n      <td>$10.8$</td>\n      </tr>\n      </tbody>\n      </table>", "distractors": ["", "", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "v", "marks": 0}, {"displayType": "radiogroup", "choices": ["Very strong", "Strong", "Moderate", "Weak", "None"], "displayColumns": 0, "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 treatments offered by the sun-creams?</p>", "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 more investigation is needed.", "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": "me[0]-tol", "maxValue": "me[0]+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "me[1]-tol", "maxValue": "me[1]+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "me[2]-tol", "maxValue": "me[2]+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "me[3]-tol", "maxValue": "me[3]+tol", "marks": 0.5}, {"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}], "type": "gapfill", "prompt": "<p style=\"text-align: center;\"><strong>Using the yardsticks</strong></p>\n      <p>Enter the sample means for the sun-creams:</p>\n      <p>W: [[0]], X:[[1]], Y:[[2]], Z:[[3]] (to 2 decimal places).</p>\n      <p>Calclate the LSD and Tukey yardsticks using the value for $\\sqrt{RMS}$ to 2 decimal places obtained above.</p>\n      <p>&nbsp;</p>\n      <p>LSD yardstick value = &nbsp; &nbsp;[[4]] (to 2 decimal places).</p>\n      <p>&nbsp;</p>\n      <p>Tukey yardstick value = [[5]] (to 2 decimal places).</p>\n      <p>&nbsp;</p>", "showCorrectAnswer": true, "marks": 0}, {"layout": {"expression": ""}, "choices": ["$W$ and $X$", "$W$ and $Y$", "$W$ and $Z$", "$X$ and $Y$", "$X$ and $Z$", "$Y$ and $Z$"], "matrix": "w", "prompt": "<p>Using these yardsticks fill in the following table indicating if there is a possible or definite significant difference between the sample means of pairs of sun-creams.</p>\n        <p>&nbsp;</p>", "type": "m_n_x", "maxAnswers": 0, "shuffleChoices": false, "marks": 0, "scripts": {}, "maxMarks": 0, "minAnswers": 0, "minMarks": 0, "shuffleAnswers": false, "showCorrectAnswer": true, "answers": ["<strong>Definite Significant Difference</strong>", "<strong>Possible Significant Difference</strong>", "<strong>No Significant Difference</strong>"]}], "statement": "<p>To test the effectiveness of sun-tan creams, five volunteers A, B, C, D, E each tried four creams W, X, Y, Z on various parts of their legs. They were then subjected to ultra-violet radiation and an estimate of the degree of burning was made (higher figures indicate greater burning). The results are given below with some totals:</p>\n  <table>\n  <tbody>\n  <tr>\n  <td style=\"vertical-align: middle; border-bottom: 1px solid; border-right: 1px solid;\">&nbsp;</td>\n  <td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>W</strong></td>\n  <td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>X</strong></td>\n  <td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>Y</strong></td>\n  <td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>Z</strong></td>\n  <td style=\"vertical-align: middle; border-bottom: 1px solid; border-left: 1px solid;\"><strong>Totals</strong></td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>A</strong></td>\n  <td>{r[0][0]}</td>\n  <td>{r[0][1]}</td>\n  <td>{r[0][2]}</td>\n  <td>{r[0][3]}</td>\n  <td style=\"vertical-align: middle; border-left: 1px solid;\">{t[0]}</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>B</strong></td>\n  <td>{r[1][0]}</td>\n  <td>{r[1][1]}</td>\n  <td>{r[1][2]}</td>\n  <td>{r[1][3]}</td>\n  <td style=\"vertical-align: middle; border-left: 1px solid;\">{t[1]}</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>C</strong></td>\n  <td>{r[2][0]}</td>\n  <td>{r[2][1]}</td>\n  <td>{r[2][2]}</td>\n  <td>{r[2][3]}</td>\n  <td style=\"vertical-align: middle; border-left: 1px solid;\">{t[2]}</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>D</strong></td>\n  <td>{r[3][0]}</td>\n  <td>{r[3][1]}</td>\n  <td>{r[3][2]}</td>\n  <td>{r[3][3]}</td>\n  <td style=\"vertical-align: middle; border-left: 1px solid;\">{t[3]}</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>E</strong></td>\n  <td>{r[4][0]}</td>\n  <td>{r[4][1]}</td>\n  <td>{r[4][2]}</td>\n  <td>{r[4][3]}</td>\n  <td style=\"vertical-align: middle; border-left: 1px solid;\">{t[4]}</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid;\"><strong>Totals</strong></td>\n  <td style=\"vertical-align: middle; border-top: 1px solid;\">{cols[0]}</td>\n  <td style=\"vertical-align: middle; border-top: 1px solid;\">{cols[1]}</td>\n  <td style=\"vertical-align: middle; border-top: 1px solid;\">{cols[2]}</td>\n  <td style=\"vertical-align: middle; border-top: 1px solid;\">{cols[3]}</td>\n  <td style=\"vertical-align: middle; border-top: 1px solid; border-left: 1px solid;\">{tot}</td>\n  </tr>\n  </tbody>\n  </table>\n  <p>&nbsp;</p>\n  <ul>\n  <li>Test the null-hypothesis using two-way ANOVA that the creams are equally effective.</li>\n  </ul>\n  <p>&nbsp;</p>\n  <ul>\n  <li>Write down the sample mean for each sun-cream together with the LSD and Tukey yardsticks so that you can see if there is any significant difference between the sample means given by these yardsticks..</li>\n  </ul>", "tags": ["ANOVA", "checked2015", "hypothesis testing", "lsd", "LSD", "PSY2010", "sample means", "significant difference", "statistics", "Tukey", "tukey ", "two-way ANOVA", "yardsticks", "Yardsticks"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>LSD and Tukey yardsticks on five treatments. Also two-way Anova test on same set of data.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h2 style=\"text-align: center;\">&nbsp;<strong>Using the Yardsticks</strong></h2>\n  <p>The mean values for each sun-cream 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\">W</th>\n  <td>$\\var{me[0]}$</td>\n  </tr>\n  <tr><th align=\"left\">X</th>\n  <td>$\\var{me[1]}$</td>\n  </tr>\n  <tr><th style=\"text-align: left;\">Y</th>\n  <td style=\"text-align: left;\">$\\var{me[2]}$</td>\n  </tr>\n  <tr><th style=\"text-align: left;\">Z</th>\n  <td style=\"text-align: left;\">$\\var{me[3]}$</td>\n  </tr>\n  </tbody>\n  </table>\n  <p>The differences between the mean values for the sun-creams are:</p>\n  <p style=\"text-align: left;\">Between $W$ and $X=\\;|\\var{me[0]}-\\var{me[1]}|=\\var{abs(me[0]-me[1])}$</p>\n  <p style=\"text-align: left;\">Between $W$ and $Y=\\;|\\var{me[0]}-\\var{me[2]}|=\\var{abs(me[0]-me[2])}$</p>\n  <p style=\"text-align: left;\">Between $W$ and $Z=\\;|\\var{me[0]}-\\var{me[3]}|=\\var{abs(me[0]-me[3])}$</p>\n  <p style=\"text-align: left;\">Between $X$ and $Y=\\;|\\var{me[1]}-\\var{me[2]}|=\\var{abs(me[1]-me[2])}$</p>\n  <p style=\"text-align: left;\">Between $X$ and $Z=\\;|\\var{me[1]}-\\var{me[3]}|=\\var{abs(me[1]-me[3])}$</p>\n  <p style=\"text-align: left;\">Between $Y$ and $Z=\\;|\\var{me[2]}-\\var{me[3]}|=\\var{abs(me[2]-me[3])}$</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 = $2.179\\times\\var{sqrms}\\times\\sqrt{2/\\var{n}}=\\var{lsd}$ to 2 decimal places, where $\\var{sqrms}$ is the value of $\\sqrt{RMS}$ found above.</p>\n  <p style=\"text-align: left;\">Tukey yardstick = $4.2\\times\\var{sqrms}\\times\\sqrt{1/\\var{n}}=\\var{tukey}$ 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 sun-creams:</p>\n  <table style=\"float: left;\" border=\"1\">\n  <tbody>\n  <tr><th align=\"left\">Pairs of Sun-creams</th><th>Definite Significant Difference</th><th>Possible Significant Difference</th><th>No Significant Difference</th></tr>\n  <tr><th align=\"left\">Means of W and X</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 W and Y</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 W and Z</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  <tr><th align=\"left\">Means of X and Y</th>\n  <td><strong>{yn[3][0]}</strong></td>\n  <td><strong>{yn[3][1]}</strong></td>\n  <td><strong>{yn[3][2]}</strong></td>\n  </tr>\n  <tr><th align=\"left\">Means of X and Z</th>\n  <td><strong>{yn[4][0]}</strong></td>\n  <td><strong>{yn[4][1]}</strong></td>\n  <td><strong>{yn[4][2]}</strong></td>\n  </tr>\n  <tr><th align=\"left\">Means of Y and Z</th>\n  <td><strong>{yn[5][0]}</strong></td>\n  <td><strong>{yn[5][1]}</strong></td>\n  <td><strong>{yn[5][2]}</strong></td>\n  </tr>\n  </tbody>\n  </table>"}]}], "showQuestionGroupNames": false, "showstudentname": true, "navigation": {"onleave": {"action": "none", "message": ""}, "allowregen": true, "reverse": true, "preventleave": true, "browse": true, "showresultspage": "oncompletion", "showfrontpage": true}, "timing": {"timedwarning": {"action": "none", "message": ""}, "timeout": {"action": "none", "message": ""}, "allowPause": true}, "duration": 0, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "extensions": ["stats"], "custom_part_types": [], "resources": []}