// Numbas version: finer_feedback_settings
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"switch(vr>=10.8,[1,0,0,0,0],vr>=5.95,[0,1,0,0,0],vr>=3.49,[0,0,1,0,0],vr>=2.61,[0,0,0,1,0],[0,0,0,0,1])", "name": "v", "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": ""}, "ssq": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(sum(map(x^2,x,r[y])),y,0..n-1))", "name": "ssq", "description": ""}, "vrbb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(msbb/rs,2)", "name": "vrbb", "description": ""}, "t90": {"templateType": "anything", "group": "Ungrouped variables", "definition": "13.36", "name": "t90", "description": ""}, "tss": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ssq-tot^2/(m*n),2)", "name": "tss", "description": ""}, "mu": {"templateType": "anything", "group": 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"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", 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"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}], "name": "LSD and Tukey yardsticks, and two-way ANOVA", "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>", "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/"}]}