// Numbas version: finer_feedback_settings
{"name": "LSD and Tukey yardsticks, and two-way ANOVA - SES2003", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "name": "LSD and Tukey yardsticks, and two-way ANOVA - SES2003", "parts": [{"sortAnswers": false, "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>", "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "gaps": [{"correctAnswerFraction": false, "maxValue": "sqrms+tol", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "minValue": "sqrms-tol", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacements": []}, {"displayColumns": 1, "displayType": "radiogroup", "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>", "maxMarks": 0, "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>"], "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "matrix": "v", "customMarkingAlgorithm": "", "shuffleChoices": false, "minMarks": 0, "extendBaseMarkingAlgorithm": true, "showCellAnswerState": true, "variableReplacements": []}, {"displayColumns": 0, "displayType": "radiogroup", "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 creams?</p>", "maxMarks": 0, "choices": ["Very strong", "Strong", "Moderate", "Weak", "None"], "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "matrix": "v", "customMarkingAlgorithm": "", "shuffleChoices": false, "minMarks": 0, "extendBaseMarkingAlgorithm": true, "showCellAnswerState": true, "variableReplacements": []}, {"displayColumns": 1, "displayType": "radiogroup", "prompt": "<p>Hence what is your decision based on the above ANOVA analysis?</p>", "maxMarks": 0, "choices": ["We reject the null hypothesis at the $0.1\\%$ level", "We reject the null hypothesis at the $1\\%$ level.", "We reject the null hypothesis at the $5\\%$ level.", "We do not reject the null hypothesis but more investigation is needed.", "We do not reject the null hypothesis."], "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "matrix": "v", "customMarkingAlgorithm": "", "shuffleChoices": false, "minMarks": 0, "extendBaseMarkingAlgorithm": true, "showCellAnswerState": true, "variableReplacements": []}, {"sortAnswers": false, "prompt": "<p style=\"text-align: center;\"><strong>Using the yardsticks</strong></p>\n<p>Enter the sample means for the 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>", "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "gaps": [{"correctAnswerFraction": false, "maxValue": "me[0]+tol", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "minValue": "me[0]-tol", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 0.5, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}, {"correctAnswerFraction": false, "maxValue": "me[1]+tol", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "minValue": "me[1]-tol", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 0.5, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}, {"correctAnswerFraction": false, "maxValue": "me[2]+tol", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "minValue": "me[2]-tol", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 0.5, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}, {"correctAnswerFraction": false, "maxValue": "me[3]+tol", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "minValue": "me[3]-tol", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 0.5, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}, {"correctAnswerFraction": false, "maxValue": "lsd+tol", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "minValue": "lsd-tol", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}, {"correctAnswerFraction": false, "maxValue": "tukey+tol", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "minValue": "tukey-tol", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacements": []}, {"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 creams.</p>\n<p>&nbsp;</p>", "maxMarks": 0, "choices": ["$W$ and $X$", "$W$ and $Y$", "$W$ and $Z$", "$X$ and $Y$", "$X$ and $Z$", "$Y$ and $Z$"], "maxAnswers": 0, "showFeedbackIcon": true, "minAnswers": 0, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "matrix": "w", "marks": 0, "minMarks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "displayType": "radiogroup", "unitTests": [], "scripts": {}, "showCorrectAnswer": true, "warningType": "none", "type": "m_n_x", "layout": {"type": "all", "expression": ""}, "answers": ["<strong>Definite Significant Difference</strong>", "<strong>Possible Significant Difference</strong>", "<strong>No Significant Difference</strong>"], "shuffleChoices": false, "showCellAnswerState": true, "shuffleAnswers": false}], "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": ["stats"], "functions": {"yeaornay": {"language": "jme", "definition": "if(n=1, \"Yes\", \"No\")", "type": "string", "parameters": [["n", "number"]]}}, "advice": "<h2 style=\"text-align: center;\">&nbsp;<strong>Using the Yardsticks</strong></h2>\n<p>The mean values for each 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>\n<th align=\"left\">W</th>\n<td>$\\var{me[0]}$</td>\n</tr>\n<tr>\n<th align=\"left\">X</th>\n<td>$\\var{me[1]}$</td>\n</tr>\n<tr>\n<th style=\"text-align: left;\">Y</th>\n<td style=\"text-align: left;\">$\\var{me[2]}$</td>\n</tr>\n<tr>\n<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 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 creams:</p>\n<table style=\"float: left;\" border=\"1\">\n<tbody>\n<tr>\n<th align=\"left\">Pairs of creams</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 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>\n<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>\n<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>\n<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>\n<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>\n<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>", "rulesets": {}, "metadata": {"description": "<p>LSD and Tukey yardsticks on five treatments. Also two-way Anova test on same set of data.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>To test the effectiveness of pain-relief creams on back pain, five volunteers (A-E) suffering with back pain tried each of four creams (W-Z). The degree of pain relief was measured on a 0-50 scale (higher figures indicate higher levels of pain relief). The results are given below with some totals:</p>\n<p></p>\n<table height=\"186\" style=\"width: 567px;\">\n<tbody>\n<tr>\n<td style=\"vertical-align: middle; border-bottom: 1px solid; border-right: 1px solid; width: 50px;\">&nbsp;</td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid; width: 67px;\"><b>A</b></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid; width: 72px;\"><b>B</b></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid; width: 71px;\"><b>C</b></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid; width: 75px;\"><b>D</b></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid; width: 76px;\"><b>E</b><strong></strong></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid; border-left: 1px solid; width: 51px;\"><strong>Totals</strong></td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid; width: 50px;\"><strong>W</strong></td>\n<td style=\"width: 67px;\">{r[0][0]}</td>\n<td style=\"width: 72px;\">{r[1][0]}</td>\n<td style=\"width: 71px;\">{r[2][0]}</td>\n<td style=\"width: 75px;\">{r[3][0]}</td>\n<td style=\"width: 76px;\">{r[4][0]}</td>\n<td style=\"vertical-align: middle; border-left: 1px solid; width: 51px;\">{cols[0]}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid; width: 50px;\"><strong>X</strong></td>\n<td style=\"width: 67px;\">{r[0][1]}</td>\n<td style=\"width: 72px;\">{r[1][1]}</td>\n<td style=\"width: 71px;\">{r[2][1]}</td>\n<td style=\"width: 75px;\">{r[3][1]}</td>\n<td style=\"width: 76px;\">{r[4][1]}</td>\n<td style=\"vertical-align: middle; border-left: 1px solid; width: 51px;\">{cols[1]}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid; width: 50px;\"><strong>Y</strong></td>\n<td style=\"width: 67px;\">{r[0][2]}</td>\n<td style=\"width: 72px;\">{r[1][2]}</td>\n<td style=\"width: 71px;\">{r[2][2]}</td>\n<td style=\"width: 75px;\">{r[3][2]}</td>\n<td style=\"width: 76px;\">{r[4][2]}</td>\n<td style=\"vertical-align: middle; border-left: 1px solid; width: 51px;\">{cols[2]}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid; width: 50px;\"><strong>Z</strong></td>\n<td style=\"width: 67px;\">{r[0][3]}</td>\n<td style=\"width: 72px;\">{r[1][3]}</td>\n<td style=\"width: 71px;\">{r[2][3]}</td>\n<td style=\"width: 75px;\">{r[3][3]}</td>\n<td style=\"width: 76px;\">{r[4][3]}</td>\n<td style=\"vertical-align: middle; border-left: 1px solid; width: 51px;\">{cols[3]}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid; border-top: 1px solid; width: 50px;\"><strong>Totals</strong></td>\n<td style=\"vertical-align: middle; border-top: 1px solid; width: 67px;\">{t[0]}</td>\n<td style=\"vertical-align: middle; border-top: 1px solid; width: 72px;\">{t[1]}</td>\n<td style=\"vertical-align: middle; border-top: 1px solid; width: 71px;\">{t[2]}</td>\n<td style=\"vertical-align: middle; border-top: 1px solid; width: 75px;\">{t[3]}</td>\n<td style=\"vertical-align: middle; border-top: 1px solid; width: 76px;\">{t[4]}</td>\n<td style=\"vertical-align: middle; border-top: 1px solid; border-left: 1px solid; width: 51px;\">{tot}</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<ul>\n<li>Test the null-hypothesis using two-way ANOVA that the creams are equally effective.</li>\n</ul>\n<ul>\n<li>Write down the sample mean for each 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>", "variables": {"rt": {"templateType": "anything", "description": "", "definition": "list(transpose(matrix(r)))", "name": "rt", "group": "Ungrouped variables"}, "sqrms": {"templateType": "anything", "description": "", "definition": "precround(sqrt(rs),2)", "name": "sqrms", "group": "Ungrouped variables"}, "me": {"templateType": "anything", "description": "", "definition": "map(mean(r1[x]),x,0..3)", 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