// Numbas version: exam_results_page_options
{"name": "Simon's copy of Perform chi-squared test on 2D frequency table", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"ch11": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ch11", "definition": "precround((f1-e11)^2/e11,3)"}, "t90": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "t90", "definition": "13.36"}, "r5": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "r5", "definition": "a2+b2+c2"}, "ch7": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ch7", "definition": "precround((a1-e7)^2/e7,3)"}, "f": {"group": "Ungrouped variables", "templateType": "anything", 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"switch(pval=0,'no',pval=1,'slight',pval=2,'moderate','strong')"}, "e14": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "e14", "definition": "precround(r5*col2/tot,3)"}, "e5": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "e5", "definition": "precround(r2*col2/tot,3)"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c1", "definition": "random(10..24)"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c", "definition": "random(10..22)"}, "e6": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "e6", "definition": "precround(r2*col3/tot,3)"}, "ch15": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ch15", "definition": "precround((c2-e15)^2/e15,3)"}, "ch6": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ch6", "definition": "precround((t-e6)^2/e6,3)"}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b1", "definition": "random(10..22)"}, "e7": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "e7", "definition": "precround(r3*col1/tot,3)"}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "d1", "definition": "random(12..20)"}, "t95": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "t95", "definition": "15.51"}, "col1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "col1", "definition": "a+d+a1+d1+a2"}, "e11": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "e11", "definition": "precround(r4*col2/tot,3)"}}, "advice": "<h2></h2>\n<p><strong>(a)</strong></p>\n<p>The expected frequencies are given by replacing a value in the table by the expected value:</p>\n<p>\\[E = \\frac{\\textrm{row total} \\times \\textrm{column total}}{\\textrm{overall total}}\\]</p>\n<p>For example, the Excellent category for Marketing &amp; Management lies in the second row (with sum $\\var{r2}$) and the first column (with sum $\\var{col1}$).</p>\n<p>So the expected frequency of Excellent Marketing &amp; Management students is:<br/>\\[E = \\simplify[]{({r2}*{col1})/({tot})} = \\var{e4}\\]</p>\n<p>Hence we get the following table of expected frequencies:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"vertical-align: middle; border-bottom: 1px solid; border-right: 1px solid;\"><span class=\"caps\">EXPECTED</span> <span class=\"caps\">FREQUENCIES</span></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>Excellent</strong></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>Strong</strong></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>Average</strong></td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Marketing</strong></td>\n<td>{e1}</td>\n<td>{e2}</td>\n<td>{e3}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Marketing &amp; Management</strong></td>\n<td>{e4}</td>\n<td>{e5}</td>\n<td>{e6}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Business Management</strong></td>\n<td>{e7}</td>\n<td>{e8}</td>\n<td>{e9}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Accounting &amp; Finance</strong></td>\n<td>{e10}</td>\n<td>{e11}</td>\n<td>{e12}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Mathematics</strong></td>\n<td>{e13}</td>\n<td>{e14}</td>\n<td>{e15}</td>\n</tr>\n</tbody>\n</table>\n<p>In order to test to see if there is an association we compare this table with the table of observed values and calculate the test statistic by looking at</p>\n<p>\\[\\chi^2 = \\sum \\frac{(O - E)^2}{E}\\]</p>\n<p>Calculating the values for Business Management and Mathematics, all to 3 decimal places, you should obtain:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"vertical-align: middle; border-bottom: 1px solid; border-right: 1px solid;\">$\\frac{(O - E)^2}{E}$</td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>Excellent</strong></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>Strong</strong></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>Average</strong></td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Marketing</strong></td>\n<td>{ch1}</td>\n<td>{ch2}</td>\n<td>{ch3}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Marketing &amp; Management</strong></td>\n<td>{ch4}</td>\n<td>{ch5}</td>\n<td>{ch6}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Business Management</strong></td>\n<td>{ch7}</td>\n<td>{ch8}</td>\n<td>{ch9}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Accounting &amp; Finance</strong></td>\n<td>{ch10}</td>\n<td>{ch11}</td>\n<td>{ch12}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Mathematics</strong></td>\n<td>{ch13}</td>\n<td>{ch14}</td>\n<td>{ch15}</td>\n</tr>\n</tbody>\n</table>\n<p>To find the $\\chi^2$ statistic you sum these fifteen values to get:</p>\n<p>\\[\\begin{eqnarray} \\chi^2 &amp;=&amp; \\var{ch1} + \\var{ch2} + \\var{ch3} + \\var{ch4} + \\var{ch5} +\\\\ &amp;&amp; \\var{ch6} + \\var{ch7} + \\var{ch8} + \\var{ch9} + \\var{ch10} +\\\\ &amp;&amp; \\var{ch11} + \\var{ch12} + \\var{ch13} + \\var{ch14} + \\var{ch15} \\\\ &amp;=&amp; \\var{ch}. \\end{eqnarray}\\]</p>\n<h2></h2>\n<p></p>\n<p><strong>(b)</strong></p>\n<p>The degrees of freedom is given by:</p>\n<p>\\[\\nu = (\\textrm{no. of rows} - 1) \\times (\\textrm{no. of columns} - 1) = 4 \\times 2 = 8\\]</p>\n<p>The following are the critical values for $\\nu = 8$:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid; border-bottom: 1px solid;\"><strong>$p$-value</strong></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>$10 \\%$</strong></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>$5 \\%$</strong></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>$1 \\%$</strong></td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Critical value</strong></td>\n<td>$\\var{t90}$</td>\n<td>$\\var{t95}$</td>\n<td>$\\var{t99}$</td>\n</tr>\n</tbody>\n</table>\n<p>Comparing these values with the the test statistic we see that the $p$-value {pResult}.</p>\n<h2></h2>\n<p></p>\n<p></p>\n<p><strong>(c)</strong></p>\n<p>As the $p$-value {pResult}, there is <strong>{evi}</strong> evidence against $H_0$.</p>\n<p>Hence we <strong>{retain}</strong>.</p>\n<p>{summary}</p>", "preamble": {"css": "", "js": ""}, "tags": [], "parts": [{"customMarkingAlgorithm": "", "marks": 0, "unitTests": [], "variableReplacementStrategy": "originalfirst", "useCustomName": false, "type": "information", "showFeedbackIcon": true, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<h3>Step 1: Null hypothesis</h3>\n<p>$H_0$: There is no association between degree subject and performance.</p>\n<h3>Step 2: Alternate hypothesis</h3>\n<p>$H_1$: There <em>is</em> an association between degree subject and performance.</p>", "scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true}, {"customMarkingAlgorithm": "", "marks": 0, "unitTests": [], "variableReplacementStrategy": "originalfirst", "useCustomName": false, "type": "gapfill", "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "showCorrectAnswer": 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"variableReplacements": [], "type": "numberentry", "maxValue": "ch+0.005", "scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true}], "prompt": "<h3>Step 3: Test statistic</h3>\n<p>You are given the expected frequencies (all to $3$ decimal places) for Marketing, Marketing &amp; Management and Accounting &amp; Finance.</p>\n<p>You have to calculate the expected frequencies for Business Management and Mathematics and put them in the following table.</p>\n<p>Input each expected frequency to $3$ decimal places.</p>\n<table>\n<tbody>\n<tr>\n<td style=\"vertical-align: middle; border-bottom: 1px solid; border-right: 1px solid;\"><strong><span class=\"caps\">EXPECTED</span> FREQUENCIES</strong></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>Excellent</strong></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>Strong</strong></td>\n<td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>Average</strong></td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Marketing</strong></td>\n<td>{e1}</td>\n<td>{e2}</td>\n<td>{e3}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Marketing &amp; Management</strong></td>\n<td>{e4}</td>\n<td>{e5}</td>\n<td>{e6}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Business Management</strong></td>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Accounting &amp; Finance</strong></td>\n<td>{e10}</td>\n<td>{e11}</td>\n<td>{e12}</td>\n</tr>\n<tr>\n<td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Mathematics</strong></td>\n<td>[[3]]</td>\n<td>[[4]]</td>\n<td>[[5]]</td>\n</tr>\n</tbody>\n</table>\n<p>In order to test to see if there is an association we compare this table with the table of observed values and calculate the test statistic by finding:</p>\n<p>\\[\\chi^2 = \\sum \\frac{(O - E)^2}{E}\\]</p>\n<p>Now calculate the test statistic $\\chi^2 = \\phantom{{}}$[[6]]&nbsp;</p>\n<p></p>", "scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true}, {"customMarkingAlgorithm": "", "marks": 0, "unitTests": [], "variableReplacementStrategy": "originalfirst", "useCustomName": false, "type": "gapfill", "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "showCorrectAnswer": true, "gaps": [{"allowFractions": false, "correctAnswerFraction": false, "customMarkingAlgorithm": "", "marks": 1, "unitTests": [], "showFractionHint": true, "minValue": "8", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "type": "numberentry", "showFeedbackIcon": true, "variableReplacements": [], "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "maxValue": "8", "scripts": {}, "mustBeReduced": false, "correctAnswerStyle": "plain", "customName": "", "extendBaseMarkingAlgorithm": true}, {"minMarks": 0, "customMarkingAlgorithm": "", "marks": 0, "unitTests": [], "maxMarks": 0, "showCellAnswerState": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "type": "1_n_2", "showFeedbackIcon": true, "choices": ["<p>$p$ is bigger than $10 \\%$</p>", "<p>$p$ lies between $10 \\%$ and $5 \\%$</p>", "<p>$p$ lies between $5 \\%$ and $1 \\%$</p>", "<p>$p$ is less than $1 \\%$</p>"], "variableReplacements": [], "showCorrectAnswer": true, "displayType": "radiogroup", "matrix": "Mm", "shuffleChoices": false, "displayColumns": 1, "scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true}], "prompt": "<h3>Step 4: $p$-value range</h3>\n<p>Calculate $\\nu$, the degrees of freedom, for this test.</p>\n<p>$\\nu = \\phantom{{}}$[[0]]</p>\n<p>Use tables to find a range for your $p$-value. Choose from the options below.</p>\n<p>[[1]]</p>", "scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true}, {"customMarkingAlgorithm": "", "marks": 0, "unitTests": [], "variableReplacementStrategy": "originalfirst", "useCustomName": false, "type": "gapfill", "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "showCorrectAnswer": true, "gaps": [{"minMarks": 0, "customMarkingAlgorithm": "", "marks": 0, "unitTests": [], "maxMarks": 0, "showCellAnswerState": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "type": "1_n_2", "showFeedbackIcon": true, "choices": ["<p>None</p>", "<p>Slight</p>", "<p>Moderate</p>", "<p>Strong</p>"], "variableReplacements": [], "showCorrectAnswer": true, "displayType": "radiogroup", "matrix": "Mm", "shuffleChoices": false, "displayColumns": 0, "scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true}, {"minMarks": 0, "customMarkingAlgorithm": "", "marks": 0, "unitTests": [], "distractors": ["", ""], "maxMarks": 0, "showCellAnswerState": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "type": "1_n_2", "showFeedbackIcon": true, "choices": ["<p>Retain $H_0$</p>", "<p>Reject $H_0$</p>"], "variableReplacements": [], "showCorrectAnswer": true, "displayType": "radiogroup", "matrix": ["if(pval<2,1,0)", "if(pval>=2,1,0)"], "shuffleChoices": false, "displayColumns": 0, "scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true}, {"minMarks": 0, "customMarkingAlgorithm": "", "marks": 0, "unitTests": [], "distractors": ["", ""], "maxMarks": 0, "showCellAnswerState": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "type": "1_n_2", "showFeedbackIcon": true, "choices": ["<p>There <strong>is</strong> evidence to suggest an association between degree subject and performance.</p>", "<p>There is <strong>insufficient&nbsp;</strong>evidence to suggest an association between degree subject and performance.</p>"], "variableReplacements": [], "showCorrectAnswer": true, "displayType": "radiogroup", "matrix": ["if(pval>=2,1,0)", "if(pval<2,1,0)"], "shuffleChoices": false, "displayColumns": 1, "scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true}], "prompt": "<h3>Step 5: Conclusion</h3>\n<p>Given the $p$-value and the range you have found, what is the strength of evidence against the null hypothesis?</p>\n<p>[[0]]</p>\n<p>Your decision:</p>\n<p>[[1]]</p>\n<p>Your conclusion:</p>\n<p>[[2]]</p>", "scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true}], "ungrouped_variables": ["f1", "ch", "presult", "a", "d", "r2", "pval", "b1", "b2", "t95", "t1", "c1", "d1", "e11", "e10", "e13", "e12", "e15", "e14", "tot", "ch15", "e3", "b", "t", "r4", "ch13", "col2", "col3", "col1", "a1", "a2", "e9", "t99", "ch9", "e5", "e4", "e7", "e6", "e1", "c2", "e2", "ch12", "evi", "c", "ch11", "r1", "ch14", "f", "mm", "ch10", "r3", "summary", "t90", "ch1", "ch2", "ch3", "ch4", "ch5", "ch6", "ch7", "ch8", "retain", "r5", "e8"], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>The human resources department of a large finance company is attempting to determine if an employee&rsquo;s performance is influenced by their undergraduate degree subject. Personnel ratings are used to judge performance and the task is to use expected frequencies and the chi-squared statistic to test the null hypothesis that there is no association.</p>"}, "rulesets": {}, "statement": "\n<p>The human resources department of a large finance company is attempting to determine if an employee&rsquo;s performance is influenced by their undergraduate degree subject.</p>\n  <p>The 5 subjects considered are: <strong>Marketing</strong>, <strong>Marketing &amp; Management</strong>, <strong>Business Management</strong>, <strong>Accounting &amp; Finance</strong> and <strong>Mathematics</strong>.</p>\n  <p>Personnel ratings are grouped as <strong>Excellent</strong>, <strong>Strong</strong> and <strong>Average</strong>.</p>\n  <p>A recent assessment gave the following results:</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>Excellent</strong></td>\n  <td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>Strong</strong></td>\n  <td style=\"vertical-align: middle; border-bottom: 1px solid;\"><strong>Average</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>Marketing</strong></td>\n  <td>{a}</td>\n  <td>{b}</td>\n  <td>{c}</td>\n  <td style=\"vertical-align: middle; border-left: 1px solid;\">{r1}</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Marketing &amp; Management</strong></td>\n  <td>{d}</td>\n  <td>{f}</td>\n  <td>{t}</td>\n  <td style=\"vertical-align: middle; border-left: 1px solid;\">{r2}</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Business Management</strong></td>\n  <td>{a1}</td>\n  <td>{b1}</td>\n  <td>{c1}</td>\n  <td style=\"vertical-align: middle; border-left: 1px solid;\">{r3}</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Accounting &amp; Finance</strong></td>\n  <td>{d1}</td>\n  <td>{f1}</td>\n  <td>{t1}</td>\n  <td style=\"vertical-align: middle; border-left: 1px solid;\">{r4}</td>\n  </tr>\n  <tr>\n  <td style=\"vertical-align: middle; border-right: 1px solid;\"><strong>Mathematics</strong></td>\n  <td>{a2}</td>\n  <td>{b2}</td>\n  <td>{c2}</td>\n  <td style=\"vertical-align: middle; border-left: 1px solid;\">{r5}</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;\">{col1}</td>\n  <td style=\"vertical-align: middle; border-top: 1px solid;\">{col2}</td>\n  <td style=\"vertical-align: middle; border-top: 1px solid;\">{col3}</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>Test the null hypothesis that there is no association between degree subject and performance.</p>\n", "variable_groups": [], "functions": {}, "extensions": ["stats"], "name": "Simon's copy of Perform chi-squared test on 2D frequency table", "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}