// Numbas version: finer_feedback_settings
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"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"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Perform chi-squared test on 2D frequency table", "functions": {}, "showQuestionGroupNames": false, "parts": [{"type": "information", "prompt": "\n\n\n<h2>Step 1: Null hypothesis</h2>\n\n\n\n<p>$H_0$: There is no association between degree subject and performance.</p>\n\n\n\n<h2>Step 2: Alternate hypothesis</h2>\n\n\n\n<p>$H_1$: There <em>is</em> an association between degree subject and performance.</p>\n\n\n  \n", "showCorrectAnswer": true, "scripts": {}, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "e7", "minValue": "e7", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "e8", "minValue": "e8", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "e9", "minValue": "e9", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "e13", "minValue": "e13", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "e14", "minValue": "e14", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "e15", "minValue": "e15", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ch+0.006", "minValue": "ch-0.006", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<h2>Step 3: Test statistic</h2>\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>Input the test statistic to $3$ decimal places.</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "8", "minValue": "8", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"displayType": "radiogroup", "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>"], "displayColumns": 1, "distractors": ["", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "Mm", "marks": 0}], "type": "gapfill", "prompt": "\n  <h2>Step 4: $p$-value range</h2>\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>\n  \n", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["<p>None</p>", "<p>Slight</p>", "<p>Moderate</p>", "<p>Strong</p>"], "displayColumns": 0, "distractors": ["", "", "", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": "Mm", "marks": 0}, {"displayType": "radiogroup", "choices": ["<p>Retain $H_0$</p>", "<p>Reject $H_0$</p>"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": ["if(pval<2,1,0)", "if(pval>=2,1,0)"], "marks": 0}, {"displayType": "radiogroup", "choices": ["<p>There <strong>is</strong> evidence to suggest an association between degree subject and performance.</p>", "<p>There is <strong>no</strong> evidence to suggest an association between degree subject and performance.</p>"], "displayColumns": 1, "distractors": ["", ""], "shuffleChoices": false, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": ["if(pval>=2,1,0)", "if(pval<2,1,0)"], "marks": 0}], "type": "gapfill", "prompt": "\n  <h2>Step 5: Conclusion</h2>\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>\n  \n", "showCorrectAnswer": true, "marks": 0}], "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", "tags": ["checked2015", "MAS1403"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n        \t\t<p><strong>3/01/2012:</strong></p>\n        \t\t                    <p>This is the example used in the mathssample exam and was translated from the iassess MAS1403 originally. The table is not created from the inbuilt table function. Included tag sc to indicate that this can be used for other applications.</p>\n        \t\t", "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>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h2>Step 3</h2>\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>Step 4</h2>\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>Step 5: Conclusion</h2>\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>", "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/"}]}