// Numbas version: exam_results_page_options
{"name": "Chi-square test: contingency tables", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "<p>\\(H_0:\\) Sporting activity and age profile are independent of one another.</p>\n<p>\\(H_1:\\)&nbsp;Sporting activity and age profile are not independent of one another.</p>\n<p></p>\n<p>The observed values are given below.</p>\n<table border=\"black\" style=\"border-color: black;\">\n<tbody>\n<tr>\n<td></td>\n<td>\\(12\\le age&lt;15\\)</td>\n<td>\\(15\\le age&lt;18\\)</td>\n<td>\\(18\\le age&lt;21\\)</td>\n<td><strong>Row Sum</strong></td>\n</tr>\n<tr>\n<td>Very Active</td>\n<td>{n11}</td>\n<td>{n12}</td>\n<td>{n13}</td>\n<td>\\(\\simplify{{n11}+{n12}+{n13}}\\)</td>\n</tr>\n<tr>\n<td>Active</td>\n<td>{n21}</td>\n<td>{n22}</td>\n<td>{n23}</td>\n<td>\\(\\simplify{{n21}+{n22}+{n23}}\\)</td>\n</tr>\n<tr>\n<td>Inactive</td>\n<td>{n31}</td>\n<td>{n32}</td>\n<td>{n33}</td>\n<td>\\(\\simplify{{n31}+{n32}+{n33}}\\)</td>\n</tr>\n<tr>\n<td><strong>Column Sum</strong></td>\n<td>\\(\\simplify{{n11}+{n21}+{n31}}\\)</td>\n<td>\\(\\simplify{{n12}+{n22}+{n32}}\\)</td>\n<td>\\(\\simplify{{n13}+{n23}+{n33}}\\)</td>\n<td></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>The expected values can be calculated using the formula:&nbsp;</p>\n<p style=\"padding-left: 90px;\">\\(e_{i,j}=\\frac{(sum\\,of\\,row\\,i)*(sum\\,of\\,column\\,j)}{Overall\\,total\\,sum}\\)</p>\n<p style=\"padding-left: 90px;\"></p>\n<table border=\"black\" style=\"margin-left: auto; margin-right: auto;\">\n<tbody style=\"padding-left: 60px;\">\n<tr style=\"padding-left: 60px;\">\n<td style=\"padding-left: 60px;\"></td>\n<td style=\"padding-left: 60px;\">\\(12\\le age&lt;15\\)</td>\n<td style=\"padding-left: 60px;\">\\(15\\le age&lt;18\\)</td>\n<td style=\"padding-left: 60px;\">\\(18\\le age&lt;21\\)</td>\n</tr>\n<tr style=\"padding-left: 60px;\">\n<td style=\"padding-left: 60px;\">Very Active</td>\n<td style=\"padding-left: 60px; text-align: center;\">{e11}</td>\n<td style=\"padding-left: 60px; text-align: center;\">{e12}</td>\n<td style=\"padding-left: 60px; text-align: center;\">{e13}</td>\n</tr>\n<tr style=\"padding-left: 60px;\">\n<td style=\"padding-left: 60px;\">Active</td>\n<td style=\"padding-left: 60px; text-align: center;\">{e21}</td>\n<td style=\"padding-left: 60px; text-align: center;\">{e22}</td>\n<td style=\"padding-left: 60px; text-align: center;\">{e23}</td>\n</tr>\n<tr style=\"padding-left: 60px;\">\n<td style=\"padding-left: 60px;\">Inactive</td>\n<td style=\"padding-left: 60px; text-align: center;\">{e31}</td>\n<td style=\"padding-left: 60px; text-align: center;\">{e32}</td>\n<td style=\"padding-left: 60px; text-align: center;\">{e33}</td>\n</tr>\n</tbody>\n</table>\n<p><br/>The formula for the t-statistic: &nbsp;</p>\n<p style=\"padding-left: 60px;\">\\(\\chi^2=\\sum{\\frac{(obs-exp)^2}{exp}}\\)</p>\n<p style=\"padding-left: 60px;\">\\(\\chi^2=\\frac{(\\var{n11}-\\var{e11})^2}{\\var{e11}}+\\frac{(\\var{n12}-\\var{e12})^2}{\\var{e12}}+\\frac{(\\var{n13}-\\var{e13})^2}{\\var{e13}}+\\frac{(\\var{n21}-\\var{e21})^2}{\\var{e21}}+\\frac{(\\var{n22}-\\var{e22})^2}{\\var{e22}}+\\frac{(\\var{n23}-\\var{e23})^2}{\\var{e23}}+\\frac{(\\var{n31}-\\var{e31})^2}{\\var{e31}}+\\frac{(\\var{n32}-\\var{e32})^2}{\\var{e32}}+\\frac{(\\var{n33}-\\var{e33})^2}{\\var{e33}}\\)</p>\n<p style=\"padding-left: 60px;\">\\(\\chi^2=\\var{test_statistic}\\)</p>\n<p>The \\(\\chi^2\\) table values will be for a two-tailed test having \\((r-1)*(c-1)=2*2=4\\)&nbsp;degrees of freedom. We only need to consider the upper value.</p>\n<p style=\"padding-left: 60px;\">\\(\\begin{array}{r|rrrr}&amp;0.10&amp;0.05&amp;0.02\\\\\\hline3&amp;\\var{Chi_90}&amp;\\var{Chi_95}&amp;\\var{Chi_98}\\end{array}\\)</p>\n<p>Compare the test statistic with the \\(\\chi^2\\)-table values and choose your conclusion.</p>", "variable_groups": [], "ungrouped_variables": ["test_statistic", "Chi_95", "scenario", "decision_matrix", "Chi_98", "Chi_90", "n11", "n12", "n13", "n21", "n31", "n23", "n22", "n32", "n33", "r1", "r2", "r3", "e11", "e21", "e31", "e12", "e22", "e32", "e13", "e23", "e33"], "functions": {}, "extensions": ["stats"], "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question", "statement": "<p>A social scientist was interested in the relationship between sporting activity and age.&nbsp;</p>\n<p>People in different age groups&nbsp;were asked to categorise their involvement in sports under three headings:</p>\n<p style=\"padding-left: 30px;\"><em>Very Active</em> if they were members of a sports team</p>\n<p style=\"padding-left: 30px;\"><em>Active</em> if they played sports on a regular basis but not a member of a sports team.</p>\n<p style=\"padding-left: 30px;\"><em>Inactive</em> if they rarely or never played sports.</p>\n<p>The data collected is presented below:</p>\n<table style=\"margin-left: auto; margin-right: auto;\" border=\"black\">\n<tbody style=\"padding-left: 60px;\">\n<tr style=\"padding-left: 60px;\">\n<td style=\"padding-left: 60px;\"></td>\n<td style=\"padding-left: 60px;\">\\(12\\le age&lt;15\\)</td>\n<td style=\"padding-left: 60px;\">\\(15\\le age&lt;18\\)</td>\n<td style=\"padding-left: 60px;\">\\(18\\le age&lt;21\\)</td>\n</tr>\n<tr style=\"padding-left: 60px;\">\n<td style=\"padding-left: 60px;\">Very Active</td>\n<td style=\"padding-left: 60px; text-align: center;\">{n11}</td>\n<td style=\"padding-left: 60px; text-align: center;\">{n12}</td>\n<td style=\"padding-left: 60px; text-align: center;\">{n13}</td>\n</tr>\n<tr style=\"padding-left: 60px;\">\n<td style=\"padding-left: 60px;\">Active</td>\n<td style=\"padding-left: 60px; text-align: center;\">{n21}</td>\n<td style=\"padding-left: 60px; text-align: center;\">{n22}</td>\n<td style=\"padding-left: 60px; text-align: center;\">{n23}</td>\n</tr>\n<tr style=\"padding-left: 60px;\">\n<td style=\"padding-left: 60px;\">Inactive</td>\n<td style=\"padding-left: 60px; text-align: center;\">{n31}</td>\n<td style=\"padding-left: 60px; text-align: center;\">{n32}</td>\n<td style=\"padding-left: 60px; text-align: center;\">{n33}</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Test&nbsp;the hypothesis that Sporting activity and age profile are independent of one another.</p>\n<p></p>", "preamble": {"js": "", "css": ""}, "tags": [], "parts": [{"showFeedbackIcon": true, "marks": 0, "scripts": {}, "prompt": "<p>Enter the expected frequencies correct to one decimal place:</p>\n<p style=\"padding-left: 60px;\"></p>\n<table style=\"margin-left: 60px;\" border=\"black\">\n<tbody style=\"padding-left: 60px;\">\n<tr style=\"padding-left: 60px;\">\n<td style=\"padding-left: 60px;\"></td>\n<td style=\"padding-left: 60px; text-align: center;\">\\(12\\le age&lt;15\\)</td>\n<td style=\"padding-left: 60px; text-align: center;\">\\(15\\le age&lt;18\\)</td>\n<td style=\"padding-left: 60px; text-align: center;\">\\(18\\le age&lt;21\\)</td>\n</tr>\n<tr style=\"padding-left: 60px;\">\n<td style=\"padding-left: 60px;\">Very Active</td>\n<td style=\"padding-left: 60px; text-align: center;\">[[0]]</td>\n<td style=\"padding-left: 60px; text-align: center;\">[[1]]</td>\n<td style=\"padding-left: 60px; text-align: center;\">[[2]]</td>\n</tr>\n<tr style=\"padding-left: 60px;\">\n<td style=\"padding-left: 60px;\">Active</td>\n<td style=\"padding-left: 60px; text-align: center;\">[[3]]</td>\n<td style=\"padding-left: 60px; text-align: center;\">[[4]]</td>\n<td style=\"padding-left: 60px; text-align: center;\">[[5]]</td>\n</tr>\n<tr style=\"padding-left: 60px;\">\n<td style=\"padding-left: 60px;\">Inactive</td>\n<td style=\"padding-left: 60px; text-align: center;\">[[6]]</td>\n<td style=\"padding-left: 60px; text-align: center;\">[[7]]</td>\n<td style=\"padding-left: 60px; text-align: center;\">[[8]]</td>\n</tr>\n</tbody>\n</table>\n<p>Enter the value for the appropriate test statistic: \\(\\chi^2\\)&nbsp;=&nbsp;[[9]]</p>", "type": "gapfill", "gaps": [{"precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "strictPrecision": false, "maxValue": "{e11}", "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precision": "1", "allowFractions": false, "minValue": "{e11}", "precisionPartialCredit": 0, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "strictPrecision": false, "maxValue": "{e12}", "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precision": "1", "allowFractions": false, "minValue": "{e12}", "precisionPartialCredit": 0, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "strictPrecision": false, "maxValue": "{e13}", "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precision": "1", "allowFractions": false, "minValue": "{e13}", "precisionPartialCredit": 0, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "strictPrecision": false, "maxValue": "{e21}", "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precision": "1", "allowFractions": false, "minValue": "{e21}", "precisionPartialCredit": 0, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "strictPrecision": false, "maxValue": "{e22}", "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": "2", "precision": "1", "allowFractions": false, "minValue": "{e22}", "precisionPartialCredit": 0, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "strictPrecision": false, "maxValue": "{e23}", "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precision": "1", "allowFractions": false, "minValue": "{e23}", "precisionPartialCredit": 0, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"precisionMessage": "<p>{e11}You have not given your answer to the correct precision.</p>", "correctAnswerStyle": "plain", "scripts": {}, "strictPrecision": false, "maxValue": "{e31}", "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precision": "1", "allowFractions": false, "minValue": "{e31}", "precisionPartialCredit": 0, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "strictPrecision": false, "maxValue": "{e32}", "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precision": "1", "allowFractions": false, "minValue": "{e32}", "precisionPartialCredit": 0, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "strictPrecision": false, "maxValue": "{e33}", "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precision": "1", "allowFractions": false, "minValue": "{e33}", "precisionPartialCredit": 0, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "strictPrecision": false, "maxValue": "test_statistic", "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precision": "1", "allowFractions": false, "minValue": "test_statistic", "precisionPartialCredit": 0, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true}, {"choices": ["<p>Reject the Null&nbsp;hypothesis that sporting activity and age profile are independent of one another.</p>", "<p>Reject the Null Hypothesis at the 5% significance level but accept the Null Hypothesis at the 2% significance level and conclude that&nbsp;sporting activity and age profile are independent of one another.</p>", "<p>Reject the Null Hypothesis at the 10% significance level but accept the Null Hypothesis at the 5% significance level and conclude that&nbsp;sporting activity and age profile are independent of one another.</p>", "<p>Accept the Null Hypothesis at the 10% significance level and conclude that&nbsp;sporting activity and age profile are independent of one another.</p>"], "maxMarks": "3", "scripts": {}, "minMarks": "3", "prompt": "<p>Having compared your test statistic with the table values for a two-tailed \\(\\chi^2\\)-test, select one&nbsp;of the following answers that best describes your conclusion.</p>", "displayType": "radiogroup", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "shuffleChoices": false, "showFeedbackIcon": true, "marks": 0, "displayColumns": "1", "matrix": "decision_matrix", "variableReplacements": [{"part": "p0", "must_go_first": false, "variable": ""}]}], "rulesets": {}, "variables": {"n11": {"templateType": "randrange", "definition": "random(120..130#1)", "description": "", "group": "Ungrouped variables", "name": "n11"}, "Chi_95": {"templateType": "number", "definition": "11.14", "description": "", "group": "Ungrouped variables", "name": "Chi_95"}, "e11": {"templateType": "anything", "definition": "precround(250*{r1}/690,1)", "description": "", "group": "Ungrouped variables", "name": "e11"}, "n32": {"templateType": "anything", "definition": "240-{n12}-{n22}", "description": "", "group": "Ungrouped variables", "name": "n32"}, "n23": {"templateType": "randrange", "definition": "random(86..95#1)", "description": "", "group": "Ungrouped variables", "name": "n23"}, "test_statistic": {"templateType": "anything", "definition": "precround(({n11}-{e11})^2/{e11}+({n12}-{e12})^2/{e12}+({n13}-{e13})^2/{e13}+({n21}-{e21})^2/{e21}+({n22}-{e22})^2/{e22}+({n23}-{e23})^2/{e23}+({n31}-{e31})^2/{e31}+({n32}-{e32})^2/{e32}+({n33}-{e33})^2/{e33},1)", "description": "", "group": "Ungrouped variables", "name": "test_statistic"}, "e23": {"templateType": "anything", "definition": "precround(200*{r2}/690,1)", "description": "", "group": "Ungrouped variables", "name": "e23"}, "e31": {"templateType": "anything", "definition": "precround(250*{r3}/690,1)", "description": "", "group": "Ungrouped variables", "name": "e31"}, "Chi_90": {"templateType": "number", "definition": "9.488", "description": "", "group": "Ungrouped variables", "name": "Chi_90"}, "e22": {"templateType": "anything", "definition": "precround(240*{r2}/690,1)", "description": "", "group": "Ungrouped variables", "name": "e22"}, "r2": {"templateType": "anything", "definition": "{n21}+{n22}+{n23}", "description": "", "group": "Ungrouped variables", "name": "r2"}, "r3": {"templateType": "anything", "definition": "{n31}+{n32}+{n33}", "description": "", "group": "Ungrouped variables", "name": "r3"}, "Chi_98": {"templateType": "number", "definition": "13.28", "description": "", "group": "Ungrouped variables", "name": "Chi_98"}, "n31": {"templateType": "anything", "definition": "250-{n11}-{n21}", "description": "", "group": "Ungrouped variables", "name": "n31"}, "e13": {"templateType": "anything", "definition": "precround(200*{r1}/690,1)", "description": "", "group": "Ungrouped variables", "name": "e13"}, "decision_matrix": {"templateType": "anything", "definition": "[\n  [1,0,0,0],\n  [0,1,0,0],\n  [0,0,1,0],\n  [0,0,0,1]\n][scenario]", "description": "", "group": "Ungrouped variables", "name": "decision_matrix"}, "n12": {"templateType": "randrange", "definition": "random(90..99#1)", "description": "", "group": "Ungrouped variables", "name": "n12"}, "e21": {"templateType": "anything", "definition": "precround(250*{r2}/690,1)", "description": "", "group": "Ungrouped variables", "name": "e21"}, "r1": {"templateType": "anything", "definition": "{n11}+{n12}+{n13}", "description": "", "group": "Ungrouped variables", "name": "r1"}, "e12": {"templateType": "anything", "definition": "precround(240*{r1}/690,1)", "description": "", "group": "Ungrouped variables", "name": "e12"}, "n22": {"templateType": "randrange", "definition": "random(112..130#1)", "description": "", "group": "Ungrouped variables", "name": "n22"}, "n21": {"templateType": "randrange", "definition": "random(95..104#1)", "description": "", "group": "Ungrouped variables", "name": "n21"}, "scenario": {"templateType": "anything", "definition": "sum(map(abs(test_statistic)<x,x,[Chi_90,Chi_95,Chi_98]))", "description": "", "group": "Ungrouped variables", "name": "scenario"}, "e33": {"templateType": "anything", "definition": "precround(200*{r3}/690,1)", "description": "", "group": "Ungrouped variables", "name": "e33"}, "n13": {"templateType": "randrange", "definition": "random(75..80#1)", "description": "", "group": "Ungrouped variables", "name": "n13"}, "n33": {"templateType": "anything", "definition": "200-{n13}-{n23}", "description": "", "group": "Ungrouped variables", "name": "n33"}, "e32": {"templateType": "anything", "definition": "precround(240*{r3}/690,1)", "description": "", "group": "Ungrouped variables", "name": "e32"}}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "name": "Chi-square test: contingency tables", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}