// Numbas version: exam_results_page_options
{"name": "Two sample Z-test on proportions", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Two sample Z-test on proportions", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>In 1990, of \\(\\var{n1}\\) men between the ages of 20 and 34 years old, \\(\\var{prop1}\\)&nbsp;were found to be overweight.</p>\n<p>Whereas, in 2010, of \\(\\var{n2}\\) men between the ages of 20 and 34 years old, \\(\\var{prop2}\\)&nbsp;were found to be overweight.</p>\n<p>Does the data provide sufficient evidence to conclude that for men between the ages of 20 and 34 years old, a higher percentage were overweight in 2010 than twenty years earlier?</p>\n<p></p>\n<p></p>", "advice": "<p>\\(H_0:\\) &nbsp;\\(p_1=p_2\\).</p>\n<p>\\(H_1:\\) \\(p_1 \\le p_2\\).</p>\n<p></p>\n<p>Given a sample of size \\(n\\) recall:</p>\n<p style=\"padding-left: 60px;\">the formula for the sample proportion: &nbsp; &nbsp;\\(\\overline{p}=\\frac{{x}}{n}\\) where \\(x\\) is the number of observations</p>\n<p style=\"padding-left: 60px;\">\\(p_1=\\frac{\\var{prop1}}{\\var{n1}}\\)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(p_2=\\frac{\\var{prop2}}{\\var{n2}}\\)</p>\n<p style=\"padding-left: 60px;\">the pooled proportion \\(\\hat{p}=\\frac{x_1+x_2}{n_1+n_2}=\\frac{\\var{prop1}+\\var{prop2}}{\\var{n1}+\\var{n2}}=\\var{pooled_p}\\)</p>\n<p style=\"padding-left: 60px;\">the formula for the Z-statistic: &nbsp; \\(Z=\\frac{{p_1}-{p_2}}{\\sqrt{{\\hat{p}(1-\\hat{p})}(\\frac{1}{n_1}+\\frac{1}{n_2})}}\\)</p>\n<p style=\"padding-left: 60px;\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(Z=\\frac{\\var{p1}-\\var{p2}}{\\sqrt{{(\\var{pooled_p})(\\simplify{1-{pooled_p}})(\\frac{1}{\\var{n1}}+\\frac{1}{\\var{n2}})}}}=\\var{test_statistic}\\)</p>\n<p>The Z-table values for a one-tailed test are given below.&nbsp;</p>\n<p style=\"padding-left: 90px;\">significance&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 10%&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 5%&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;1%</p>\n<p style=\"padding-left: 90px;\">&nbsp; &nbsp; &nbsp; limits&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\(-1.28\\)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(-1.65\\)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(-1.96\\)</p>\n<p>Compare the test statistic with the Z-table values and choose your conclusion.</p>", "rulesets": {}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p2": {"name": "p2", "group": "Ungrouped variables", "definition": "precround({prop2}/{n2},2)", "description": "", "templateType": "anything", "can_override": false}, "prop2": {"name": "prop2", "group": "Ungrouped variables", "definition": "random(120 .. 150#5)", "description": "", "templateType": "randrange", "can_override": false}, "pooled_p": {"name": "pooled_p", "group": "Ungrouped variables", "definition": "precround(({prop1}+{prop2})/({n1}+{n2}),2)", "description": "", "templateType": "anything", "can_override": false}, "prop1": {"name": "prop1", "group": "Ungrouped variables", "definition": "random(100 .. 125#1)", "description": "", "templateType": "randrange", "can_override": false}, "test_statistic": {"name": "test_statistic", "group": "Ungrouped variables", "definition": "precround((p1-p2)/sqrt(pooled_p*(1-pooled_p)*(1/{n1}+1/{n2})),2)", "description": "", "templateType": "anything", "can_override": false}, "n1": {"name": "n1", "group": "Ungrouped variables", "definition": "random(360 .. 400#10)", "description": "", "templateType": "randrange", "can_override": true}, "Z90": {"name": "Z90", "group": "Ungrouped variables", "definition": "1.28", "description": "", "templateType": "number", "can_override": false}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "precround({prop1}/{n1},2)", "description": "", "templateType": "anything", "can_override": false}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "random(420 .. 470#10)", "description": "", "templateType": "randrange", "can_override": false}, "Z95": {"name": "Z95", "group": "Ungrouped variables", "definition": "1.65", "description": "", "templateType": "number", "can_override": false}, "atvalue": {"name": "atvalue", "group": "Ungrouped variables", "definition": "abs(test_statistic)", "description": "", "templateType": "anything", "can_override": false}, "Z99": {"name": "Z99", "group": "Ungrouped variables", "definition": "1.96", "description": "", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "if(atvalue>=Z99,[1,0,0,0],if(atvalue>=Z95,[0,1,0,0],if(atvalue>=Z90,[0,0,1,0],[0,0,0,1])))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["Z95", "Z90", "test_statistic", "n1", "p1", "prop1", "n2", "prop2", "p2", "pooled_p", "atvalue", "Z99", "v"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Enter the value of the pooled&nbsp;proportion: \\(p=\\) [[1]]</p>\n<p>Enter the value for the appropriate test statistic: Z =&nbsp;[[0]]</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "sample_mean_2", "part": "p0", "must_go_first": true}, {"variable": "sample_stdev_2", "part": "p0", "must_go_first": true}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "test_statistic", "maxValue": "test_statistic", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{pooled_p}", "maxValue": "{pooled_p}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "test_statistic", "part": "p0g0", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Having compared your test statistic with the table values for a one-tailed Z-test, select one&nbsp;of the foll owing conclusions that best describes your conclusion.</p>", "minMarks": "2", "maxMarks": "2", "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": "1", "showCellAnswerState": true, "choices": ["<p>Reject the Null Hypothesis and conclude conclude that the proportion of men in 2010 that are overweight is greater than the proportion of men in 1990 that were overweight.&nbsp;</p>", "<p>Reject the Null Hypothesis at the 5% significance level but accept the Null Hypothesis at the 1% significance level and conclude that there is no significant difference between the proportion of men that are overweight in 2010 and the proportion of men that&nbsp;were overweight in 1990. &nbsp; &nbsp;&nbsp;</p>", "<p>Reject the Null Hypothesis at the 10% significance level but accept the Null Hypothesis at the 5% significance level and conclude that there is no significant difference between the proportion of men that are overweight in 2010 and the proportion of men that&nbsp;were overweight in 1990.</p>", "<p>Accept the Null Hypothesis at the 10% significance level and conclude that there is no significant difference between the proportion of men that are overweight in 2010 and the proportion of men that&nbsp;were overweight in 1990.</p>"], "matrix": "v"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}