// Numbas version: finer_feedback_settings
{"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": [{"tags": [], "functions": {}, "variables": {"pooled_p": {"group": "Ungrouped variables", "description": "", "definition": "precround(({prop1}+{prop2})/({n1}+{n2}),2)", "name": "pooled_p", "templateType": "anything"}, "Z90": {"group": "Ungrouped variables", "description": "", "definition": "1.28", "name": "Z90", "templateType": "number"}, "n1": {"group": "Ungrouped variables", "description": "", "definition": "random(360..450#10)", "name": "n1", "templateType": "randrange"}, "prop2": {"group": "Ungrouped variables", "description": "", "definition": "random(120..180#5)", "name": "prop2", "templateType": "randrange"}, "Z99": {"group": "Ungrouped variables", "description": "", "definition": "2.33", "name": "Z99", "templateType": "number"}, "prop1": {"group": "Ungrouped variables", "description": "", "definition": "random(100..125#1)", "name": "prop1", "templateType": "randrange"}, "Z95": {"group": "Ungrouped variables", "description": "", "definition": "1.65", "name": "Z95", "templateType": "number"}, "n2": {"group": "Ungrouped variables", "description": "", "definition": "random(360..520#10)", "name": "n2", "templateType": "randrange"}, "p1": {"group": "Ungrouped variables", "description": "", "definition": "precround({prop1}/{n1},2)", "name": "p1", "templateType": "anything"}, "test_statistic": {"group": "Ungrouped variables", "description": "", "definition": "precround((p1-p2)/sqrt(pooled_p*(1-pooled_p)*(1/{n1}+1/{n2})),2)", "name": "test_statistic", "templateType": "anything"}, "scenario": {"group": "Ungrouped variables", "description": "", "definition": "sum(map(abs(test_statistic)<x,x,[Z90,Z95,Z99]))", "name": "scenario", "templateType": "anything"}, "decision_matrix": {"group": "Ungrouped variables", "description": "", "definition": "[\n  [1,0,0,0],\n  [0,1,0,0],\n  [0,0,1,0],\n  [0,0,0,1]\n][scenario]", "name": "decision_matrix", "templateType": "anything"}, "p2": {"group": "Ungrouped variables", "description": "", "definition": "precround({prop2}/{n2},2)", "name": "p2", "templateType": "anything"}}, "variable_groups": [], "ungrouped_variables": ["Z95", "scenario", "decision_matrix", "Z99", "Z90", "test_statistic", "n1", "p1", "prop1", "n2", "prop2", "p2", "pooled_p"], "parts": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "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>", "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": "1", "precisionType": "dp", "precision": "2", "allowFractions": false, "showPrecisionHint": true, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReducedPC": 0, "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "type": "numberentry", "maxValue": "test_statistic", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "minValue": "test_statistic", "variableReplacements": [{"variable": "sample_mean_2", "part": "p0", "must_go_first": true}, {"variable": "sample_stdev_2", "part": "p0", "must_go_first": true}], "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 1, "precisionType": "dp", "precision": "2", "allowFractions": false, "showPrecisionHint": true, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "mustBeReducedPC": 0, "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "type": "numberentry", "maxValue": "{pooled_p}", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "minValue": "{pooled_p}", "variableReplacements": [], "mustBeReduced": false}], "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "maxMarks": "2", "marks": 0, "shuffleChoices": false, "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>"], "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>", "showCorrectAnswer": true, "displayColumns": "1", "minMarks": "2", "displayType": "radiogroup", "type": "1_n_2", "matrix": "decision_matrix", "showFeedbackIcon": true, "variableReplacements": [{"variable": "test_statistic", "part": "p0g0", "must_go_first": false}], "scripts": {}}], "preamble": {"css": "", "js": ""}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "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;\\(-2.33\\)</p>\n<p>Compare the test statistic with the Z-table values and choose your conclusion.</p>", "type": "question", "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>", "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "extensions": ["stats"], "name": "Two sample Z-test on proportions", "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/"}]}