// Numbas version: finer_feedback_settings
{"name": "David's copy of Z-test on sample mean", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "David's copy of Z-test on sample mean", "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "gaps": [{"extendBaseMarkingAlgorithm": true, "variableReplacements": [], "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "sample_mean_2", "correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "marks": 1, "precisionPartialCredit": 0, "unitTests": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "mustBeReduced": false, "precisionMessage": "You have not given your answer to the correct precision.", "maxValue": "sample_mean_2", "precisionType": "dp", "type": "numberentry", "showPrecisionHint": false, "precision": "2", "strictPrecision": false, "showFeedbackIcon": true, "showCorrectAnswer": true}, {"extendBaseMarkingAlgorithm": true, "variableReplacements": [], "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "sample_stdev_2", "correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "marks": 1, "precisionPartialCredit": 0, "unitTests": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "mustBeReduced": false, "precisionMessage": "You have not given your answer to the correct precision.", "maxValue": "sample_stdev_2", "precisionType": "dp", "type": "numberentry", "showPrecisionHint": false, "precision": "2", "strictPrecision": false, "showFeedbackIcon": true, "showCorrectAnswer": true}, {"extendBaseMarkingAlgorithm": true, "variableReplacements": [{"must_go_first": true, "variable": "sample_mean_2", "part": "p0g0"}, {"must_go_first": true, "variable": "sample_stdev_2", "part": "p0g1"}], "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "test_statistic", "correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "marks": "1", "precisionPartialCredit": 0, "unitTests": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "mustBeReduced": false, "precisionMessage": "You have not given your answer to the correct precision.", "maxValue": "test_statistic", "precisionType": "dp", "type": "numberentry", "showPrecisionHint": false, "precision": "2", "strictPrecision": false, "showFeedbackIcon": true, "showCorrectAnswer": true}], "sortAnswers": false, "extendBaseMarkingAlgorithm": true, "prompt": "<p>Give all your answers correct to two decimal places.</p>\n<p>Input the sample mean:&nbsp; &nbsp;\\(\\bar{x}=\\)&nbsp;[[0]]</p>\n<p>Input the sample standard deviation:&nbsp; &nbsp;\\(s=\\) [[1]]</p>\n<p>Enter the value for the test statistic:&nbsp; &nbsp;\\(Z=\\)&nbsp;[[2]]</p>\n<p></p>", "type": "gapfill", "variableReplacements": [], "customMarkingAlgorithm": "", "marks": 0, "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "minMarks": "2", "displayType": "radiogroup", "displayColumns": "1", "maxMarks": "2", "extendBaseMarkingAlgorithm": true, "matrix": "decision_matrix", "prompt": "<p>Having compared your test statistic with the table values for a two-tailed Z-test, select one&nbsp;of the following conclusions that best describes your conclusion.</p>", "type": "1_n_2", "variableReplacements": [{"must_go_first": false, "variable": "test_statistic", "part": "p0g2"}], "customMarkingAlgorithm": "", "marks": 0, "shuffleChoices": false, "choices": ["<p>Reject the Null Hypothesis and conclude that mean lifespan for an Irish man&nbsp;is not \\(\\simplify{{mu1}-{k}}\\).</p>", "<p>Reject the Null Hypothesis at the 5% significance level but accept the Null Hypothesis at the 1% significance level and conclude that mean lifespan for an Irish man&nbsp;is \\(\\simplify{{mu1}-{k}}\\).</p>", "<p>Reject the Null Hypothesis at the 10% significance level but accept the Null Hypothesis at the 5% significance level and conclude that mean lifespan for an Irish man&nbsp;is \\(\\simplify{{mu1}-{k}}\\).</p>", "<p>Accept the Null Hypothesis at the 10% significance level and conclude that meanlifespan for an Irish man&nbsp;is \\(\\simplify{{mu1}-{k}}\\).</p>"], "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true}], "preamble": {"js": "", "css": ""}, "variable_groups": [], "rulesets": {}, "advice": "<p>In this example we&nbsp;have a single sample of 30 data values. The sample is just large enough to apply a Z-test</p>\n<p>\\(H_0:\\) The mean \\(=\\simplify{{mu1}-{k}}\\).</p>\n<p>\\(H_1:\\)&nbsp;The mean \\(\\ne \\simplify{{mu1}-{k}}\\).</p>\n<p>Given a sample of size \\(n\\) recall:</p>\n<p style=\"padding-left: 60px;\">the formula for the sample mean: &nbsp; &nbsp;\\(\\overline{x}=\\frac{\\sum {x}}{n}=\\var{sample_mean_2}\\)</p>\n<p style=\"padding-left: 60px;\">the formula for the sample standard deviation: &nbsp; \\(s=\\sqrt{\\frac{\\sum{(x-\\overline{x})^2}}{n-1}}=\\var{sample_stdev_2}\\)</p>\n<p style=\"padding-left: 60px;\">the formula for the Z-statistic: &nbsp; \\(Z=\\frac{\\overline{x}-\\mu}{\\frac{s}{\\sqrt{n}}}=\\frac{\\var{sample_mean_2}-\\simplify{{mu1}-{k}}}{\\frac{\\var{sample_stdev_2}}{\\sqrt{\\var{sample_size}}}}=\\var{test_statistic}\\)</p>\n<p>The Z-table values will be for a two-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; \\(\\pm1.65\\)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(\\pm1.96\\)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(\\pm2.58\\)</p>\n<p>Compare the test statistic with the Z-table values and choose your conclusion.</p>", "tags": [], "extensions": ["stats"], "variables": {"r1": {"description": "", "name": "r1", "templateType": "anything", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu1,sigm1)),sample_size)"}, "t99": {"description": "", "name": "t99", "templateType": "number", "group": "Ungrouped variables", "definition": "2.58"}, "sigm1": {"description": "", "name": "sigm1", "templateType": "randrange", "group": "Ungrouped variables", "definition": "random(5..8#0.5)"}, "sample_stdev_2": {"description": "", "name": "sample_stdev_2", "templateType": "anything", "group": "Ungrouped variables", "definition": "precround(sqrt(30*stdev(r1)^2/29),2)"}, "t90": {"description": "", "name": "t90", "templateType": "number", "group": "Ungrouped variables", "definition": "1.65"}, "sample_mean_2": {"description": "", "name": "sample_mean_2", "templateType": "anything", "group": "Ungrouped variables", "definition": "precround(mean(r1),2)"}, "decision_matrix": {"description": "", "name": "decision_matrix", "templateType": "anything", "group": "Ungrouped variables", "definition": "[\n  [1,0,0,0],\n  [0,1,0,0],\n  [0,0,1,0],\n  [0,0,0,1]\n][scenario]"}, "test_statistic": {"description": "", "name": "test_statistic", "templateType": "anything", "group": "Ungrouped variables", "definition": "precround((sample_mean_2-mu1+k)/(sample_stdev_2/sqrt(sample_size)),2)"}, "mu1": {"description": "", "name": "mu1", "templateType": "randrange", "group": "Ungrouped variables", "definition": "random(72..75#0.5)"}, "k": {"description": "", "name": "k", "templateType": "randrange", "group": "Ungrouped variables", "definition": "random(1..2.5#0.25)"}, "sample_size": {"description": "", "name": "sample_size", "templateType": "number", "group": "Ungrouped variables", "definition": "30"}, "t95": {"description": "", "name": "t95", "templateType": "number", "group": "Ungrouped variables", "definition": "1.96"}, "scenario": {"description": "", "name": "scenario", "templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(abs(test_statistic)<x,x,[t90,t95,t99]))"}}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "functions": {}, "statement": "<p>A sample of 30 deaths of men in Ireland during the past year was selected at random and the ages of each man at the time of death recorded. &nbsp;The data collected is presented below.</p>\n<table width=\"1185\" height=\"64\">\n<tbody>\n<tr>\n<td style=\"width: 63.5417px;\">{r1[0]}</td>\n<td style=\"width: 63.5417px;\">{r1[1]}</td>\n<td style=\"width: 63.5417px;\">{r1[2]}</td>\n<td style=\"width: 63.5417px;\">{r1[3]}</td>\n<td style=\"width: 63.5417px;\">{r1[4]}</td>\n<td style=\"width: 63.5417px;\">{r1[5]}</td>\n<td style=\"width: 63.5417px;\">{r1[6]}</td>\n<td style=\"width: 63.5417px;\">{r1[7]}</td>\n<td style=\"width: 63.5417px;\">{r1[8]}</td>\n<td style=\"width: 63.5417px;\">{r1[9]}</td>\n<td style=\"width: 63.5417px;\">{r1[10]}</td>\n<td style=\"width: 63.5417px;\">{r1[11]}</td>\n<td style=\"width: 63.5417px;\">{r1[12]}</td>\n<td style=\"width: 63.5417px;\">{r1[13]}</td>\n<td style=\"width: 63.5417px;\">{r1[14]}</td>\n</tr>\n<tr>\n<td style=\"width: 63.5417px;\">{r1[15]}</td>\n<td style=\"width: 63.5417px;\">{r1[16]}</td>\n<td style=\"width: 63.5417px;\">{r1[17]}</td>\n<td style=\"width: 63.5417px;\">{r1[18]}</td>\n<td style=\"width: 63.5417px;\">{r1[19]}</td>\n<td style=\"width: 63.5417px;\">{r1[20]}</td>\n<td style=\"width: 63.5417px;\">{r1[21]}</td>\n<td style=\"width: 63.5417px;\">{r1[22]}</td>\n<td style=\"width: 63.5417px;\">{r1[23]}</td>\n<td style=\"width: 63.5417px;\">{r1[24]}</td>\n<td style=\"width: 63.5417px;\">{r1[25]}</td>\n<td style=\"width: 63.5417px;\">{r1[26]}</td>\n<td style=\"width: 63.5417px;\">{r1[27]}</td>\n<td style=\"width: 63.5417px;\">{r1[28]}</td>\n<td style=\"width: 63.5417px;\">{r1[29]}</td>\n</tr>\n</tbody>\n</table>\n<p>It is believed that the mean lifespan for Irish men is&nbsp;\\(\\simplify{{mu1}-{k}}\\) years.</p>\n<p>Does the data support this theory?&nbsp;</p>\n<p></p>", "ungrouped_variables": ["sample_size", "r1", "decision_matrix", "sample_mean_2", "sample_stdev_2", "mu1", "scenario", "sigm1", "t95", "t99", "k", "t90", "test_statistic"], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "David Goulding", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2365/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "David Goulding", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2365/"}]}