// Numbas version: exam_results_page_options
{"name": "Small sample t-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": [{"metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "type": "question", "preamble": {"css": "", "js": ""}, "advice": "<p></p>\n<p>\\(H_0:\\) The mean \\(=\\var{mu1}\\).</p>\n<p>\\(H_1:\\)&nbsp;The mean \\(\\ne\\var{mu1}\\).</p>\n<p>This is a two-sided test.</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 t-statistic: &nbsp; \\(t=\\frac{\\overline{x}-\\mu}{\\frac{s}{\\sqrt{n}}}=\\frac{\\var{sample_mean_2}-\\var{mu1}}{\\frac{\\var{sample_stdev_2}}{\\sqrt{\\var{sample_size}}}}=\\var{test_statistic}\\)</p>\n<p>The t-table values will be for a two-tailed test and will have \\(n-1=13\\) degrees of freedom looking this up gives:</p>\n<p>\\[\\begin{array}{r|rrrr}&amp;0.10&amp;0.05&amp;0.01\\\\\\hline13&amp;\\pm\\var{t90}&amp;\\pm\\var{t95}&amp;\\pm\\var{t99}\\end{array}\\]</p>\n<p>Compare the test statistic with the t-table values and choose your conclusion.</p>", "rulesets": {}, "extensions": ["stats"], "name": "Small sample t-test on sample mean", "ungrouped_variables": ["sample_size", "r1", "sample_mean_2", "sample_stdev_2", "mu1", "sigm1", "test_statistic", "t95", "scenario", "decision_matrix", "t99", "t90", "t999"], "functions": {}, "tags": [], "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "variables": {"t90": {"name": "t90", "group": "Ungrouped variables", "definition": "1.771", "description": "", "templateType": "number"}, "sample_stdev_2": {"name": "sample_stdev_2", "group": "Ungrouped variables", "definition": "precround(sqrt(14*stdev(r1)^2/13),3)", "description": "", "templateType": "anything"}, "scenario": {"name": "scenario", "group": "Ungrouped variables", "definition": "sum(map(abs(test_statistic)<x,x,[t90,t95,t99]))", "description": "", "templateType": "anything"}, "mu1": {"name": "mu1", "group": "Ungrouped variables", "definition": "random(58..74#0.5)", "description": "", "templateType": "randrange"}, "t999": {"name": "t999", "group": "Ungrouped variables", "definition": "4.221", "description": "", "templateType": "anything"}, "test_statistic": {"name": "test_statistic", "group": "Ungrouped variables", "definition": "precround((sample_mean_2-mu1)/(sample_stdev_2/sqrt(sample_size)),3)", "description": "<p>t</p>", "templateType": "anything"}, "t95": {"name": "t95", "group": "Ungrouped variables", "definition": "2.16", "description": "", "templateType": "number"}, "sample_size": {"name": "sample_size", "group": "Ungrouped variables", "definition": "14", "description": "", "templateType": "number"}, "t99": {"name": "t99", "group": "Ungrouped variables", "definition": "3.012", "description": "", "templateType": "number"}, "r1": {"name": "r1", "group": "Ungrouped variables", "definition": "repeat(round(normalsample(mu1,sigm1)),sample_size)", "description": "", "templateType": "anything"}, "sample_mean_2": {"name": "sample_mean_2", "group": "Ungrouped variables", "definition": "precround(mean(r1),3)", "description": "", "templateType": "anything"}, "sigm1": {"name": "sigm1", "group": "Ungrouped variables", "definition": "random(1..5#0.5)", "description": "", "templateType": "randrange"}, "decision_matrix": {"name": "decision_matrix", "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]", "description": "", "templateType": "anything"}}, "statement": "<p>The following&nbsp;set of \\(\\var{sample_size}\\) numbers is believed to be drawn from a normal population:&nbsp;</p>\n<table>\n<tbody>\n<tr>\n<td>{r1[0]}</td>\n<td>{r1[1]}</td>\n<td>{r1[2]}</td>\n<td>{r1[3]}</td>\n<td>{r1[4]}</td>\n<td>{r1[5]}</td>\n<td>{r1[6]}</td>\n<td>{r1[7]}</td>\n<td>{r1[8]}</td>\n<td>{r1[9]}</td>\n<td>{r1[10]}</td>\n<td>{r1[11]}</td>\n<td>{r1[12]}</td>\n<td>{r1[13]}</td>\n</tr>\n</tbody>\n</table>\n<p>We want to test the hypothesis that the population mean &nbsp;= \\(\\var{mu1}\\).</p>\n<p></p>", "parts": [{"type": "gapfill", "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "<p>Input the sample mean:&nbsp;[[0]]</p>\n<p>Input the sample standard deviation:[[1]]</p>\n<p>Enter the value for the test statistic: t =&nbsp;[[2]]</p>\n<p></p>", "showCorrectAnswer": true, "gaps": [{"precision": "2", "type": "numberentry", "showPrecisionHint": true, "maxValue": "sample_mean_2", "strictPrecision": false, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "precisionType": "dp", "correctAnswerFraction": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "mustBeReduced": false, "showFeedbackIcon": true, "marks": 1, "minValue": "sample_mean_2", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacements": [], "allowFractions": false}, {"precision": "2", "type": "numberentry", "showPrecisionHint": true, "maxValue": "sample_stdev_2", "strictPrecision": false, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "precisionType": "dp", "correctAnswerFraction": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "mustBeReduced": false, "showFeedbackIcon": true, "marks": 1, "minValue": "sample_stdev_2", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacements": [], "allowFractions": false}, {"precision": "2", "type": "numberentry", "showPrecisionHint": true, "maxValue": "(sample_mean_2-mu1)/(sample_stdev_2/sqrt(sample_size))", "strictPrecision": false, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "precisionType": "dp", "correctAnswerFraction": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "mustBeReduced": false, "showFeedbackIcon": true, "marks": 1, "minValue": "(sample_mean_2-mu1)/(sample_stdev_2/sqrt(sample_size))", "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacements": [{"variable": "sample_mean_2", "part": "p0g0", "must_go_first": true}, {"variable": "sample_stdev_2", "part": "p0g1", "must_go_first": true}], "allowFractions": false}], "showFeedbackIcon": true, "variableReplacements": []}, {"type": "1_n_2", "maxMarks": "2", "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "displayColumns": "1", "minMarks": "2", "choices": ["<p>Reject the Null Hypothesis and conclude that mean value is not \\(\\var{mu1}\\)&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 mean value is \\(\\var{mu1}\\).</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 value is \\(\\var{mu1}\\).</p>", "<p>Accept the Null Hypothesis at the 10% significance level and conclude that mean value is \\(\\var{mu1}\\) .</p>"], "shuffleChoices": false, "scripts": {}, "matrix": "decision_matrix", "displayType": "radiogroup", "showCorrectAnswer": true, "prompt": "<p>Having compared your test statistic with the table values for a two-tailed t-test with&nbsp;13 degrees of freedom, select one&nbsp;of the following conclusions that best describes your conclusion.</p>", "variableReplacements": [{"variable": "test_statistic", "part": "p0g2", "must_go_first": false}]}], "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/"}]}