// Numbas version: finer_feedback_settings
{"name": "One sample t-test -SES", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"t95": {"description": "", "group": "Ungrouped variables", "templateType": "number", "definition": "2.145", "name": "t95"}, "mu1": {"description": "", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(17..17.5#0.01)", "name": "mu1"}, "sigm1": {"description": "", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(0..1#0.05)", "name": "sigm1"}, "sample_stdev_2": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "precround(sqrt(15*stdev(r1)^2/14),2)", "name": "sample_stdev_2"}, "test_statistic": {"description": "<p>t</p>", "group": "Ungrouped variables", "templateType": "anything", "definition": "precround((sample_mean_2-mu1)/(sample_stdev_2/sqrt(sample_size)),2)", "name": "test_statistic"}, "t99": {"description": "", "group": "Ungrouped variables", "templateType": "number", "definition": "2.977", "name": "t99"}, "t90": {"description": "", "group": "Ungrouped variables", "templateType": "number", "definition": "1.761", "name": "t90"}, "sample_size": {"description": "", "group": "Ungrouped variables", "templateType": "number", "definition": "15", "name": "sample_size"}, "scenario": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "sum(map(abs(test_statistic)<x,x,[t90,t95,t99]))", "name": "scenario"}, "decision_matrix": {"description": "", "group": "Ungrouped variables", "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]", "name": "decision_matrix"}, "r1": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "repeat(((precround(normalsample(mu1,sigm1),2),sample_size)))", "name": "r1"}, "sample_mean_2": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "precround(mean(r1),2)", "name": "sample_mean_2"}, "t999": {"description": "", "group": "Ungrouped variables", "templateType": "number", "definition": "4.14", "name": "t999"}}, "advice": "<p>The hypotheses that we are testing are:</p>\n<p style=\"text-align: left;\">&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; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$ H_0: \\mu=\\var{mu1} $ ,&nbsp; &nbsp; $ H_1: \\mu \\neq&nbsp;\\var{mu1}&nbsp;$</p>\n<p></p>\n<p>The test statistic for a one-sample t-test is given by:</p>\n<p></p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$ |t| = \\frac{\\bar{x}-\\mu_0}{s/\\sqrt{n}} $</p>\n<p>where $\\bar{x}$ is the sample mean, given by:</p>\n<p></p>\n<p>&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; $\\bar{x}= \\sum_{i=1}^{i=15} \\frac{x_i}{15} =&nbsp;\\var{sample_mean_2}$ $(2dp)$&nbsp;</p>\n<p>and $s$ is the sample standard deviation, given by:</p>\n<p></p>\n<p>&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; &nbsp;$ s = \\sqrt{\\frac{\\left (\\sum (x-\\bar{x})^2\\right)}{n-1}} =&nbsp;\\var{sample_stdev_2}$&nbsp; &nbsp;$(2dp)$</p>\n<p>Therefore, the test statistic is:</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$ |{t}| = \\frac{\\var{sample_mean_2}-\\var{mu1}}{\\var{sample_stdev_2}/\\sqrt{15}}$</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$&nbsp; &nbsp;= \\var{abs({test_statistic})}&nbsp;$&nbsp;$(2dp)$</p>\n<p></p>\n<p>The t-table values will be for a two-tailed test and will have $ n-1 = 14$ degrees of freedom:</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>If $ |t| &lt; t_{14}(0.1) $, then we accept $ H_0$ at the 10% level.&nbsp;</p>\n<p>If&nbsp;$ |t| &gt; t_{14}(0.1)$, then we reject $H_0$ at&nbsp;the 10% level.</p>\n<p>If $ |t| &gt; t_{14}(0.05) $, we reject $H_0$ at the $5%$ level.</p>\n<p>If $ |t| &gt; t_{14}(0.01)$, then we reject $H_0$ at the 1% level.</p>", "tags": [], "functions": {}, "rulesets": {}, "preamble": {"js": "", "css": ""}, "extensions": ["stats"], "variable_groups": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "ungrouped_variables": ["sample_size", "r1", "sample_mean_2", "sample_stdev_2", "mu1", "sigm1", "test_statistic", "t95", "scenario", "decision_matrix", "t99", "t90", "t999"], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>The following&nbsp;is a set of BMI's (Body Mass Index) for 15 overweight children aged 5.</p>\n<p>The data is thought to follow a normal distribution.</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 equals \\(\\var{mu1}\\) .</p>\n<p></p>", "parts": [{"showFeedbackIcon": true, "maxValue": "sample_mean_2", "variableReplacementStrategy": "originalfirst", "unitTests": [], "type": "numberentry", "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "prompt": "<p>Calculate the sample mean.</p>", "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "scripts": {}, "correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "showCorrectAnswer": true, "minValue": "sample_mean_2"}, {"showFeedbackIcon": true, "maxValue": "sample_stdev_2", "variableReplacementStrategy": "originalfirst", "unitTests": [], "type": "numberentry", "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "prompt": "<p>Calculate the sample standard deviation.</p>", "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "scripts": {}, "correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "showCorrectAnswer": true, "minValue": "sample_stdev_2"}, {"showFeedbackIcon": true, "maxValue": "abs{test_statistic}", "variableReplacementStrategy": "originalfirst", "unitTests": [], "type": "numberentry", "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "stepsPenalty": 0, "prompt": "<p>Calculate the test statistic.</p>", "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "scripts": {}, "correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "showCorrectAnswer": true, "steps": [{"showFeedbackIcon": true, "scripts": {}, "unitTests": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "information", "customMarkingAlgorithm": "", "marks": 0, "prompt": "<p>For a one sample t-test, the test statistic is given by:&nbsp; &nbsp;&nbsp;</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $&nbsp; |t| =&nbsp; \\frac{\\bar{x}-\\mu}{s/\\sqrt{n}} $&nbsp;</p>", "variableReplacements": [], "showCorrectAnswer": true}], "minValue": "abs{test_statistic}"}, {"matrix": "decision_matrix", "showFeedbackIcon": false, "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>"], "unitTests": [], "extendBaseMarkingAlgorithm": true, "type": "1_n_2", "displayColumns": "1", "prompt": "<p>Using t-tables, compare your test statistic to the appropriate t-value, taking into account the correct degrees of freedom. From this comparison select one of the following statements that best describes your conclusion.</p>", "minMarks": "2", "variableReplacements": [{"must_go_first": false, "part": "p0", "variable": "test_statistic"}], "customMarkingAlgorithm": "", "displayType": "radiogroup", "scripts": {}, "maxMarks": "2", "marks": 0, "showCorrectAnswer": true, "shuffleChoices": false, "variableReplacementStrategy": "originalfirst"}], "name": "One sample t-test -SES", "type": "question", "contributors": [{"name": "Lauren Frances Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2490/"}]}]}], "contributors": [{"name": "Lauren Frances Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2490/"}]}