// Numbas version: finer_feedback_settings
{"name": "Brad's copy of Two sample t-test: comparison of means", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "tags": [], "parts": [{"unitTests": [], "marks": 0, "prompt": "<p>Test to see if there has been a change in sales on average at the 5% level of significance.&nbsp;</p>\n<p>What is the sample mean?:&nbsp;[[0]]</p>\n<p>What is the population mean?:&nbsp;[[1]]</p>\n<p>What is the sample standard deviation?:&nbsp;[[3]]</p>\n<p>Enter the value for the test statistic: t =&nbsp;[[4]]</p>\n<p></p>", "type": "gapfill", "scripts": {}, "variableReplacements": [], "showCorrectAnswer": true, "customMarkingAlgorithm": "", "sortAnswers": false, "useCustomName": false, "customName": "", "extendBaseMarkingAlgorithm": true, "gaps": [{"unitTests": [], "mustBeReducedPC": 0, "marks": 1, "variableReplacements": [], "showPrecisionHint": true, "showCorrectAnswer": true, "precisionType": "dp", "correctAnswerStyle": "plain", "customName": "", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "strictPrecision": false, "minValue": "sample_mean_1", "type": "numberentry", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "allowFractions": false, "precision": "2", "useCustomName": false, "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "maxValue": "sample_mean_1", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"unitTests": [], "mustBeReducedPC": 0, "marks": 1, "variableReplacements": [], "showPrecisionHint": true, "showCorrectAnswer": true, "precisionType": "dp", "correctAnswerStyle": "plain", "customName": "", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "strictPrecision": false, "minValue": "sample_stdev_1", "type": "numberentry", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "allowFractions": false, "precision": "2", "useCustomName": false, "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "maxValue": "sample_stdev_1", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"unitTests": [], "mustBeReducedPC": 0, "marks": 1, "variableReplacements": [{"variable": "sample_mean_2", "part": "p0g0", "must_go_first": true}, {"variable": "sample_stdev_2", "part": "p0g1", "must_go_first": true}], "showPrecisionHint": true, "showCorrectAnswer": true, "precisionType": "dp", "correctAnswerStyle": "plain", "customName": "", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "strictPrecision": false, "minValue": "test_statistic", "type": "numberentry", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "allowFractions": false, "precision": "2", "useCustomName": false, "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "maxValue": "test_statistic", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"unitTests": [], "mustBeReducedPC": 0, "marks": 1, "variableReplacements": [], "showPrecisionHint": true, "showCorrectAnswer": true, "precisionType": "dp", "correctAnswerStyle": "plain", "customName": "", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "strictPrecision": false, "minValue": "sample_mean_2", "type": "numberentry", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "allowFractions": false, "precision": "2", "useCustomName": false, "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "maxValue": "sample_mean_2", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"unitTests": [], "mustBeReducedPC": 0, "marks": 1, "variableReplacements": [], "showPrecisionHint": true, "showCorrectAnswer": true, "precisionType": "dp", "correctAnswerStyle": "plain", "customName": "", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "strictPrecision": true, "minValue": "sample_stdev_2", "type": "numberentry", "correctAnswerFraction": false, "customMarkingAlgorithm": "", "allowFractions": false, "precision": "2", "useCustomName": false, "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "maxValue": "sample_stdev_2", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"unitTests": [], "displayColumns": "1", "marks": 0, "prompt": "<p>Having compared your test statistic with the table values for a one-tailed t-test with 28 degrees of freedom, select one&nbsp;of the following conclusions that best describes your conclusion.</p>", "type": "1_n_2", "scripts": {}, "variableReplacements": [{"variable": "r1", "part": "p0g2", "must_go_first": false}], "shuffleChoices": false, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "useCustomName": false, "customName": "", "minMarks": "3", "displayType": "radiogroup", "choices": ["<p>Reject the Null Hypothesis and conclude that mean hydrocarbon emission rate has decreased.&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 hydrocarbon emission rate is the same for both models.</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 hydrocarbon emission rate is the same for both models.</p>", "<p>Accept the Null Hypothesis at the 10% significance level and conclude that mean hydrocarbon emission rate is the same for both models.</p>"], "extendBaseMarkingAlgorithm": true, "matrix": "decision_matrix", "maxMarks": "3", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCellAnswerState": true}], "name": "Brad's copy of Two sample t-test: comparison of means", "rulesets": {}, "statement": "<p>A company has been testing a new marketing plan. Previous data gave a mean of &pound;[0] a week per store. A sample of 12 stores had a mean of &pound;[1] and a standard deviation of &pound;[2].</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</p>", "preamble": {"css": "", "js": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"sigm1": {"description": "", "definition": "random(5 .. 10#1)", "group": "Ungrouped variables", "name": "sigm1", "templateType": "randrange"}, "sigm2": {"description": "", "definition": "random(4 .. 8#1)", "group": "Ungrouped variables", "name": "sigm2", "templateType": "randrange"}, "t95": {"description": "", "definition": "1.701", "group": "Ungrouped variables", "name": "t95", "templateType": "number"}, "r2": {"description": "", "definition": "repeat(round(normalsample(mu2,sigm2)),sample_size)", "group": "Ungrouped variables", "name": "r2", "templateType": "anything"}, "sample_stdev_2": {"description": "", "definition": "precround(sqrt(15*stdev(r2)^2/14),2)", "group": "Ungrouped variables", "name": "sample_stdev_2", "templateType": "anything"}, "mu2": {"description": "", "definition": "mu1-k", "group": "Ungrouped variables", "name": "mu2", "templateType": "anything"}, "t90": {"description": "", "definition": "1.313", "group": "Ungrouped variables", "name": "t90", "templateType": "number"}, "sample_mean_1": {"description": "", "definition": "precround(mean(r1),0\n", "group": "Ungrouped variables", "name": "sample_mean_1", "templateType": "anything"}, "decision_matrix": {"description": "", "definition": "[\n  [1,0,0,0],\n  [0,1,0,0],\n  [0,0,1,0],\n  [0,0,0,1]\n][scenario]", "group": "Ungrouped variables", "name": "decision_matrix", "templateType": "anything"}, "k": {"description": "", "definition": "random(3 .. 6#1)", "group": "Ungrouped variables", "name": "k", "templateType": "randrange"}, "pooled_s": {"description": "", "definition": "precround(sqrt((sample_stdev_1^2+sample_stdev_2^2)/2),2)", "group": "Ungrouped variables", "name": "pooled_s", "templateType": "anything"}, "test_statistic": {"description": "<p>t</p>", "definition": "precround((sample_mean_1-sample_mean_2)/(pooled_s*sqrt(2/15)),2)", "group": "Ungrouped variables", "name": "test_statistic", "templateType": "anything"}, "t99": {"description": "", "definition": "2.467", "group": "Ungrouped variables", "name": "t99", "templateType": "number"}, "mu1": {"description": "", "definition": "random(30000 .. 50000#50)", "group": "Ungrouped variables", "name": "mu1", "templateType": "randrange"}, "sample_mean_2": {"description": "", "definition": "precround(mean(r2),2)", "group": "Ungrouped variables", "name": "sample_mean_2", "templateType": "anything"}, "scenario": {"description": "", "definition": "sum(map(abs(test_statistic)<x,x,[t90,t95,t99]))", "group": "Ungrouped variables", "name": "scenario", "templateType": "anything"}, "sample_size": {"description": "", "definition": "15", "group": "Ungrouped variables", "name": "sample_size", "templateType": "number"}, "sample_stdev_1": {"description": "", "definition": "precround(sqrt(15*stdev(r1)^2/14),2)", "group": "Ungrouped variables", "name": "sample_stdev_1", "templateType": "anything"}}, "advice": "<p>In this example we are comparing the means of two distict samples.&nbsp;</p>\n<p>\\(H_0:\\) \\(\\mu_1=\\mu_2\\)&nbsp; &nbsp; i.e. The mean emission rate&nbsp;is the same for both sets of cars.</p>\n<p>\\(H_1:\\)&nbsp;\\(\\mu_1&gt;\\mu_2\\)&nbsp; &nbsp; i.e.&nbsp; The mean emission rate of the older set of cars is greater than the mean emission rate of the newer set of cars.</p>\n<p>For each sample we must evaluate the sample mean and the sample standard deviation.</p>\n<p>Recall when given a sample of size \\(n\\) :</p>\n<p style=\"padding-left: 60px;\">the formula for the sample mean: &nbsp; &nbsp;\\(\\overline{x}=\\frac{\\sum {x}}{n}\\)</p>\n<p style=\"padding-left: 240px;\">&nbsp;\\(\\overline{x_1}=\\var{sample_mean_1}\\)&nbsp; &nbsp;and&nbsp; &nbsp;&nbsp;\\(\\overline{x_2}=\\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}}\\)</p>\n<p style=\"padding-left: 240px;\">\\(s_1=\\var{sample_stdev_1}\\)&nbsp; &nbsp; &nbsp;and&nbsp; &nbsp; &nbsp;\\(s_2=\\var{sample_stdev_2}\\)</p>\n<p>We must evaluate the pooled sample standard deviation using the two sample standard deviartions \\(s_1\\) and \\(s_2\\) and the formula</p>\n<p style=\"padding-left: 60px;\">\\(s_p=\\sqrt{\\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}=\\sqrt{\\frac{14(\\var{sample_stdev_1})^2+14(\\var{sample_stdev_2})^2}{15+15-2}}=\\var{pooled_s}\\)</p>\n<p>The formula for the t-statistic: &nbsp;</p>\n<p style=\"padding-left: 60px;\">\\(t=\\frac{\\overline{x_1}-\\overline{x_2}}{s_p\\sqrt{\\frac{1}{n_1}+\\frac{1}{n_2}}}=\\frac{\\var{sample_mean_1}-\\var{sample_mean_2}}{\\var{pooled_s}\\sqrt{\\frac{1}{14}+\\frac{1}{14}}}=\\var{test_statistic}\\)</p>\n<p>The t-table value will be for a one-tailed test and will have \\(n_1+n_2-2=28\\) degrees of freedom. Because of the alternative hypothesis the t-value chosen will be positive.&nbsp;</p>\n<p style=\"padding-left: 60px;\">\\[\\begin{array}{r|rrrr}&amp;0.10&amp;0.05&amp;0.01\\\\\\hline28&amp;\\var{t90}&amp;\\var{t95}&amp;\\var{t99}\\end{array}\\]</p>\n<p>Compare the test statistic with the t-table values and choose your conclusion.</p>\n<p></p>", "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "functions": {}, "ungrouped_variables": ["sample_mean_1", "sample_stdev_1", "mu1", "sigm1", "test_statistic", "t95", "scenario", "decision_matrix", "t99", "t90", "r2", "mu2", "sigm2", "sample_mean_2", "sample_stdev_2", "pooled_s", "sample_size", "k"], "extensions": ["stats"], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Brad Allison", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3394/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Brad Allison", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3394/"}]}