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{"name": "Two sample t-test: Paired data", "extensions": ["stats", "polynomials"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Two sample t-test: Paired data", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<script id=\"th-iframe-script\" src=\"chrome-extension://ofdopmlmgifpfkijadehmhjccbefaeec/assets/comms/commsiframe.js\"></script>\n<p>\n<script id=\"th-iframe-script\" src=\"chrome-extension://ofdopmlmgifpfkijadehmhjccbefaeec/assets/comms/commsiframe.js\"></script>\n</p>\n<p>\n<script id=\"th-iframe-script\" src=\"chrome-extension://ofdopmlmgifpfkijadehmhjccbefaeec/assets/comms/commsiframe.js\"></script>\n</p>\n<p>idence intervals[Mean1{0}]</p>", "advice": "<p>In this example we are dealing with paired data.</p>\n<p>\\(H_0:\\) \\(\\mu_d=0\\)&nbsp; &nbsp; i.e. The mean zinc concentration for surface water and bottom water is the same.</p>\n<p>\\(H_1:\\)&nbsp;\\(\\mu_d&lt;0\\)&nbsp; &nbsp; i.e&nbsp; The mean zinc concentration for surface water is less than the&nbsp;mean zinc concentration for bottom&nbsp;water.</p>\n<p>We must evaluate the differences: for each pair of values \\(d=x_1-x_2\\)</p>\n<p style=\"padding-left: 90px;\">\\(\\var{diff}\\)</p>\n<p>We now have a&nbsp;sample of \\(n=\\var{sample_size}\\) differences.</p>\n<p>Recall:</p>\n<p style=\"padding-left: 60px;\">the formula for the mean difference: &nbsp; &nbsp;\\(\\overline{d}=\\frac{\\sum {d}}{n}=\\var{diff_mean}\\)</p>\n<p style=\"padding-left: 60px;\">the formula for the standard deviation: &nbsp; \\(s=\\sqrt{\\frac{\\sum{(d-\\overline{d})^2}}{n-1}}=\\var{diff_stdev}\\)</p>\n<p style=\"padding-left: 60px;\">the formula for the t-statistic: &nbsp; \\(t=\\frac{\\overline{d}}{\\frac{s}{\\sqrt{n}}}=\\frac{\\var{diff_mean}}{\\frac{\\var{diff_stdev}}{\\sqrt{12}}}=\\var{test_statistic}\\)</p>\n<p>The t-table value will be for a one-tailed test and will have \\(n-1=11\\) degrees of freedom. Because of the alternative hypothesis the t-value chosen will be negative.&nbsp;</p>\n<p style=\"padding-left: 60px;\">\\[\\begin{array}{r|rrrr}&amp;0.10&amp;0.05&amp;0.01\\\\\\hline11&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>", "rulesets": {}, "extensions": ["polynomials", "stats"], "variables": {"Mean1": {"name": "Mean1", "group": "Ungrouped variables", "definition": "random(30000 .. 50000#25)", "description": "", "templateType": "randrange"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["Mean1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, 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false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "diff_stdev", "maxValue": "diff_stdev", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, 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"sortAnswers": false}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [{"variable": "Mean1", "part": "p0g2", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<script id=\"th-iframe-script\" src=\"chrome-extension://ofdopmlmgifpfkijadehmhjccbefaeec/assets/comms/commsiframe.js\"></script>\n<p>Having compared your test statistic with the table value, select one&nbsp;of the following conclusions that best describes your conclusion.</p>", "minMarks": "2", "maxMarks": "2", "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": "1", "showCellAnswerState": true, "choices": ["<p>Reject the Null Hypothesis and conclude that the mean zinc concentration is greater&nbsp;for bottom water than surface water</p>", "<p>Reject the Null Hypothesis at the 5% significance level but accept the Null Hypothesis at the 1% significance level and conclude that the mean zinc concentration is the same for bottom water and surface water</p>", "<p>Reject the Null Hypothesis at the 10% significance level but accept the Null Hypothesis at the 5% significance level and conclude that the mean zinc concentration is the same for bottom water and surface water</p>", "<p>Accept the Null Hypothesis at the 10% significance level and conclude that the mean zinc concentration is the same for bottom water and surface water</p>"], "matrix": "decision_matrix"}], "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": 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