// 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}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"t999": {"templateType": "anything", "group": "Ungrouped variables", "definition": "4.221", "description": "", "name": "t999"}, "sigm1": {"templateType": "randrange", "group": "Ungrouped variables", "definition": "random(1..5#0.5)", "description": "", "name": "sigm1"}, "mu1": {"templateType": "randrange", "group": "Ungrouped variables", "definition": "random(58..74#0.5)", "description": "", "name": "mu1"}, "t95": {"templateType": "number", "group": "Ungrouped variables", "definition": "2.16", "description": "", "name": "t95"}, "scenario": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(abs(test_statistic)t

", "name": "test_statistic"}}, "name": "Small sample t-test on sample mean", "functions": {}, "statement": "

The following set of \$$\\var{sample_size}\$$ numbers is believed to be drawn from a normal population:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 {r1[0]} {r1[1]} {r1[2]} {r1[3]} {r1[4]} {r1[5]} {r1[6]} {r1[7]} {r1[8]} {r1[9]} {r1[10]} {r1[11]} {r1[12]} {r1[13]}
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We want to test the hypothesis that the population mean  = \$$\\var{mu1}\$$.

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", "extensions": ["stats"], "preamble": {"css": "", "js": ""}, "tags": [], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

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\$$H_0:\$$ The mean \$$=\\var{mu1}\$$.

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\$$H_1:\$$ The mean \$$\\ne\\var{mu1}\$$.

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This is a two-sided test.

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Given a sample of size \$$n\$$ recall:

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the formula for the sample mean:    \$$\\overline{x}=\\frac{\\sum {x}}{n}=\\var{sample_mean_2}\$$

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the formula for the sample standard deviation:   \$$s=\\sqrt{\\frac{\\sum{(x-\\overline{x})^2}}{n-1}}=\\var{sample_stdev_2}\$$

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the formula for the t-statistic:   \$$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}\$$

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The t-table values will be for a two-tailed test and will have \$$n-1=13\$$ degrees of freedom looking this up gives:

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\$\\begin{array}{r|rrrr}&0.10&0.05&0.01\\\\\\hline13&\\pm\\var{t90}&\\pm\\var{t95}&\\pm\\var{t99}\\end{array}\$

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Compare the test statistic with the t-table values and choose your conclusion.

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Input the sample mean: [[0]]

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Input the sample standard deviation:[[1]]

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Enter the value for the test statistic: t = [[2]]

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Reject the Null Hypothesis and conclude that mean value is not \$$\\var{mu1}\$$

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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}\$$.

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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}\$$.

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Accept the Null Hypothesis at the 10% significance level and conclude that mean value is \$$\\var{mu1}\$$ .

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Having compared your test statistic with the table values for a two-tailed t-test with 13 degrees of freedom, select one of the following conclusions that best describes your conclusion.

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