// Numbas version: exam_results_page_options
{"name": "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": [{"name": "Two sample t-test: comparison of means", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>It is believed that the hydrocarbon emissions of cars have decreased in the first decade of the new millenium. A study was conducted to compare the hydrocarbon emissions at idling speed, in parts per million (ppm), of cars built in 2000 and cars built in 2010. Fifteen&nbsp;cars from each year were randomly selected on the year of their production and their hydrocarbon emission levels were recorded.</p>\n<p>The data gathered is tabled below:</p>\n<table style=\"width: 1157px;\">\n<tbody>\n<tr>\n<td style=\"width: 94.7755px;\"><strong>2000 </strong>Model</td>\n<td style=\"width: 10px;\">{r1[0]}</td>\n<td style=\"width: 54px;\">{r1[1]}</td>\n<td style=\"width: 54px;\">{r1[2]}</td>\n<td style=\"width: 54px;\">{r1[3]}</td>\n<td style=\"width: 54px;\">{r1[4]}</td>\n<td style=\"width: 54px;\">{r1[5]}</td>\n<td style=\"width: 54px;\">{r1[6]}</td>\n<td style=\"width: 54px;\">{r1[7]}</td>\n<td style=\"width: 54px;\">{r1[8]}</td>\n<td style=\"width: 54px;\">{r1[9]}</td>\n<td style=\"width: 63px;\">{r1[10]}</td>\n<td style=\"width: 63px;\">{r1[11]}</td>\n<td style=\"width: 63px;\">{r1[12]}</td>\n<td style=\"width: 63px;\">{r1[13]}</td>\n<td style=\"width: 63px;\">{r1[14]}</td>\n</tr>\n<tr>\n<td style=\"width: 94.7755px;\"><strong>2010 </strong>Model</td>\n<td style=\"width: 10px;\">{r2[0]}</td>\n<td style=\"width: 54px;\">{r2[1]}</td>\n<td style=\"width: 54px;\">{r2[2]}</td>\n<td style=\"width: 54px;\">{r2[3]}</td>\n<td style=\"width: 54px;\">{r2[4]}</td>\n<td style=\"width: 54px;\">{r2[5]}</td>\n<td style=\"width: 54px;\">{r2[6]}</td>\n<td style=\"width: 54px;\">{r2[7]}</td>\n<td style=\"width: 54px;\">{r2[8]}</td>\n<td style=\"width: 54px;\">{r2[9]}</td>\n<td style=\"width: 63px;\">{r2[10]}</td>\n<td style=\"width: 63px;\">{r2[11]}</td>\n<td style=\"width: 63px;\">{r2[12]}</td>\n<td style=\"width: 63px;\">{r2[13]}</td>\n<td style=\"width: 63px;\">{r2[14]}</td>\n</tr>\n</tbody>\n</table>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</p>", "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>", "rulesets": {}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"t99": {"name": "t99", "group": "Ungrouped variables", "definition": "2.467", "description": "", "templateType": "number", "can_override": false}, "sample_size": {"name": "sample_size", "group": "Ungrouped variables", "definition": "15", "description": "", "templateType": "number", "can_override": false}, "sigm1": {"name": "sigm1", "group": "Ungrouped variables", "definition": "random(5 .. 10#1)", "description": "", "templateType": "randrange", "can_override": false}, "t90": {"name": "t90", "group": "Ungrouped 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