{"id":1207,"date":"2023-06-22T02:09:06","date_gmt":"2023-06-22T02:09:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/module-11-cheat-sheet\/"},"modified":"2025-02-11T00:30:34","modified_gmt":"2025-02-11T00:30:34","slug":"module-11-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/module-11-cheat-sheet\/","title":{"raw":"Module 10: Cheat Sheet","rendered":"Module 10: Cheat Sheet"},"content":{"raw":"<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Statistics+Exemplar\/Cheat+Sheets\/Module+10_+Cheat+Sheet.pdf\" target=\"_blank\" rel=\"noopener\">Download a PDF of this page here.<\/a><\/span><\/p>\r\n<h2>Essential Concepts<\/h2>\r\n<h3>One-Sample Z-Test of Proportions<\/h3>\r\n<ol>\r\n\t<li>Write out the null and alternative hypotheses.\r\n\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">The null hypothesis, [latex] H_{0} [\/latex], is what we assume to be true to begin with. It is often a statement of no change from the previous value or from what is expected (e.g., we expect a coin to be fair).\r\n\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">The null hypothesis, [latex] H_{0} [\/latex], is always given in the form: [latex] p = value [\/latex] for population proportions.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li style=\"font-weight: 400;\">The alternative hypothesis, [latex] H_{A} [\/latex], is what we consider to be plausible if the null hypothesis is false. Often, it is a change from the null hypothesis that we would like to test the accuracy of. With a null value of [latex]a[\/latex], the alternative hypothesis, [latex] H_{A} [\/latex], is written as an inequality:\r\n\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">[latex]H_{A} : p&gt;a[\/latex],<\/li>\r\n\t<li style=\"font-weight: 400;\">[latex]H_{A}: p \\lt a[\/latex], or<\/li>\r\n\t<li style=\"font-weight: 400;\">[latex]H_{A}: p\\neq a[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Check the conditions for the hypothesis test. For testing a one-sample z-test for proportions, we require:\r\n\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">Large counts: Check that [latex]np\\ge10[\/latex] and [latex]n(1-p)\\ge10[\/latex].<\/li>\r\n\t<li style=\"font-weight: 400;\">Random samples\/assignment: Check that the sample is a random sample.<\/li>\r\n\t<li style=\"font-weight: 400;\">10% population size: Check that the sample size, [latex]n[\/latex], is less than 10% of the population size, [latex]N[\/latex]: [latex]n&lt;0.10(N)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>If not using technology, calculate a test statistic.<\/li>\r\n<\/ol>\r\n<ul>\r\n\t<li style=\"list-style-type: none;\">\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">A test statistic measures the distance between the sample statistic and the null hypothesis value in terms of the standard error of the null hypothesis value.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 80px;\">[latex] \\text{test statistic }= \\frac{\\text{sample statistic }-\\text{ null hypothesis value}}{\\text{standard error of the null hypothesis value}}=\\frac{\\stackrel{\u02c6}{p}-p}{\\sqrt{\\frac{p(1-p)}{n}}}[\/latex]<\/p>\r\n<p style=\"padding-left: 80px;\">where [latex]\\stackrel{\u02c6}{p}[\/latex] is the sample statistics and [latex]p[\/latex] is the null hypothesis value.<\/p>\r\n<p style=\"padding-left: 40px;\">4. Use technology to calculate a P-value.<\/p>\r\n<ul>\r\n\t<li style=\"list-style-type: none;\">\r\n<ul>\r\n\t<li>We define the P-value as the probability of obtaining a test statistic at least as extreme (in the direction of the alternative hypothesis) as the one that is actually seen if the null hypothesis is true.<\/li>\r\n\t<li>The smaller the probability, the less likely it is that the sample occurred by chance alone, the more evidence we have <em>against<\/em> the null hypothesis.<\/li>\r\n\t<li>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.\r\n\r\n<ul>\r\n\t<li>The significance level, [latex] \\alpha [\/latex], is the cut-off for P-values at which we have enough evidence to reject the null hypothesis. Typically, small significance levels such as 1%,\u00a0 5%, or 10% are used in hypothesis testing.<br \/>\r\n<table style=\"border-collapse: collapse; width: 100%; height: 150px;\">\r\n<thead>\r\n<tr>\r\n<th style=\"width: 39.8067%;\">Decision<\/th>\r\n<th style=\"width: 60.1054%;\">Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 39.8067%;\">If P-value [latex]\\le\\alpha[\/latex], there is enough evidence to reject the null hypothesis.<\/td>\r\n<td style=\"width: 60.1054%;\">At the [latex]\\alpha\\times[\/latex]100% significance level, the data provide convincing evidence in support of the alternative hypothesis.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 39.8067%;\">If P-value [latex]\\gt\\alpha[\/latex], there is not enough evidence to reject the null hypothesis.<\/td>\r\n<td style=\"width: 60.1054%;\">At the [latex]\\alpha\\times[\/latex]100% significance level, the data do not provide convincing evidence in support of the alternative hypothesis.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>We never accept the null hypothesis.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 40px;\">5. Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).\u00a0<\/p>\r\n<ul>\r\n\t<li style=\"list-style-type: none;\">\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">If a hypothesis test results in rejecting the null hypothesis because the P-value is less than the significance level, we say we have statistical significance in favor of the alternative hypothesis.<\/li>\r\n\t<li style=\"font-weight: 400;\">If the results are meaningful, we say that the results have practical significance. Having practical significance usually means the results show a significant improvement!<\/li>\r\n\t<li style=\"font-weight: 400;\">Sometimes, due to chance, the result of the hypothesis test does not align with reality. If we reject a correct null hypothesis, we are committing a type I error. If we do not reject a null hypothesis that is actually incorrect, we are committing a type II error.<\/li>\r\n\t<li>\r\n<table style=\"border-collapse: collapse; width: 57.8692%; height: 66px;\">\r\n<tbody>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 19.1044%; height: 22px; padding-left: 80px; text-align: left;\">\u00a0<\/td>\r\n<td style=\"width: 17.435%; height: 22px; text-align: left;\"><strong>Reject the null hypothesis<\/strong><\/td>\r\n<td style=\"width: 21.3301%; height: 22px; text-align: left;\"><strong>Fail to reject the null hypothesis<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 19.1044%; height: 22px; text-align: left;\"><strong>Null hypothesis is correct<\/strong><\/td>\r\n<td style=\"width: 17.435%; height: 22px; text-align: left;\">Type I error<\/td>\r\n<td style=\"width: 21.3301%; height: 22px; text-align: left;\">No error<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 19.1044%; height: 22px; text-align: left;\"><strong>Null hypothesis is incorrect<\/strong><\/td>\r\n<td style=\"width: 17.435%; height: 22px; text-align: left;\">No error<\/td>\r\n<td style=\"width: 21.3301%; height: 22px; text-align: left;\">Type II error<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h3>Two-Sample Z-Test of Proportions<\/h3>\r\n<ol>\r\n\t<li style=\"font-weight: 400;\">Write out the null and alternative hypotheses.\r\n\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">Null hypothesis: [latex]H_0: p_1=p_2[\/latex] or [latex]H_0: p_1-p_2=0[\/latex]<\/li>\r\n\t<li style=\"font-weight: 400;\">Alternative hypothesis:\r\n\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">[latex]H_A: p_1\\lt p_2[\/latex] or [latex]H_A: p_1-p_2\\lt 0[\/latex]<\/li>\r\n\t<li style=\"font-weight: 400;\">[latex]H_A: p_1&gt;p_2[\/latex] or [latex]H_A: p_1-p_2&gt;0[\/latex]<\/li>\r\n\t<li style=\"font-weight: 400;\">[latex]H_A: p_1\\ne p_2[\/latex] or [latex]H_A: p_1-p_2\\ne0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li style=\"font-weight: 400;\">Check the conditions for the hypothesis test. For testing a one-sample z-test for proportions, we require:\r\n\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">Large counts: For [latex]\\hat{p}_c = \\frac{x_1+x_2}{n_1+n_2}[\/latex], check that: [latex]n_1\\hat{p}_c \\ge 10[\/latex], [latex]n_2\\hat{p}_c \\ge 10[\/latex], [latex]n_1(1-\\hat{p}_c) \\ge 10[\/latex], and [latex]n_2(1-\\hat{p}_c) \\ge 10[\/latex].<\/li>\r\n\t<li style=\"font-weight: 400;\">Random samples\/assignment: Check that the two samples are independent and random samples or that they come from randomly assigned groups in an experiment.<\/li>\r\n\t<li style=\"font-weight: 400;\">10%: Check that [latex]n_1&lt;0.10(N_1)[\/latex] and [latex]n_2&lt;0.10(N_2)[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li style=\"font-weight: 400;\">Calculate a test statistic.<\/li>\r\n\t<li style=\"font-weight: 400;\">Calculate a P-value.<\/li>\r\n\t<li style=\"font-weight: 400;\">Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<\/li>\r\n\t<li style=\"font-weight: 400;\">Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).<\/li>\r\n<\/ol>\r\n<ul>\r\n\t<li>Typically, the conclusion drawn from a two-tailed confidence interval is usually the same as the conclusion drawn from a two-tailed hypothesis test. If a confidence interval contains the hypothesized parameter, a hypothesis test at the 0.05 level will almost always fail to reject the null hypothesis. If the 95% confidence interval does not contain the hypothesized parameter, a hypothesis test at the 0.05 level will almost always reject the null hypothesis. While this does not always hold for tests of proportions, a confidence interval typically provides more information about reasonable values of the parameter.<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<p><strong>pooled sample proportion<\/strong><\/p>\r\n<p>[latex]\\hat{p}_c = \\frac{x_1+x_2}{n_1+n_2}[\/latex]<\/p>\r\n<p><strong>test statistic<\/strong><\/p>\r\n<p>[latex] \\text{test statistic } =\u00a0 z =\\frac{\\stackrel{\u02c6}{p}-p}{\\sqrt{\\frac{p(1-p)}{n}}}[\/latex]<\/p>\r\n<h2>Glossary<\/h2>\r\n<p><strong>hypothesis testing<\/strong><\/p>\r\n<p>the process of forming hypotheses, collecting data, and using the data to draw a conclusion about the hypotheses<\/p>\r\n<p><strong>outcomes of hypothesis tests<\/strong><\/p>\r\n<p>reject the null hypothesis, fail to reject the null hypothesis<\/p>\r\n<p><strong>practical significance<\/strong><\/p>\r\n<p>the results of a hypothesis test are meaningful<\/p>\r\n<p><strong>P-value<\/strong><\/p>\r\n<p>the probability of obtaining a test statistic at least as extreme (in the direction of the alternative hypothesis) as the one that is actually seen if the null hypothesis is true<\/p>\r\n<p><strong>significance level<\/strong><\/p>\r\n<p>the cut-off for P-values at which we have enough evidence to reject the null hypothesis<\/p>\r\n<p><strong>statistical significance<\/strong><\/p>\r\n<p>enough evidence against the null hypothesis to convince us to reject the null hypothesis<\/p>\r\n<p><strong>type I error<\/strong><\/p>\r\n<p>rejecting a correct null hypothesis<\/p>\r\n<p><strong>type II error<\/strong><\/p>\r\n<p>not rejecting an incorrect null hypothesis<\/p>","rendered":"<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Statistics+Exemplar\/Cheat+Sheets\/Module+10_+Cheat+Sheet.pdf\" target=\"_blank\" rel=\"noopener\">Download a PDF of this page here.<\/a><\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<h3>One-Sample Z-Test of Proportions<\/h3>\n<ol>\n<li>Write out the null and alternative hypotheses.\n<ul>\n<li style=\"font-weight: 400;\">The null hypothesis, [latex]H_{0}[\/latex], is what we assume to be true to begin with. It is often a statement of no change from the previous value or from what is expected (e.g., we expect a coin to be fair).\n<ul>\n<li style=\"font-weight: 400;\">The null hypothesis, [latex]H_{0}[\/latex], is always given in the form: [latex]p = value[\/latex] for population proportions.<\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400;\">The alternative hypothesis, [latex]H_{A}[\/latex], is what we consider to be plausible if the null hypothesis is false. Often, it is a change from the null hypothesis that we would like to test the accuracy of. With a null value of [latex]a[\/latex], the alternative hypothesis, [latex]H_{A}[\/latex], is written as an inequality:\n<ul>\n<li style=\"font-weight: 400;\">[latex]H_{A} : p>a[\/latex],<\/li>\n<li style=\"font-weight: 400;\">[latex]H_{A}: p \\lt a[\/latex], or<\/li>\n<li style=\"font-weight: 400;\">[latex]H_{A}: p\\neq a[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Check the conditions for the hypothesis test. For testing a one-sample z-test for proportions, we require:\n<ul>\n<li style=\"font-weight: 400;\">Large counts: Check that [latex]np\\ge10[\/latex] and [latex]n(1-p)\\ge10[\/latex].<\/li>\n<li style=\"font-weight: 400;\">Random samples\/assignment: Check that the sample is a random sample.<\/li>\n<li style=\"font-weight: 400;\">10% population size: Check that the sample size, [latex]n[\/latex], is less than 10% of the population size, [latex]N[\/latex]: [latex]n<0.10(N)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>If not using technology, calculate a test statistic.<\/li>\n<\/ol>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"font-weight: 400;\">A test statistic measures the distance between the sample statistic and the null hypothesis value in terms of the standard error of the null hypothesis value.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 80px;\">[latex]\\text{test statistic }= \\frac{\\text{sample statistic }-\\text{ null hypothesis value}}{\\text{standard error of the null hypothesis value}}=\\frac{\\stackrel{\u02c6}{p}-p}{\\sqrt{\\frac{p(1-p)}{n}}}[\/latex]<\/p>\n<p style=\"padding-left: 80px;\">where [latex]\\stackrel{\u02c6}{p}[\/latex] is the sample statistics and [latex]p[\/latex] is the null hypothesis value.<\/p>\n<p style=\"padding-left: 40px;\">4. Use technology to calculate a P-value.<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>We define the P-value as the probability of obtaining a test statistic at least as extreme (in the direction of the alternative hypothesis) as the one that is actually seen if the null hypothesis is true.<\/li>\n<li>The smaller the probability, the less likely it is that the sample occurred by chance alone, the more evidence we have <em>against<\/em> the null hypothesis.<\/li>\n<li>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.\n<ul>\n<li>The significance level, [latex]\\alpha[\/latex], is the cut-off for P-values at which we have enough evidence to reject the null hypothesis. Typically, small significance levels such as 1%,\u00a0 5%, or 10% are used in hypothesis testing.<br \/>\n<table style=\"border-collapse: collapse; width: 100%; height: 150px;\">\n<thead>\n<tr>\n<th style=\"width: 39.8067%;\">Decision<\/th>\n<th style=\"width: 60.1054%;\">Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 39.8067%;\">If P-value [latex]\\le\\alpha[\/latex], there is enough evidence to reject the null hypothesis.<\/td>\n<td style=\"width: 60.1054%;\">At the [latex]\\alpha\\times[\/latex]100% significance level, the data provide convincing evidence in support of the alternative hypothesis.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 39.8067%;\">If P-value [latex]\\gt\\alpha[\/latex], there is not enough evidence to reject the null hypothesis.<\/td>\n<td style=\"width: 60.1054%;\">At the [latex]\\alpha\\times[\/latex]100% significance level, the data do not provide convincing evidence in support of the alternative hypothesis.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>We never accept the null hypothesis.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">5. Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).\u00a0<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"font-weight: 400;\">If a hypothesis test results in rejecting the null hypothesis because the P-value is less than the significance level, we say we have statistical significance in favor of the alternative hypothesis.<\/li>\n<li style=\"font-weight: 400;\">If the results are meaningful, we say that the results have practical significance. Having practical significance usually means the results show a significant improvement!<\/li>\n<li style=\"font-weight: 400;\">Sometimes, due to chance, the result of the hypothesis test does not align with reality. If we reject a correct null hypothesis, we are committing a type I error. If we do not reject a null hypothesis that is actually incorrect, we are committing a type II error.<\/li>\n<li>\n<table style=\"border-collapse: collapse; width: 57.8692%; height: 66px;\">\n<tbody>\n<tr style=\"height: 22px;\">\n<td style=\"width: 19.1044%; height: 22px; padding-left: 80px; text-align: left;\">\u00a0<\/td>\n<td style=\"width: 17.435%; height: 22px; text-align: left;\"><strong>Reject the null hypothesis<\/strong><\/td>\n<td style=\"width: 21.3301%; height: 22px; text-align: left;\"><strong>Fail to reject the null hypothesis<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 19.1044%; height: 22px; text-align: left;\"><strong>Null hypothesis is correct<\/strong><\/td>\n<td style=\"width: 17.435%; height: 22px; text-align: left;\">Type I error<\/td>\n<td style=\"width: 21.3301%; height: 22px; text-align: left;\">No error<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 19.1044%; height: 22px; text-align: left;\"><strong>Null hypothesis is incorrect<\/strong><\/td>\n<td style=\"width: 17.435%; height: 22px; text-align: left;\">No error<\/td>\n<td style=\"width: 21.3301%; height: 22px; text-align: left;\">Type II error<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h3>Two-Sample Z-Test of Proportions<\/h3>\n<ol>\n<li style=\"font-weight: 400;\">Write out the null and alternative hypotheses.\n<ul>\n<li style=\"font-weight: 400;\">Null hypothesis: [latex]H_0: p_1=p_2[\/latex] or [latex]H_0: p_1-p_2=0[\/latex]<\/li>\n<li style=\"font-weight: 400;\">Alternative hypothesis:\n<ul>\n<li style=\"font-weight: 400;\">[latex]H_A: p_1\\lt p_2[\/latex] or [latex]H_A: p_1-p_2\\lt 0[\/latex]<\/li>\n<li style=\"font-weight: 400;\">[latex]H_A: p_1>p_2[\/latex] or [latex]H_A: p_1-p_2>0[\/latex]<\/li>\n<li style=\"font-weight: 400;\">[latex]H_A: p_1\\ne p_2[\/latex] or [latex]H_A: p_1-p_2\\ne0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400;\">Check the conditions for the hypothesis test. For testing a one-sample z-test for proportions, we require:\n<ul>\n<li style=\"font-weight: 400;\">Large counts: For [latex]\\hat{p}_c = \\frac{x_1+x_2}{n_1+n_2}[\/latex], check that: [latex]n_1\\hat{p}_c \\ge 10[\/latex], [latex]n_2\\hat{p}_c \\ge 10[\/latex], [latex]n_1(1-\\hat{p}_c) \\ge 10[\/latex], and [latex]n_2(1-\\hat{p}_c) \\ge 10[\/latex].<\/li>\n<li style=\"font-weight: 400;\">Random samples\/assignment: Check that the two samples are independent and random samples or that they come from randomly assigned groups in an experiment.<\/li>\n<li style=\"font-weight: 400;\">10%: Check that [latex]n_1<0.10(N_1)[\/latex] and [latex]n_2<0.10(N_2)[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400;\">Calculate a test statistic.<\/li>\n<li style=\"font-weight: 400;\">Calculate a P-value.<\/li>\n<li style=\"font-weight: 400;\">Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<\/li>\n<li style=\"font-weight: 400;\">Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).<\/li>\n<\/ol>\n<ul>\n<li>Typically, the conclusion drawn from a two-tailed confidence interval is usually the same as the conclusion drawn from a two-tailed hypothesis test. If a confidence interval contains the hypothesized parameter, a hypothesis test at the 0.05 level will almost always fail to reject the null hypothesis. If the 95% confidence interval does not contain the hypothesized parameter, a hypothesis test at the 0.05 level will almost always reject the null hypothesis. While this does not always hold for tests of proportions, a confidence interval typically provides more information about reasonable values of the parameter.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<p><strong>pooled sample proportion<\/strong><\/p>\n<p>[latex]\\hat{p}_c = \\frac{x_1+x_2}{n_1+n_2}[\/latex]<\/p>\n<p><strong>test statistic<\/strong><\/p>\n<p>[latex]\\text{test statistic } =\u00a0 z =\\frac{\\stackrel{\u02c6}{p}-p}{\\sqrt{\\frac{p(1-p)}{n}}}[\/latex]<\/p>\n<h2>Glossary<\/h2>\n<p><strong>hypothesis testing<\/strong><\/p>\n<p>the process of forming hypotheses, collecting data, and using the data to draw a conclusion about the hypotheses<\/p>\n<p><strong>outcomes of hypothesis tests<\/strong><\/p>\n<p>reject the null hypothesis, fail to reject the null hypothesis<\/p>\n<p><strong>practical significance<\/strong><\/p>\n<p>the results of a hypothesis test are meaningful<\/p>\n<p><strong>P-value<\/strong><\/p>\n<p>the probability of obtaining a test statistic at least as extreme (in the direction of the alternative hypothesis) as the one that is actually seen if the null hypothesis is true<\/p>\n<p><strong>significance level<\/strong><\/p>\n<p>the cut-off for P-values at which we have enough evidence to reject the null hypothesis<\/p>\n<p><strong>statistical significance<\/strong><\/p>\n<p>enough evidence against the null hypothesis to convince us to reject the null hypothesis<\/p>\n<p><strong>type I error<\/strong><\/p>\n<p>rejecting a correct null hypothesis<\/p>\n<p><strong>type II error<\/strong><\/p>\n<p>not rejecting an incorrect null hypothesis<\/p>\n","protected":false},"author":8,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1205,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1207"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":19,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1207\/revisions"}],"predecessor-version":[{"id":6251,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1207\/revisions\/6251"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1205"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1207\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1207"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1207"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1207"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1207"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}