{"id":286,"date":"2021-07-14T15:59:06","date_gmt":"2021-07-14T15:59:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/introduction-hypothesis-testing-with-two-samples\/"},"modified":"2022-03-28T17:55:17","modified_gmt":"2022-03-28T17:55:17","slug":"why-it-matters-hypothesis-testing-with-two-samples","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/why-it-matters-hypothesis-testing-with-two-samples\/","title":{"raw":"Why It Matters: Hypothesis Testing with Two Samples","rendered":"Why It Matters: Hypothesis Testing with Two Samples"},"content":{"raw":"<div>\r\n<div class=\"media-body\"><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Why learn to make inferences about two populations?<\/span><\/div>\r\n<\/div>\r\nThe concepts discussed in the module <em>Hypothesis Testing with One Sample<\/em> can be applied to situations involving two samples. The reason we can do this is due to the following big ideas:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Random samples vary. When we use a sample proportion or sample mean to make an inference about a population proportion or population mean, there is uncertainty. For this reason, inference involves probability.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Under certain conditions, we can model the variability in sample proportions or sample means with a normal curve. We use the normal curve to make probability-based decisions about population values.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">We can estimate a population proportion or a population mean with a confidence interval. The confidence interval is an actual sample proportion or sample mean plus or minus a margin of error. We state our confidence in the accuracy of these intervals using probability.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">We can test a hypothesis about a population proportion or population mean using a sample proportion or a sample mean. Again, we base our conclusion on probability using a P-value. The P-value describes the strength of our evidence in rejecting a hypothesis about the population.<\/li>\r\n<\/ul>\r\nIn <em>Hypothesis Testing with Two Samples<\/em>, we extend these big ideas to make inferences that compare two populations (or two treatments). An example of such an inference follows:\r\n<ol>\r\n \t<li style=\"list-style-type: none;\"><\/li>\r\n<\/ol>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<h2>The Abecedarian Early Intervention Project<\/h2>\r\nIn the 1970s, Abecedarian Early Intervention Project studied the long-term effects of early childhood education for poor children.\r\n\r\n<strong>Research question: <\/strong><em>Does early childhood education increase the likelihood of college attendance for poor children?<\/em>\r\n<ol>\r\n \t<li><strong>Produce Data:<\/strong> <em>Determine what to measure, then collect the data.\u00a0<\/em>In this experiment, researchers selected 111 high-risk infants on the basis of the mothers\u2019 education, family income, and other factors. They randomly assigned 57 infants to receive 5 years of high-quality preschool. The remaining 54 infants were a control group. All children received nutritional supplements, social services, and health care to control the effects of these confounding factors on the outcomes of the experiment.<\/li>\r\n \t<li><strong>Exploratory Data Analysis:<\/strong> <em>Analyze and summarize the data<\/em>. By the age of 21 a much higher percentage of the treatment group enrolled in college, 42% vs. 20%.<\/li>\r\n \t<li><strong>Draw a Conclusion:<\/strong> <em>Use data, probability, and statistical inference to draw a conclusion about the populations<\/em>. Is this difference statistically significant? In other words, is this difference due to the pre-school experience or due to chance? We will test the claim that a larger proportion of children who attend pre-school will attend college.<\/li>\r\n<\/ol>\r\nThe following figure summarizes this investigation in the Big Picture.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032410\/m9_inference_two_proportion_topic_9_1_m9_intro__dist_diff_sample_proportion_1_image2.png\" alt=\"The Big Picture applied to the Abecedarian Early Intervention Project\" width=\"728\" height=\"720\" \/>\r\n\r\n<\/div>","rendered":"<div>\n<div class=\"media-body\"><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Why learn to make inferences about two populations?<\/span><\/div>\n<\/div>\n<p>The concepts discussed in the module <em>Hypothesis Testing with One Sample<\/em> can be applied to situations involving two samples. The reason we can do this is due to the following big ideas:<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Random samples vary. When we use a sample proportion or sample mean to make an inference about a population proportion or population mean, there is uncertainty. For this reason, inference involves probability.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Under certain conditions, we can model the variability in sample proportions or sample means with a normal curve. We use the normal curve to make probability-based decisions about population values.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">We can estimate a population proportion or a population mean with a confidence interval. The confidence interval is an actual sample proportion or sample mean plus or minus a margin of error. We state our confidence in the accuracy of these intervals using probability.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">We can test a hypothesis about a population proportion or population mean using a sample proportion or a sample mean. Again, we base our conclusion on probability using a P-value. The P-value describes the strength of our evidence in rejecting a hypothesis about the population.<\/li>\n<\/ul>\n<p>In <em>Hypothesis Testing with Two Samples<\/em>, we extend these big ideas to make inferences that compare two populations (or two treatments). An example of such an inference follows:<\/p>\n<ol>\n<li style=\"list-style-type: none;\"><\/li>\n<\/ol>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<h2>The Abecedarian Early Intervention Project<\/h2>\n<p>In the 1970s, Abecedarian Early Intervention Project studied the long-term effects of early childhood education for poor children.<\/p>\n<p><strong>Research question: <\/strong><em>Does early childhood education increase the likelihood of college attendance for poor children?<\/em><\/p>\n<ol>\n<li><strong>Produce Data:<\/strong> <em>Determine what to measure, then collect the data.\u00a0<\/em>In this experiment, researchers selected 111 high-risk infants on the basis of the mothers\u2019 education, family income, and other factors. They randomly assigned 57 infants to receive 5 years of high-quality preschool. The remaining 54 infants were a control group. All children received nutritional supplements, social services, and health care to control the effects of these confounding factors on the outcomes of the experiment.<\/li>\n<li><strong>Exploratory Data Analysis:<\/strong> <em>Analyze and summarize the data<\/em>. By the age of 21 a much higher percentage of the treatment group enrolled in college, 42% vs. 20%.<\/li>\n<li><strong>Draw a Conclusion:<\/strong> <em>Use data, probability, and statistical inference to draw a conclusion about the populations<\/em>. Is this difference statistically significant? In other words, is this difference due to the pre-school experience or due to chance? We will test the claim that a larger proportion of children who attend pre-school will attend college.<\/li>\n<\/ol>\n<p>The following figure summarizes this investigation in the Big Picture.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032410\/m9_inference_two_proportion_topic_9_1_m9_intro__dist_diff_sample_proportion_1_image2.png\" alt=\"The Big Picture applied to the Abecedarian Early Intervention Project\" width=\"728\" height=\"720\" \/><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-286\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li><strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Concepts in Statistics. <strong>Provided by<\/strong>: Open Learning Initiative. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/oli.cmu.edu\">http:\/\/oli.cmu.edu<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Concepts in Statistics\",\"author\":\"\",\"organization\":\"Open Learning Initiative\",\"url\":\"http:\/\/oli.cmu.edu\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-286","chapter","type-chapter","status-publish","hentry"],"part":285,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/286","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/286\/revisions"}],"predecessor-version":[{"id":3884,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/286\/revisions\/3884"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/285"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/286\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=286"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=286"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=286"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=286"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}