{"id":602,"date":"2017-04-15T03:28:35","date_gmt":"2017-04-15T03:28:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/conceptstest1\/chapter\/test-of-independence-3-of-3\/"},"modified":"2017-05-31T04:41:52","modified_gmt":"2017-05-31T04:41:52","slug":"test-of-independence-3-of-3","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/chapter\/test-of-independence-3-of-3\/","title":{"raw":"Test of Independence (3 of 3)","rendered":"Test of Independence (3 of 3)"},"content":{"raw":"&nbsp;\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Conduct a chi-square test of independence. Interpret the conclusion in context.<\/li>\r\n<\/ul>\r\n<\/div>\r\nOn this page, we practice the chi-square test for independence in its entirety and learn how to use statistical software to conduct this test. We also investigate the effect of sample size on the chi-square test statistic.\r\n<div class=\"textbox exercises\">\r\n<h3>Learn By Doing<\/h3>\r\n<h2>A Real Court Case<\/h2>\r\nIn the early 1970s, a young man challenged an Oklahoma state law that prohibited the sale of 3.2% beer to males under age 21 but allowed its sale to females in the same age group. The case (<em>Craig v. Boren<\/em>, 429 U.S. 190, 1976) was ultimately heard by the U.S. Supreme Court. The state of Oklahoma argued that the law improved traffic safety. One of the three main pieces of data presented to the court was the result of a \u201crandom roadside survey.\u201d This survey gathered information on gender and whether or not the driver had been drinking alcohol in the previous 2 hours. A total of 619 drivers under 21 years of age were included in the survey.\r\n\r\nPlease <a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/chi_squared_cal\/chisquared1.html\" target=\"new\">click here to open the simulation<\/a> for use in the following activity.\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3809\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3810\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3811\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3812\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3813\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3814\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3815\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3816\r\n\r\n<\/div>\r\n<h3>Comment: The Effect of Sample Size on Chi-Square<\/h3>\r\nWith other hypothesis tests, we have seen that sample size can affect the P-value and our conclusion. This is also true for chi-square. To illustrate this idea, we multiplied all of the counts in the Oklahoma data by 3.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032832\/m11_chi_square_tests_topic_11_2_m11_test_independence_3_image8.png\" alt=\"Conditional percents of the data from the Oklahoma case\" width=\"521\" height=\"295\" \/>\r\n\r\nNotice that the conditional percentages do not change, so the new \u201cdata\u201d shows the same relationship between gender and drinking before driving. The probability that a driver under the age of 21 drinks alcohol before driving is still about 15.0% (279\/1857). Males are still more likely to consume alcohol before driving (231\/1443 = 16.0%) than are females (48\/414 = 11.6%), with the same difference of 4.4% that we saw in the original data.\r\n\r\nWe used technology to find expected counts and the chi-square test statistic.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032834\/m11_chi_square_tests_topic_11_2_m11_test_independence_3_image9.png\" alt=\"The chi-square value for the original data is 1.637. The chi-square value for the original data multiplied by 3 is 4.91\" width=\"575\" height=\"263\" \/>\r\n\r\nNotice that multiplying the observed counts by 3 also triples the expected counts and the chi-square value. This increase in the chi-square value gives a statistically significant P-value of 0.0267, which changes our conclusion. With this larger sample, the evidence is strong enough to reject the null hypothesis. We conclude that gender is associated with drinking alcohol before driving. The variables are dependent for drivers under the age of 21 in Oklahoma. With this sample size, the data provides evidence in support of the Oklahoma law that forbids sale of 3.2% beer to males and permits it to females with the goal of improving traffic safety.\r\n\r\n<strong>What\u2019s the point?<\/strong> We see once again that sample size affects the P-value in a hypothesis test. This means that a small sample may not detect a relationship that exists between two categorical variables in a population. Conversely, a large sample may indicate that a relationship is statistically significant on the basis of differences in observed and expected counts that are not important in a practical sense.\r\n\r\n&nbsp;","rendered":"<p>&nbsp;<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Conduct a chi-square test of independence. Interpret the conclusion in context.<\/li>\n<\/ul>\n<\/div>\n<p>On this page, we practice the chi-square test for independence in its entirety and learn how to use statistical software to conduct this test. We also investigate the effect of sample size on the chi-square test statistic.<\/p>\n<div class=\"textbox exercises\">\n<h3>Learn By Doing<\/h3>\n<h2>A Real Court Case<\/h2>\n<p>In the early 1970s, a young man challenged an Oklahoma state law that prohibited the sale of 3.2% beer to males under age 21 but allowed its sale to females in the same age group. The case (<em>Craig v. Boren<\/em>, 429 U.S. 190, 1976) was ultimately heard by the U.S. Supreme Court. The state of Oklahoma argued that the law improved traffic safety. One of the three main pieces of data presented to the court was the result of a \u201crandom roadside survey.\u201d This survey gathered information on gender and whether or not the driver had been drinking alcohol in the previous 2 hours. A total of 619 drivers under 21 years of age were included in the survey.<\/p>\n<p>Please <a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/Concepts+in+Statistics\/interactives\/chi_squared_cal\/chisquared1.html\" target=\"new\">click here to open the simulation<\/a> for use in the following activity.<\/p>\n<p>\t<iframe id=\"lumen_assessment_3809\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3809&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3809\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3810\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3810&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3810\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3811\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3811&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3811\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3812\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3812&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3812\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3813\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3813&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3813\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3814\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3814&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3814\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3815\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3815&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3815\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3816\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3816&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3816\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<h3>Comment: The Effect of Sample Size on Chi-Square<\/h3>\n<p>With other hypothesis tests, we have seen that sample size can affect the P-value and our conclusion. This is also true for chi-square. To illustrate this idea, we multiplied all of the counts in the Oklahoma data by 3.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032832\/m11_chi_square_tests_topic_11_2_m11_test_independence_3_image8.png\" alt=\"Conditional percents of the data from the Oklahoma case\" width=\"521\" height=\"295\" \/><\/p>\n<p>Notice that the conditional percentages do not change, so the new \u201cdata\u201d shows the same relationship between gender and drinking before driving. The probability that a driver under the age of 21 drinks alcohol before driving is still about 15.0% (279\/1857). Males are still more likely to consume alcohol before driving (231\/1443 = 16.0%) than are females (48\/414 = 11.6%), with the same difference of 4.4% that we saw in the original data.<\/p>\n<p>We used technology to find expected counts and the chi-square test statistic.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032834\/m11_chi_square_tests_topic_11_2_m11_test_independence_3_image9.png\" alt=\"The chi-square value for the original data is 1.637. The chi-square value for the original data multiplied by 3 is 4.91\" width=\"575\" height=\"263\" \/><\/p>\n<p>Notice that multiplying the observed counts by 3 also triples the expected counts and the chi-square value. This increase in the chi-square value gives a statistically significant P-value of 0.0267, which changes our conclusion. With this larger sample, the evidence is strong enough to reject the null hypothesis. We conclude that gender is associated with drinking alcohol before driving. The variables are dependent for drivers under the age of 21 in Oklahoma. With this sample size, the data provides evidence in support of the Oklahoma law that forbids sale of 3.2% beer to males and permits it to females with the goal of improving traffic safety.<\/p>\n<p><strong>What\u2019s the point?<\/strong> We see once again that sample size affects the P-value in a hypothesis test. This means that a small sample may not detect a relationship that exists between two categorical variables in a population. Conversely, a large sample may indicate that a relationship is statistically significant on the basis of differences in observed and expected counts that are not important in a practical sense.<\/p>\n<p>&nbsp;<\/p>\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-602\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><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":163,"menu_order":8,"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\":\"\"}]","CANDELA_OUTCOMES_GUID":"0ac61310-cf5d-4e25-a23c-ccb125efc153","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-602","chapter","type-chapter","status-web-only","hentry"],"part":570,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/602","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/users\/163"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/602\/revisions"}],"predecessor-version":[{"id":1525,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/602\/revisions\/1525"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/parts\/570"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/602\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/media?parent=602"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapter-type?post=602"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/contributor?post=602"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/license?post=602"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}