{"id":1333,"date":"2023-06-22T02:17:42","date_gmt":"2023-06-22T02:17:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-two-population-means-independent-samples-apply-it-1\/"},"modified":"2025-05-16T22:32:02","modified_gmt":"2025-05-16T22:32:02","slug":"comparing-two-population-means-independent-samples-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-two-population-means-independent-samples-apply-it-1\/","title":{"raw":"Comparing Two Population Means (Independent Samples): Apply It 1","rendered":"Comparing Two Population Means (Independent Samples): Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Complete a two-sample t-test for independent population means from hypotheses to conclusions&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Complete a two-sample [latex]t[\/latex]-test for independent population means from hypotheses to conclusions<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Hate Crime in California<\/h2>\r\n<p><img class=\"wp-image-8281 size-medium alignright\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22021741\/Stop-Asian-Hate-Yellow-Blue-Red-Quotes-Bright-and-Positive-Illustrative-Instagram-Post-1.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p>\r\n<p>A hate crime is defined as a criminal offense that is motivated, in whole or in part, by the offender\u2019s bias(es) against a person based on race, ethnicity, ancestry, religion, sexual orientation, disability, gender, and gender identity. [footnote]https:\/\/www.justice.gov\/crs\/highlights\/2020-hate-crimes-statistics[\/footnote] In the year 2020, [latex]62\\%[\/latex] of victims were targeted because of the offenders\u2019 bias toward race\/ethnicity\/ancestry, which continues to be the largest bias motivation category. Participating agencies reported [latex]5,227[\/latex] race\/ethnicity\/ancestry-based incidents in 2020, a [latex]32\\%[\/latex] increase from 2019. Anti-Black or African American hate crimes continue to be the largest bias incident victim category, with [latex]2,871[\/latex] incidents in 2020, a [latex]49\\%[\/latex] increase since 2019. Additionally, there were [latex]279[\/latex] anti-Asian incidents reported in 2020, a [latex]77\\%[\/latex] increase since 2019. The other largest categories of hate crimes include anti-Hispanic or Latino incidents, with [latex]517[\/latex], and anti-White incidents, with [latex]869[\/latex] in total.<\/p>\r\n<p>Suppose we wanted to study the increase in hate crime from 2019 to 2020, specifically in the state of California.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]5514[\/ohm2_question]<\/section>\r\n<p>Previously, you have conducted a one-sample [latex]t[\/latex]-test (a hypothesis test about a population mean). We can apply the same steps to analyze and test a hypothesis about the difference in means for two independent populations. A hypothesis test for comparing two population means is often referred to as a <strong>two-sample [latex]t[\/latex]-test.<\/strong><\/p>\r\n<p>[reveal-answer q=\"630785\"]<strong>Steps for Hypothesis Testing<\/strong>[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"630785\"]<\/p>\r\n<ol>\r\n\t<li>Write out the null and alternative hypotheses.<\/li>\r\n\t<li>Check the conditions for the hypothesis test.<\/li>\r\n\t<li>Calculate a test statistic.<\/li>\r\n\t<li>Calculate a P-value.<\/li>\r\n\t<li>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<\/li>\r\n\t<li>Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]5521[\/ohm2_question]<\/section>\r\n<p>Note: Previously, we have verified that the assumptions for a two-sample [latex]t[\/latex]-test were met for the Hate Crime data set for the state of California. We know that the two samples were independent, representative of the populations of interest, and large.<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Complete a two-sample t-test for independent population means from hypotheses to conclusions&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Complete a two-sample [latex]t[\/latex]-test for independent population means from hypotheses to conclusions<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Hate Crime in California<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-8281 size-medium alignright\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22021741\/Stop-Asian-Hate-Yellow-Blue-Red-Quotes-Bright-and-Positive-Illustrative-Instagram-Post-1.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p>\n<p>A hate crime is defined as a criminal offense that is motivated, in whole or in part, by the offender\u2019s bias(es) against a person based on race, ethnicity, ancestry, religion, sexual orientation, disability, gender, and gender identity. <a class=\"footnote\" title=\"https:\/\/www.justice.gov\/crs\/highlights\/2020-hate-crimes-statistics\" id=\"return-footnote-1333-1\" href=\"#footnote-1333-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> In the year 2020, [latex]62\\%[\/latex] of victims were targeted because of the offenders\u2019 bias toward race\/ethnicity\/ancestry, which continues to be the largest bias motivation category. Participating agencies reported [latex]5,227[\/latex] race\/ethnicity\/ancestry-based incidents in 2020, a [latex]32\\%[\/latex] increase from 2019. Anti-Black or African American hate crimes continue to be the largest bias incident victim category, with [latex]2,871[\/latex] incidents in 2020, a [latex]49\\%[\/latex] increase since 2019. Additionally, there were [latex]279[\/latex] anti-Asian incidents reported in 2020, a [latex]77\\%[\/latex] increase since 2019. The other largest categories of hate crimes include anti-Hispanic or Latino incidents, with [latex]517[\/latex], and anti-White incidents, with [latex]869[\/latex] in total.<\/p>\n<p>Suppose we wanted to study the increase in hate crime from 2019 to 2020, specifically in the state of California.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm5514\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=5514&theme=lumen&iframe_resize_id=ohm5514&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Previously, you have conducted a one-sample [latex]t[\/latex]-test (a hypothesis test about a population mean). We can apply the same steps to analyze and test a hypothesis about the difference in means for two independent populations. A hypothesis test for comparing two population means is often referred to as a <strong>two-sample [latex]t[\/latex]-test.<\/strong><\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q630785\"><strong>Steps for Hypothesis Testing<\/strong><\/button><\/p>\n<div id=\"q630785\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Write out the null and alternative hypotheses.<\/li>\n<li>Check the conditions for the hypothesis test.<\/li>\n<li>Calculate a test statistic.<\/li>\n<li>Calculate a P-value.<\/li>\n<li>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<\/li>\n<li>Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm5521\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=5521&theme=lumen&iframe_resize_id=ohm5521&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Note: Previously, we have verified that the assumptions for a two-sample [latex]t[\/latex]-test were met for the Hate Crime data set for the state of California. We know that the two samples were independent, representative of the populations of interest, and large.<\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1333-1\">https:\/\/www.justice.gov\/crs\/highlights\/2020-hate-crimes-statistics <a href=\"#return-footnote-1333-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":8,"menu_order":23,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1309,"module-header":"apply_it","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\/1333"}],"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":5,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1333\/revisions"}],"predecessor-version":[{"id":6818,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1333\/revisions\/6818"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1309"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1333\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1333"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1333"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1333"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1333"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}