{"id":1331,"date":"2023-06-22T02:17:39","date_gmt":"2023-06-22T02:17:39","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-two-population-means-independent-samples-learn-it-3\/"},"modified":"2025-05-16T22:30:26","modified_gmt":"2025-05-16T22:30:26","slug":"comparing-two-population-means-independent-samples-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/comparing-two-population-means-independent-samples-learn-it-3\/","title":{"raw":"Comparing Two Population Means (Independent Samples): Learn It 3","rendered":"Comparing Two Population Means (Independent Samples): Learn It 3"},"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<p>Let's visualize the difference in means.<\/p>\r\n<p>[reveal-answer q=\"240453\"]Hate Crime Data Set[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"240453\"]<\/p>\r\n<table style=\"width: 254px;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 101.484px;\"><strong>County in California<\/strong><\/td>\r\n<td style=\"width: 47.3004px;\"><strong>Year 2019<\/strong><\/td>\r\n<td style=\"width: 52.3264px;\"><strong>Year 2020<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Alameda<\/td>\r\n<td style=\"width: 47.3004px;\">2.6<\/td>\r\n<td style=\"width: 52.3264px;\">3.1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Amador<\/td>\r\n<td style=\"width: 47.3004px;\">6.3<\/td>\r\n<td style=\"width: 52.3264px;\">7.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Butte<\/td>\r\n<td style=\"width: 47.3004px;\">2.3<\/td>\r\n<td style=\"width: 52.3264px;\">1.4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Colusa<\/td>\r\n<td style=\"width: 47.3004px;\">4.7<\/td>\r\n<td style=\"width: 52.3264px;\">4.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Contra Costa<\/td>\r\n<td style=\"width: 47.3004px;\">3<\/td>\r\n<td style=\"width: 52.3264px;\">2.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Del Notre<\/td>\r\n<td style=\"width: 47.3004px;\">3.6<\/td>\r\n<td style=\"width: 52.3264px;\">3.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">El Dorado<\/td>\r\n<td style=\"width: 47.3004px;\">0.5<\/td>\r\n<td style=\"width: 52.3264px;\">2.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Fresno<\/td>\r\n<td style=\"width: 47.3004px;\">1.6<\/td>\r\n<td style=\"width: 52.3264px;\">1.7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Glenn<\/td>\r\n<td style=\"width: 47.3004px;\">3.5<\/td>\r\n<td style=\"width: 52.3264px;\">3.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Humboldt<\/td>\r\n<td style=\"width: 47.3004px;\">7.4<\/td>\r\n<td style=\"width: 52.3264px;\">2.2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Imperial<\/td>\r\n<td style=\"width: 47.3004px;\">0.6<\/td>\r\n<td style=\"width: 52.3264px;\">1.1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Kern<\/td>\r\n<td style=\"width: 47.3004px;\">0.8<\/td>\r\n<td style=\"width: 52.3264px;\">1.2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Kings<\/td>\r\n<td style=\"width: 47.3004px;\">0.7<\/td>\r\n<td style=\"width: 52.3264px;\">0.7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Lake<\/td>\r\n<td style=\"width: 47.3004px;\">4.7<\/td>\r\n<td style=\"width: 52.3264px;\">4.7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Lassen<\/td>\r\n<td style=\"width: 47.3004px;\">3.3<\/td>\r\n<td style=\"width: 52.3264px;\">3.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Los Angeles<\/td>\r\n<td style=\"width: 47.3004px;\">4.1<\/td>\r\n<td style=\"width: 52.3264px;\">5.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Marin<\/td>\r\n<td style=\"width: 47.3004px;\">5<\/td>\r\n<td style=\"width: 52.3264px;\">8.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Mendocino<\/td>\r\n<td style=\"width: 47.3004px;\">2.3<\/td>\r\n<td style=\"width: 52.3264px;\">1.2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Merced<\/td>\r\n<td style=\"width: 47.3004px;\">0.4<\/td>\r\n<td style=\"width: 52.3264px;\">1.1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Monterey<\/td>\r\n<td style=\"width: 47.3004px;\">1.6<\/td>\r\n<td style=\"width: 52.3264px;\">2.1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Napa<\/td>\r\n<td style=\"width: 47.3004px;\">1.4<\/td>\r\n<td style=\"width: 52.3264px;\">1.4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Nevada<\/td>\r\n<td style=\"width: 47.3004px;\">1<\/td>\r\n<td style=\"width: 52.3264px;\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Orange<\/td>\r\n<td style=\"width: 47.3004px;\">2.3<\/td>\r\n<td style=\"width: 52.3264px;\">2.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Placer<\/td>\r\n<td style=\"width: 47.3004px;\">1.5<\/td>\r\n<td style=\"width: 52.3264px;\">2.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Riverside<\/td>\r\n<td style=\"width: 47.3004px;\">1.5<\/td>\r\n<td style=\"width: 52.3264px;\">1.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Sacramento<\/td>\r\n<td style=\"width: 47.3004px;\">1<\/td>\r\n<td style=\"width: 52.3264px;\">1.4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">San Benito<\/td>\r\n<td style=\"width: 47.3004px;\">1.6<\/td>\r\n<td style=\"width: 52.3264px;\">1.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">San Bernardino<\/td>\r\n<td style=\"width: 47.3004px;\">1.1<\/td>\r\n<td style=\"width: 52.3264px;\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">San Diego<\/td>\r\n<td style=\"width: 47.3004px;\">3<\/td>\r\n<td style=\"width: 52.3264px;\">3.7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">San Fransisco<\/td>\r\n<td style=\"width: 47.3004px;\">7.4<\/td>\r\n<td style=\"width: 52.3264px;\">6.2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">San Joaquin<\/td>\r\n<td style=\"width: 47.3004px;\">0.7<\/td>\r\n<td style=\"width: 52.3264px;\">1.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">San Luis Obispo<\/td>\r\n<td style=\"width: 47.3004px;\">4.6<\/td>\r\n<td style=\"width: 52.3264px;\">4.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">San Mateo<\/td>\r\n<td style=\"width: 47.3004px;\">2.2<\/td>\r\n<td style=\"width: 52.3264px;\">2.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Santa Barbara<\/td>\r\n<td style=\"width: 47.3004px;\">0.2<\/td>\r\n<td style=\"width: 52.3264px;\">0.9<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Santa Clara<\/td>\r\n<td style=\"width: 47.3004px;\">2.6<\/td>\r\n<td style=\"width: 52.3264px;\">6.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Santa Cruz<\/td>\r\n<td style=\"width: 47.3004px;\">3.7<\/td>\r\n<td style=\"width: 52.3264px;\">4.8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Shasta<\/td>\r\n<td style=\"width: 47.3004px;\">0.6<\/td>\r\n<td style=\"width: 52.3264px;\">3.9<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Solano<\/td>\r\n<td style=\"width: 47.3004px;\">1.6<\/td>\r\n<td style=\"width: 52.3264px;\">3.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Sonoma<\/td>\r\n<td style=\"width: 47.3004px;\">2.2<\/td>\r\n<td style=\"width: 52.3264px;\">5.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Stanislaus<\/td>\r\n<td style=\"width: 47.3004px;\">1.3<\/td>\r\n<td style=\"width: 52.3264px;\">3.1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Sutter<\/td>\r\n<td style=\"width: 47.3004px;\">1<\/td>\r\n<td style=\"width: 52.3264px;\">4.2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Tehama<\/td>\r\n<td style=\"width: 47.3004px;\">3.1<\/td>\r\n<td style=\"width: 52.3264px;\">6.2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Tulare<\/td>\r\n<td style=\"width: 47.3004px;\">0.2<\/td>\r\n<td style=\"width: 52.3264px;\">0.4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Tuolumne<\/td>\r\n<td style=\"width: 47.3004px;\">1.8<\/td>\r\n<td style=\"width: 52.3264px;\">1.8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Ventura<\/td>\r\n<td style=\"width: 47.3004px;\">0.8<\/td>\r\n<td style=\"width: 52.3264px;\">1.8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Yolo<\/td>\r\n<td style=\"width: 47.3004px;\">2.3<\/td>\r\n<td style=\"width: 52.3264px;\">8.2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 101.484px;\">Yuba<\/td>\r\n<td style=\"width: 47.3004px;\">1.3<\/td>\r\n<td style=\"width: 52.3264px;\">2.5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>Note: Hate crimes data are limited as they do not fully capture all hate\u00a0incidents.[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"203470\"]Year 2019[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"203470\"]<\/p>\r\n<table style=\"border-collapse: collapse; width: 29px;\" border=\"0\" width=\"87\" cellspacing=\"0\" cellpadding=\"0\">\r\n<tbody>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18.0pt;\">\r\n<td style=\"height: 18pt; width: 237.448px;\" align=\"right\" height=\"24\">6.3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18.0pt;\">\r\n<td style=\"height: 18pt; width: 237.448px;\" align=\"right\" height=\"24\">2.3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18.0pt;\">\r\n<td style=\"height: 18pt; box-sizing: border-box; width: 278.003px;\" align=\"right\" height=\"24\">4.7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; box-sizing: border-box; width: 278.003px;\" align=\"right\" height=\"21\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">0.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">7.4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">0.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">0.8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">0.7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">4.7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3.3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">4.1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">0.4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">7.4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">0.7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">4.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">0.2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3.7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">0.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3.1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">0.2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">0.8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"628393\"]Year 2020[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"628393\"]<\/p>\r\n<table style=\"border-collapse: collapse; width: 79px;\" border=\"0\" width=\"87\" cellspacing=\"0\" cellpadding=\"0\">\r\n<tbody>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3.1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18.0pt;\">\r\n<td style=\"height: 18pt; width: 237.448px;\" align=\"right\" height=\"24\">7.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18.0pt;\">\r\n<td style=\"height: 18pt; width: 237.448px;\" align=\"right\" height=\"24\">1.4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18.0pt;\">\r\n<td style=\"height: 18pt; box-sizing: border-box; width: 278.003px;\" align=\"right\" height=\"24\">4.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; box-sizing: border-box; width: 278.003px;\" align=\"right\" height=\"21\">2.3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">0.7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">4.7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3.3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">5.3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">8.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3.7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">6.2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">4.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">0.9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">6.3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">4.8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3.9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3.6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">5.3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">3.1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">4.2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">6.2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">0.4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">1.8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">8.2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16.0pt;\">\r\n<td style=\"height: 16pt; width: 237.448px;\" align=\"right\" height=\"21\">2.5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/ \" width=\"100%\" height=\"900\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><\/iframe><br \/>\r\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]5486[\/ohm2_question]<\/section>\r\n<p>The analysis above uses descriptive statistics only. How can we make an inference about the increase in hate crimes in the United States?<\/p>\r\n<p>We can make an inference using a hypothesis test to provide evidence that there is an increase in the number of hate crimes between 2019 and 2020. A hypothesis test is needed to go beyond the visualizations and show that the difference is not simply sampling variability.<\/p>\r\n<h3>Standard Error of the Difference of Means<\/h3>\r\n<p>We can use a hypothesis test to determine if the observed difference in sample means is consistent with a hypothesized difference in population means.<\/p>\r\n<p>To do this, we use what we know about the sampling distribution of [latex]\\bar{x}_1-\\bar{x}_2[\/latex] and, in particular, its estimated standard deviation (the standard error). Recall that you learned that the difference in the sample means, [latex]\\bar{x}_1-\\bar{x}_2[\/latex], also has an approximately normal distribution, centered at the difference of the population means, [latex]\\mu_1-\\mu_2[\/latex].<\/p>\r\n<p>The standard deviation is given by the following formula: [latex]\\sqrt{\\frac{\\sigma_1^2}{n_1}+\\frac{\\sigma_2^2}{n_2}}[\/latex]<\/p>\r\n<p>In practice, we will have to estimate the standard deviation because it depends on the unknown population standard deviations. Replacing [latex]\\sigma_1[\/latex] and [latex]\\sigma_2[\/latex] by the sample standard deviations [latex]s_1[\/latex] and [latex]s_2[\/latex], we will get the <strong>standard error <\/strong>of the difference.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>standard error of the difference of means<\/h3>\r\n<p><strong>standard error<\/strong> of [latex]\\bar{x}_1-\\bar{x}_2[\/latex]: [latex]\\sqrt{\\dfrac{s_1^2}{n_1}+\\dfrac{s_2^2}{n_2}}[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]5498[\/ohm2_question]<\/section>\r\n<p>Now that we have the standard error, we can calculate the test statistic, P-value, and make an inference about the population. We can leverage technology to generate our statistical output and use it to interpret our results.<\/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<p>Let&#8217;s visualize the difference in means.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q240453\">Hate Crime Data Set<\/button><\/p>\n<div id=\"q240453\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"width: 254px;\">\n<tbody>\n<tr>\n<td style=\"width: 101.484px;\"><strong>County in California<\/strong><\/td>\n<td style=\"width: 47.3004px;\"><strong>Year 2019<\/strong><\/td>\n<td style=\"width: 52.3264px;\"><strong>Year 2020<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Alameda<\/td>\n<td style=\"width: 47.3004px;\">2.6<\/td>\n<td style=\"width: 52.3264px;\">3.1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Amador<\/td>\n<td style=\"width: 47.3004px;\">6.3<\/td>\n<td style=\"width: 52.3264px;\">7.5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Butte<\/td>\n<td style=\"width: 47.3004px;\">2.3<\/td>\n<td style=\"width: 52.3264px;\">1.4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Colusa<\/td>\n<td style=\"width: 47.3004px;\">4.7<\/td>\n<td style=\"width: 52.3264px;\">4.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Contra Costa<\/td>\n<td style=\"width: 47.3004px;\">3<\/td>\n<td style=\"width: 52.3264px;\">2.3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Del Notre<\/td>\n<td style=\"width: 47.3004px;\">3.6<\/td>\n<td style=\"width: 52.3264px;\">3.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">El Dorado<\/td>\n<td style=\"width: 47.3004px;\">0.5<\/td>\n<td style=\"width: 52.3264px;\">2.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Fresno<\/td>\n<td style=\"width: 47.3004px;\">1.6<\/td>\n<td style=\"width: 52.3264px;\">1.7<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Glenn<\/td>\n<td style=\"width: 47.3004px;\">3.5<\/td>\n<td style=\"width: 52.3264px;\">3.5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Humboldt<\/td>\n<td style=\"width: 47.3004px;\">7.4<\/td>\n<td style=\"width: 52.3264px;\">2.2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Imperial<\/td>\n<td style=\"width: 47.3004px;\">0.6<\/td>\n<td style=\"width: 52.3264px;\">1.1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Kern<\/td>\n<td style=\"width: 47.3004px;\">0.8<\/td>\n<td style=\"width: 52.3264px;\">1.2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Kings<\/td>\n<td style=\"width: 47.3004px;\">0.7<\/td>\n<td style=\"width: 52.3264px;\">0.7<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Lake<\/td>\n<td style=\"width: 47.3004px;\">4.7<\/td>\n<td style=\"width: 52.3264px;\">4.7<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Lassen<\/td>\n<td style=\"width: 47.3004px;\">3.3<\/td>\n<td style=\"width: 52.3264px;\">3.3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Los Angeles<\/td>\n<td style=\"width: 47.3004px;\">4.1<\/td>\n<td style=\"width: 52.3264px;\">5.3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Marin<\/td>\n<td style=\"width: 47.3004px;\">5<\/td>\n<td style=\"width: 52.3264px;\">8.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Mendocino<\/td>\n<td style=\"width: 47.3004px;\">2.3<\/td>\n<td style=\"width: 52.3264px;\">1.2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Merced<\/td>\n<td style=\"width: 47.3004px;\">0.4<\/td>\n<td style=\"width: 52.3264px;\">1.1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Monterey<\/td>\n<td style=\"width: 47.3004px;\">1.6<\/td>\n<td style=\"width: 52.3264px;\">2.1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Napa<\/td>\n<td style=\"width: 47.3004px;\">1.4<\/td>\n<td style=\"width: 52.3264px;\">1.4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Nevada<\/td>\n<td style=\"width: 47.3004px;\">1<\/td>\n<td style=\"width: 52.3264px;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Orange<\/td>\n<td style=\"width: 47.3004px;\">2.3<\/td>\n<td style=\"width: 52.3264px;\">2.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Placer<\/td>\n<td style=\"width: 47.3004px;\">1.5<\/td>\n<td style=\"width: 52.3264px;\">2.5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Riverside<\/td>\n<td style=\"width: 47.3004px;\">1.5<\/td>\n<td style=\"width: 52.3264px;\">1.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Sacramento<\/td>\n<td style=\"width: 47.3004px;\">1<\/td>\n<td style=\"width: 52.3264px;\">1.4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">San Benito<\/td>\n<td style=\"width: 47.3004px;\">1.6<\/td>\n<td style=\"width: 52.3264px;\">1.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">San Bernardino<\/td>\n<td style=\"width: 47.3004px;\">1.1<\/td>\n<td style=\"width: 52.3264px;\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">San Diego<\/td>\n<td style=\"width: 47.3004px;\">3<\/td>\n<td style=\"width: 52.3264px;\">3.7<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">San Fransisco<\/td>\n<td style=\"width: 47.3004px;\">7.4<\/td>\n<td style=\"width: 52.3264px;\">6.2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">San Joaquin<\/td>\n<td style=\"width: 47.3004px;\">0.7<\/td>\n<td style=\"width: 52.3264px;\">1.3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">San Luis Obispo<\/td>\n<td style=\"width: 47.3004px;\">4.6<\/td>\n<td style=\"width: 52.3264px;\">4.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">San Mateo<\/td>\n<td style=\"width: 47.3004px;\">2.2<\/td>\n<td style=\"width: 52.3264px;\">2.5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Santa Barbara<\/td>\n<td style=\"width: 47.3004px;\">0.2<\/td>\n<td style=\"width: 52.3264px;\">0.9<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Santa Clara<\/td>\n<td style=\"width: 47.3004px;\">2.6<\/td>\n<td style=\"width: 52.3264px;\">6.3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Santa Cruz<\/td>\n<td style=\"width: 47.3004px;\">3.7<\/td>\n<td style=\"width: 52.3264px;\">4.8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Shasta<\/td>\n<td style=\"width: 47.3004px;\">0.6<\/td>\n<td style=\"width: 52.3264px;\">3.9<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Solano<\/td>\n<td style=\"width: 47.3004px;\">1.6<\/td>\n<td style=\"width: 52.3264px;\">3.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Sonoma<\/td>\n<td style=\"width: 47.3004px;\">2.2<\/td>\n<td style=\"width: 52.3264px;\">5.3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Stanislaus<\/td>\n<td style=\"width: 47.3004px;\">1.3<\/td>\n<td style=\"width: 52.3264px;\">3.1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Sutter<\/td>\n<td style=\"width: 47.3004px;\">1<\/td>\n<td style=\"width: 52.3264px;\">4.2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Tehama<\/td>\n<td style=\"width: 47.3004px;\">3.1<\/td>\n<td style=\"width: 52.3264px;\">6.2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Tulare<\/td>\n<td style=\"width: 47.3004px;\">0.2<\/td>\n<td style=\"width: 52.3264px;\">0.4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Tuolumne<\/td>\n<td style=\"width: 47.3004px;\">1.8<\/td>\n<td style=\"width: 52.3264px;\">1.8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Ventura<\/td>\n<td style=\"width: 47.3004px;\">0.8<\/td>\n<td style=\"width: 52.3264px;\">1.8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Yolo<\/td>\n<td style=\"width: 47.3004px;\">2.3<\/td>\n<td style=\"width: 52.3264px;\">8.2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 101.484px;\">Yuba<\/td>\n<td style=\"width: 47.3004px;\">1.3<\/td>\n<td style=\"width: 52.3264px;\">2.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Note: Hate crimes data are limited as they do not fully capture all hate\u00a0incidents.<\/p><\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q203470\">Year 2019<\/button><\/p>\n<div id=\"q203470\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"border-collapse: collapse; width: 29px; width: 87px; border-spacing: 0px;\" cellpadding=\"0\">\n<tbody>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.6<\/td>\n<\/tr>\n<tr style=\"height: 18.0pt;\">\n<td style=\"height: 18pt; width: 237.448px; height: 24px;\" align=\"right\">6.3<\/td>\n<\/tr>\n<tr style=\"height: 18.0pt;\">\n<td style=\"height: 18pt; width: 237.448px; height: 24px;\" align=\"right\">2.3<\/td>\n<\/tr>\n<tr style=\"height: 18.0pt;\">\n<td style=\"height: 18pt; box-sizing: border-box; width: 278.003px; height: 24px;\" align=\"right\">4.7<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; box-sizing: border-box; width: 278.003px; height: 21px;\" align=\"right\">3<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">0.5<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3.5<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">7.4<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">0.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">0.8<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">0.7<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">4.7<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3.3<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">4.1<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">5<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.3<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">0.4<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.4<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.3<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.5<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.5<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.1<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">7.4<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">0.7<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">4.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.2<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">0.2<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3.7<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">0.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.2<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.3<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3.1<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">0.2<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.8<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">0.8<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.3<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q628393\">Year 2020<\/button><\/p>\n<div id=\"q628393\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"border-collapse: collapse; width: 79px; width: 87px; border-spacing: 0px;\" cellpadding=\"0\">\n<tbody>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3.1<\/td>\n<\/tr>\n<tr style=\"height: 18.0pt;\">\n<td style=\"height: 18pt; width: 237.448px; height: 24px;\" align=\"right\">7.5<\/td>\n<\/tr>\n<tr style=\"height: 18.0pt;\">\n<td style=\"height: 18pt; width: 237.448px; height: 24px;\" align=\"right\">1.4<\/td>\n<\/tr>\n<tr style=\"height: 18.0pt;\">\n<td style=\"height: 18pt; box-sizing: border-box; width: 278.003px; height: 24px;\" align=\"right\">4.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; box-sizing: border-box; width: 278.003px; height: 21px;\" align=\"right\">2.3<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.7<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3.5<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.2<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.1<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.2<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">0.7<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">4.7<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3.3<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">5.3<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">8.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.2<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.1<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.1<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.4<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.5<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.4<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3.7<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">6.2<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.3<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">4.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.5<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">0.9<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">6.3<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">4.8<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3.9<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3.6<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">5.3<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">3.1<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">4.2<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">6.2<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">0.4<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.8<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">1.8<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">8.2<\/td>\n<\/tr>\n<tr style=\"height: 16.0pt;\">\n<td style=\"height: 16pt; width: 237.448px; height: 21px;\" align=\"right\">2.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" width=\"100%\" height=\"900\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><\/iframe><br \/>\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/eda_quantitative\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm5486\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=5486&theme=lumen&iframe_resize_id=ohm5486&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>The analysis above uses descriptive statistics only. How can we make an inference about the increase in hate crimes in the United States?<\/p>\n<p>We can make an inference using a hypothesis test to provide evidence that there is an increase in the number of hate crimes between 2019 and 2020. A hypothesis test is needed to go beyond the visualizations and show that the difference is not simply sampling variability.<\/p>\n<h3>Standard Error of the Difference of Means<\/h3>\n<p>We can use a hypothesis test to determine if the observed difference in sample means is consistent with a hypothesized difference in population means.<\/p>\n<p>To do this, we use what we know about the sampling distribution of [latex]\\bar{x}_1-\\bar{x}_2[\/latex] and, in particular, its estimated standard deviation (the standard error). Recall that you learned that the difference in the sample means, [latex]\\bar{x}_1-\\bar{x}_2[\/latex], also has an approximately normal distribution, centered at the difference of the population means, [latex]\\mu_1-\\mu_2[\/latex].<\/p>\n<p>The standard deviation is given by the following formula: [latex]\\sqrt{\\frac{\\sigma_1^2}{n_1}+\\frac{\\sigma_2^2}{n_2}}[\/latex]<\/p>\n<p>In practice, we will have to estimate the standard deviation because it depends on the unknown population standard deviations. Replacing [latex]\\sigma_1[\/latex] and [latex]\\sigma_2[\/latex] by the sample standard deviations [latex]s_1[\/latex] and [latex]s_2[\/latex], we will get the <strong>standard error <\/strong>of the difference.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>standard error of the difference of means<\/h3>\n<p><strong>standard error<\/strong> of [latex]\\bar{x}_1-\\bar{x}_2[\/latex]: [latex]\\sqrt{\\dfrac{s_1^2}{n_1}+\\dfrac{s_2^2}{n_2}}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm5498\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=5498&theme=lumen&iframe_resize_id=ohm5498&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Now that we have the standard error, we can calculate the test statistic, P-value, and make an inference about the population. We can leverage technology to generate our statistical output and use it to interpret our results.<\/p>\n","protected":false},"author":8,"menu_order":22,"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":"learn_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\/1331"}],"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":4,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1331\/revisions"}],"predecessor-version":[{"id":6816,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1331\/revisions\/6816"}],"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\/1331\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1331"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1331"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1331"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1331"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}