{"id":1393,"date":"2023-06-22T02:22:38","date_gmt":"2023-06-22T02:22:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/chi-square-test-of-homogeneity-background-youll-need-2\/"},"modified":"2025-05-16T23:04:25","modified_gmt":"2025-05-16T23:04:25","slug":"chi-square-test-of-homogeneity-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/chi-square-test-of-homogeneity-background-youll-need-2\/","title":{"raw":"Module 14: Background You'll Need 3","rendered":"Module 14: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Find expected counts based on certain proportions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Flight Frequencies (continued.)<\/h2>\r\n<p>The <strong>expected count<\/strong> for each category is the number of trials of the experiment multiplied by the proportion\/probability of that particular category.<\/p>\r\n<section class=\"textbox example\">Delta Airlines would have [latex]92.246962\\%[\/latex] of its flights arrive on time. Since Delta Airlines had [latex]13,651[\/latex] flights arrive in Atlanta in total in March 2021,\r\n\r\n<p style=\"text-align: center;\">[latex]92.246962\\% \\text{ of } 13,651 = (0.92246962) * 13,651 \\approx 12,592.633[\/latex]<\/p>\r\n<p>is the number of flights that we would expect to be on time. This is called the <strong>expected count<\/strong> of on-time flights if Delta Airlines\u2019 distribution matched the overall proportions.<\/p>\r\n<p>Similarly, for Southwest Airlines, we would expect to have [latex](0.92246962)*2,562 \\approx 2,363.367[\/latex] on-time flights if its distribution matched the overall proportions. Notice that these expected counts do not have to be whole numbers because they are theoretical values.<\/p>\r\n<p>Notice also that there were [latex]14,956[\/latex] total on-time flights for these two airlines in March 2021, so once we knew that Delta Airlines would be expected to have [latex]12,592.633[\/latex] on-time flights if its distribution matched the overall proportions, we could have found the expected number of on-time flights for Southwest Airlines by subtracting:<\/p>\r\n<p style=\"text-align: center;\">[latex]14,956 - 12,592.633 = 2,363.367[\/latex]<\/p>\r\n<p>We see that we get the same expected count as we did when we used the percentage.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2383[\/ohm2_question]<\/section>\r\n<p>We can also compare the observed and expected counts by calculating the difference between the observed count and the expected count. (So, [latex]\\text{observed} - \\text{expected}[\/latex] for each cell of the table.)<\/p>\r\n<section class=\"textbox example\">Recall the two-way table that gives the counts for each value of the variable <em>flight status<\/em> for Delta Airlines and Southwest Airlines arrivals at the Atlanta airport in March 2021.\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td><strong>On-Time Flights<\/strong><\/td>\r\n<td><strong>Delayed Flights<\/strong><\/td>\r\n<td><strong>Canceled Flights<\/strong><\/td>\r\n<td><strong>Diverted Flights<\/strong><\/td>\r\n<td><strong>Total<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Delta Airlines<\/strong><\/td>\r\n<td>12,716<\/td>\r\n<td>904<\/td>\r\n<td>23<\/td>\r\n<td>8<\/td>\r\n<td>13,651<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Southwest Airlines<\/strong><\/td>\r\n<td>2,240<\/td>\r\n<td>299<\/td>\r\n<td>22<\/td>\r\n<td>1<\/td>\r\n<td>2,562<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>Notice that for Delta Airlines\u2019 on-time flights, the difference is [latex]12,716 - 12,592.633 = 123.367[\/latex].<\/p>\r\n<p>So, Delta Airlines had [latex]123.367[\/latex] more on-time flights than would be expected if Delta Airlines\u2019 distribution matched the overall proportions.<\/p>\r\n<\/section>\r\n<section>\r\n<section class=\"textbox proTip\">\r\n<ul>\r\n\t<li>When the difference between an observed count and the corresponding expected count is <strong>positive<\/strong>, it means the expected count was smaller than the observed count, so there were more observed values than expected.<\/li>\r\n\t<li>When the difference between an observed count and the corresponding expected count is <strong>negative<\/strong>, it means the expected count was larger than the observed count, so there were fewer observed values than expected.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2385[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2387[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Find expected counts based on certain proportions<\/li>\n<\/ul>\n<\/section>\n<h2>Flight Frequencies (continued.)<\/h2>\n<p>The <strong>expected count<\/strong> for each category is the number of trials of the experiment multiplied by the proportion\/probability of that particular category.<\/p>\n<section class=\"textbox example\">Delta Airlines would have [latex]92.246962\\%[\/latex] of its flights arrive on time. Since Delta Airlines had [latex]13,651[\/latex] flights arrive in Atlanta in total in March 2021,<\/p>\n<p style=\"text-align: center;\">[latex]92.246962\\% \\text{ of } 13,651 = (0.92246962) * 13,651 \\approx 12,592.633[\/latex]<\/p>\n<p>is the number of flights that we would expect to be on time. This is called the <strong>expected count<\/strong> of on-time flights if Delta Airlines\u2019 distribution matched the overall proportions.<\/p>\n<p>Similarly, for Southwest Airlines, we would expect to have [latex](0.92246962)*2,562 \\approx 2,363.367[\/latex] on-time flights if its distribution matched the overall proportions. Notice that these expected counts do not have to be whole numbers because they are theoretical values.<\/p>\n<p>Notice also that there were [latex]14,956[\/latex] total on-time flights for these two airlines in March 2021, so once we knew that Delta Airlines would be expected to have [latex]12,592.633[\/latex] on-time flights if its distribution matched the overall proportions, we could have found the expected number of on-time flights for Southwest Airlines by subtracting:<\/p>\n<p style=\"text-align: center;\">[latex]14,956 - 12,592.633 = 2,363.367[\/latex]<\/p>\n<p>We see that we get the same expected count as we did when we used the percentage.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2383\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2383&theme=lumen&iframe_resize_id=ohm2383&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>We can also compare the observed and expected counts by calculating the difference between the observed count and the expected count. (So, [latex]\\text{observed} - \\text{expected}[\/latex] for each cell of the table.)<\/p>\n<section class=\"textbox example\">Recall the two-way table that gives the counts for each value of the variable <em>flight status<\/em> for Delta Airlines and Southwest Airlines arrivals at the Atlanta airport in March 2021.<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td><strong>On-Time Flights<\/strong><\/td>\n<td><strong>Delayed Flights<\/strong><\/td>\n<td><strong>Canceled Flights<\/strong><\/td>\n<td><strong>Diverted Flights<\/strong><\/td>\n<td><strong>Total<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Delta Airlines<\/strong><\/td>\n<td>12,716<\/td>\n<td>904<\/td>\n<td>23<\/td>\n<td>8<\/td>\n<td>13,651<\/td>\n<\/tr>\n<tr>\n<td><strong>Southwest Airlines<\/strong><\/td>\n<td>2,240<\/td>\n<td>299<\/td>\n<td>22<\/td>\n<td>1<\/td>\n<td>2,562<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that for Delta Airlines\u2019 on-time flights, the difference is [latex]12,716 - 12,592.633 = 123.367[\/latex].<\/p>\n<p>So, Delta Airlines had [latex]123.367[\/latex] more on-time flights than would be expected if Delta Airlines\u2019 distribution matched the overall proportions.<\/p>\n<\/section>\n<section>\n<section class=\"textbox proTip\">\n<ul>\n<li>When the difference between an observed count and the corresponding expected count is <strong>positive<\/strong>, it means the expected count was smaller than the observed count, so there were more observed values than expected.<\/li>\n<li>When the difference between an observed count and the corresponding expected count is <strong>negative<\/strong>, it means the expected count was larger than the observed count, so there were fewer observed values than expected.<\/li>\n<\/ul>\n<\/section>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2385\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2385&theme=lumen&iframe_resize_id=ohm2385&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2387\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2387&theme=lumen&iframe_resize_id=ohm2387&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1388,"module-header":"background_you_need","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\/1393"}],"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":9,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1393\/revisions"}],"predecessor-version":[{"id":6857,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1393\/revisions\/6857"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1388"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1393\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1393"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1393"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1393"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1393"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}