{"id":1358,"date":"2023-06-22T02:20:25","date_gmt":"2023-06-22T02:20:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/introduction-to-one-way-anova-fresh-take\/"},"modified":"2025-05-16T22:52:06","modified_gmt":"2025-05-16T22:52:06","slug":"introduction-to-one-way-anova-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/introduction-to-one-way-anova-fresh-take\/","title":{"raw":"Introduction to One-Way ANOVA \u2013 Fresh Take","rendered":"Introduction to One-Way ANOVA \u2013 Fresh Take"},"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;Write a null and alternative hypothesis for a one-way ANOVA hypothesis test&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12801,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:10}\">Write a null and alternative hypothesis for a one-way ANOVA hypothesis test<\/span><\/li>\r\n\t<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Discuss the error sum of squares and group sum of squares&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:14720,&quot;10&quot;:2,&quot;11&quot;:4,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:9}\">Discuss the error sum of squares and group sum of squares<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>The purpose of a one-way ANOVA test is to determine the existence of a statistically significant difference among several group means. The test actually uses\u00a0<strong>variances<\/strong>\u00a0to help determine if the means are equal or not.<\/p>\r\n<p>The null hypothesis is simply that all the group population means are the same. The alternative hypothesis is that at least one pair of means is different.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>hypotheses<\/h3>\r\n<p>The <strong>null hypothesis<\/strong> for a one-way ANOVA states that all the group\/population means are the same. This can be written as:<\/p>\r\n<p style=\"text-align: center;\">[latex]H_0: \\mu_1 = \\mu_2 = ... = \\mu_k[\/latex]<\/p>\r\n<p>where [latex]k[\/latex] is the number of independent groups or samples.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The <strong>alternative hypothesis<\/strong> for a one-way ANOVA should be written as:<\/p>\r\n<p style=\"text-align: center;\">[latex]H_{A}:[\/latex] At least two of the group means are different.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">If the null hypothesis is false, then the variance of the combined data is larger, which is caused by the different means as shown in the second graph (green boxplots).<img class=\"aligncenter wp-image-1625 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22022024\/e7c6e5207804219a79aecc8f47cf4458ec8e8e6d.jpeg\" alt=\"3 boxplots that are orange labeled (a), and 3 boxplots that are green labeled (b). The boxplots correspond to the description below.\" width=\"487\" height=\"397\" \/><br \/>\r\n(a) [latex]H_0[\/latex] is true. All means are the same; the differences are due to random variation. (b) [latex]H_0[\/latex] is not true. All means are not the same; the differences are too large to be due to random variation.<\/section>\r\n<p>But what does it mean when we reject the null hypothesis? Remember that an ANOVA only tells us that there is a difference, not which group(s) are different. Let\u2019s use colors to understand it better.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2031[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Write a null and alternative hypothesis for a one-way ANOVA hypothesis test&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12801,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:10}\">Write a null and alternative hypothesis for a one-way ANOVA hypothesis test<\/span><\/li>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Discuss the error sum of squares and group sum of squares&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:14720,&quot;10&quot;:2,&quot;11&quot;:4,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:9}\">Discuss the error sum of squares and group sum of squares<\/span><\/li>\n<\/ul>\n<\/section>\n<p>The purpose of a one-way ANOVA test is to determine the existence of a statistically significant difference among several group means. The test actually uses\u00a0<strong>variances<\/strong>\u00a0to help determine if the means are equal or not.<\/p>\n<p>The null hypothesis is simply that all the group population means are the same. The alternative hypothesis is that at least one pair of means is different.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>hypotheses<\/h3>\n<p>The <strong>null hypothesis<\/strong> for a one-way ANOVA states that all the group\/population means are the same. This can be written as:<\/p>\n<p style=\"text-align: center;\">[latex]H_0: \\mu_1 = \\mu_2 = ... = \\mu_k[\/latex]<\/p>\n<p>where [latex]k[\/latex] is the number of independent groups or samples.<\/p>\n<p>&nbsp;<\/p>\n<p>The <strong>alternative hypothesis<\/strong> for a one-way ANOVA should be written as:<\/p>\n<p style=\"text-align: center;\">[latex]H_{A}:[\/latex] At least two of the group means are different.<\/p>\n<\/section>\n<section class=\"textbox example\">If the null hypothesis is false, then the variance of the combined data is larger, which is caused by the different means as shown in the second graph (green boxplots).<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1625 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22022024\/e7c6e5207804219a79aecc8f47cf4458ec8e8e6d.jpeg\" alt=\"3 boxplots that are orange labeled (a), and 3 boxplots that are green labeled (b). The boxplots correspond to the description below.\" width=\"487\" height=\"397\" \/><br \/>\n(a) [latex]H_0[\/latex] is true. All means are the same; the differences are due to random variation. (b) [latex]H_0[\/latex] is not true. All means are not the same; the differences are too large to be due to random variation.<\/section>\n<p>But what does it mean when we reject the null hypothesis? Remember that an ANOVA only tells us that there is a difference, not which group(s) are different. Let\u2019s use colors to understand it better.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2031\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2031&theme=lumen&iframe_resize_id=ohm2031&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":14,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1348,"module-header":"fresh_take","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\/1358"}],"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":7,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1358\/revisions"}],"predecessor-version":[{"id":6840,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1358\/revisions\/6840"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1348"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1358\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1358"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1358"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1358"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1358"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}