{"id":1352,"date":"2023-06-22T02:20:20","date_gmt":"2023-06-22T02:20:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/introduction-to-one-way-anova-learn-it-2\/"},"modified":"2025-05-16T22:49:16","modified_gmt":"2025-05-16T22:49:16","slug":"introduction-to-one-way-anova-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/introduction-to-one-way-anova-learn-it-2\/","title":{"raw":"Introduction to One-Way ANOVA \u2013 Learn It 2","rendered":"Introduction to One-Way ANOVA \u2013 Learn It 2"},"content":{"raw":"<section>\r\n<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<\/section>\r\n<h2>The Null and Alternative Hypotheses<\/h2>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>null hypothesis<\/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<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2009[\/ohm2_question]<\/section>\r\n<p class=\"student12ptnumberlist2\" style=\"margin-left: 0in; text-indent: 0in;\"><span style=\"font-family: 'Arial',sans-serif;\">The alternative hypothesis for a one-way ANOVA is a bit different than the alternative hypothesis we used when comparing only two group means (i.e., two-sample [latex]t[\/latex]-test).<\/span><\/p>\r\n<p>When there were only two group means to consider, the null hypothesis that the two means were the same was [latex]H_{0}:\\mu_{1}=\\mu_{2}[\/latex]. If you wanted to show that the two means were different or not equal, the alternative hypothesis would be [latex]H_{A}:\\mu_{1}\\neq\\mu_{2}[\/latex]. If we rejected the null hypothesis, we would be able to conclude that the two means were statistically different.<\/p>\r\n<p>When we reject the null hypothesis for a one-way ANOVA, we cannot simply state that all of the means are not equal. That is, when we reject the null hypothesis, [latex]H_{0}:\\mu_{1}=\\mu_{2}=\\ldots=\\mu_{k}[\/latex], we are not able to differentiate whether one of the means is different from the others, whether two of the means are different from the others, whether three of the means are different from the others, etc.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>alternative hypothesis<\/h3>\r\n<p>To provide flexibility and to account for the multiple outcomes associated with rejecting the null hypothesis, 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 tryIt\">[ohm2_question hide_question_numbers=1]2010[\/ohm2_question]<\/section>","rendered":"<section>\n<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<\/section>\n<h2>The Null and Alternative Hypotheses<\/h2>\n<section class=\"textbox keyTakeaway\">\n<h3>null hypothesis<\/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<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2009\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2009&theme=lumen&iframe_resize_id=ohm2009&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p class=\"student12ptnumberlist2\" style=\"margin-left: 0in; text-indent: 0in;\"><span style=\"font-family: 'Arial',sans-serif;\">The alternative hypothesis for a one-way ANOVA is a bit different than the alternative hypothesis we used when comparing only two group means (i.e., two-sample [latex]t[\/latex]-test).<\/span><\/p>\n<p>When there were only two group means to consider, the null hypothesis that the two means were the same was [latex]H_{0}:\\mu_{1}=\\mu_{2}[\/latex]. If you wanted to show that the two means were different or not equal, the alternative hypothesis would be [latex]H_{A}:\\mu_{1}\\neq\\mu_{2}[\/latex]. If we rejected the null hypothesis, we would be able to conclude that the two means were statistically different.<\/p>\n<p>When we reject the null hypothesis for a one-way ANOVA, we cannot simply state that all of the means are not equal. That is, when we reject the null hypothesis, [latex]H_{0}:\\mu_{1}=\\mu_{2}=\\ldots=\\mu_{k}[\/latex], we are not able to differentiate whether one of the means is different from the others, whether two of the means are different from the others, whether three of the means are different from the others, etc.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>alternative hypothesis<\/h3>\n<p>To provide flexibility and to account for the multiple outcomes associated with rejecting the null hypothesis, 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 tryIt\"><iframe loading=\"lazy\" id=\"ohm2010\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2010&theme=lumen&iframe_resize_id=ohm2010&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":9,"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":"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\/1352"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1352\/revisions"}],"predecessor-version":[{"id":6836,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1352\/revisions\/6836"}],"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\/1352\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1352"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1352"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1352"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1352"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}