{"id":2060,"date":"2023-07-26T00:55:19","date_gmt":"2023-07-26T00:55:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=2060"},"modified":"2025-05-16T03:37:08","modified_gmt":"2025-05-16T03:37:08","slug":"one-sample-hypothesis-test-for-proportions-learn-it-3-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/one-sample-hypothesis-test-for-proportions-learn-it-3-2\/","title":{"raw":"One-Sample Hypothesis Test for Proportions: Learn It 3","rendered":"One-Sample Hypothesis Test for Proportions: 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 one-sample z-test for proportions 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 one-sample [latex]z[\/latex]-test for proportions from hypotheses to conclusions.<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use a P-value to explain the conclusions of a completed z-test for proportions&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;}\">Use a P-value to explain the conclusions of a completed [latex]z[\/latex]-test for proportions.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>P-value<\/h2>\r\n<p>At what point in data collection and testing does the test statistic seem unusual? How do we measure \"unusual?\" In a statistical hypothesis test, the evidence used is probability.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>P-value<\/h3>\r\n<p>We define the <strong>P-value <\/strong>as the probability of obtaining a test statistic at least as extreme (in the direction of the alternative hypothesis) as the one that is actually seen if the null hypothesis is true.<\/p>\r\n<\/section>\r\n<p>The statistical evidence that we gather is always evidence in support of the alternative hypothesis and against the null hypothesis. We ask ourselves the question, \u201cDo we have enough evidence to reject the null hypothesis?\u201d The P-value answers the question: <strong>\u201c<\/strong>How <strong>unlikely<\/strong> is the sample data given that the null hypothesis is true?<strong>\u201d<\/strong><\/p>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Consider a hypothesis test with a test statistic of [latex]z=1[\/latex]. The P-value is the area under the curve of the standard normal distribution in the direction of the alternative hypothesis:<\/p>\r\n<table style=\"border-collapse: collapse; width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">If [latex]H_a: p \\lt z^*[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">If [latex]H_a: p&gt; z^*[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">If [latex]H_a: p\\ne z^*[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><img class=\"alignnone size-full wp-image-6307\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173511\/Screenshot-2025-02-21-at-10.35.05%E2%80%AFAM.png\" alt=\"A normal distribution curve labeled where z = 1 with the left side shaded. The probability is shown as 84.13%.\" width=\"1878\" height=\"702\" \/><\/td>\r\n<td style=\"width: 33.3333%;\"><img class=\"alignnone size-full wp-image-6308\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173659\/Screenshot-2025-02-21-at-10.36.52%E2%80%AFAM.png\" alt=\"A standard normal distribution with z = 1 labeled and shaded to the right. The probability is listed as 15.87%.\" width=\"1906\" height=\"712\" \/><\/td>\r\n<td style=\"width: 33.3333%;\"><img class=\"alignnone size-full wp-image-6309\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173753\/Screenshot-2025-02-21-at-10.37.46%E2%80%AFAM.png\" alt=\"A standard normal distribution with z = -1 and z = 1 labeled. The graph is shaded to the left of z = -1 and to the right of z = 1. The percentage is shown as 31.73%.\" width=\"1878\" height=\"708\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">P-value = [latex].8413[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">P-value = [latex].1587[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">P-value = [latex].3173[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>The P-value is the probability of the shaded area, typically written as a decimal to the nearest thousandth.<\/p>\r\n<\/section>\r\n<p>It is important to remember that a P-value is a probability, which means that it is a number between [latex]0[\/latex] and [latex]1[\/latex].<\/p>\r\n<section class=\"textbox proTip\">\r\n<ul>\r\n\t<li>The <strong>smaller the P-value<\/strong> is, the more <strong>unlikely<\/strong> it is to observe the sample data given that the null hypothesis is true. Thus, the evidence against the null hypothesis is stronger and is in favor of the alternative hypothesis.<\/li>\r\n\t<li>The <strong>larger the P-value<\/strong> is, the more <strong>likely<\/strong> it is to observe the sample data. Thus, the evidence against the null hypothesis is weaker.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p><strong>Important note:<\/strong><\/p>\r\n<p>A hypothesis test can be <strong>one-sided <\/strong>or <strong>two-sided<\/strong>. The example about internet access was a one-sided hypothesis test. The P-value was the area of the right (upper) tail because we want to test if the percentage has increased since 2020.<\/p>\r\n<section class=\"textbox proTip\">If the inequality in the alternative hypothesis is [latex]&lt;[\/latex] or [latex]&gt;[\/latex], the test is one-sided.\r\n\r\n<ul>\r\n\t<li>If the inequality in the alternative hypothesis is [latex]&lt;[\/latex], you have a lower-tailed test.<\/li>\r\n\t<li>If the inequality in the alternative hypothesis is [latex]&gt;[\/latex], you have an upper-tailed test.<\/li>\r\n<\/ul>\r\n<p>If the inequality is [latex]\u2260[\/latex], the test is two-sided, or two-tailed. The P-value for a two-tailed test is [latex] 2*P(|z|&gt;z^*) [\/latex] or double the corresponding one-tailed test.<\/p>\r\n<\/section>\r\n<p>Let's calculate our P-value. Use the statistical tool to answer the next question.<\/p>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\r\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/\" 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]11074[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Complete a one-sample z-test for proportions 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 one-sample [latex]z[\/latex]-test for proportions from hypotheses to conclusions.<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use a P-value to explain the conclusions of a completed z-test for proportions&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;}\">Use a P-value to explain the conclusions of a completed [latex]z[\/latex]-test for proportions.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>P-value<\/h2>\n<p>At what point in data collection and testing does the test statistic seem unusual? How do we measure &#8220;unusual?&#8221; In a statistical hypothesis test, the evidence used is probability.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>P-value<\/h3>\n<p>We define the <strong>P-value <\/strong>as the probability of obtaining a test statistic at least as extreme (in the direction of the alternative hypothesis) as the one that is actually seen if the null hypothesis is true.<\/p>\n<\/section>\n<p>The statistical evidence that we gather is always evidence in support of the alternative hypothesis and against the null hypothesis. We ask ourselves the question, \u201cDo we have enough evidence to reject the null hypothesis?\u201d The P-value answers the question: <strong>\u201c<\/strong>How <strong>unlikely<\/strong> is the sample data given that the null hypothesis is true?<strong>\u201d<\/strong><\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Consider a hypothesis test with a test statistic of [latex]z=1[\/latex]. The P-value is the area under the curve of the standard normal distribution in the direction of the alternative hypothesis:<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\">If [latex]H_a: p \\lt z^*[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">If [latex]H_a: p> z^*[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">If [latex]H_a: p\\ne z^*[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6307\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173511\/Screenshot-2025-02-21-at-10.35.05%E2%80%AFAM.png\" alt=\"A normal distribution curve labeled where z = 1 with the left side shaded. The probability is shown as 84.13%.\" width=\"1878\" height=\"702\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173511\/Screenshot-2025-02-21-at-10.35.05%E2%80%AFAM.png 1878w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173511\/Screenshot-2025-02-21-at-10.35.05%E2%80%AFAM-300x112.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173511\/Screenshot-2025-02-21-at-10.35.05%E2%80%AFAM-1024x383.png 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173511\/Screenshot-2025-02-21-at-10.35.05%E2%80%AFAM-768x287.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173511\/Screenshot-2025-02-21-at-10.35.05%E2%80%AFAM-1536x574.png 1536w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173511\/Screenshot-2025-02-21-at-10.35.05%E2%80%AFAM-65x24.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173511\/Screenshot-2025-02-21-at-10.35.05%E2%80%AFAM-225x84.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173511\/Screenshot-2025-02-21-at-10.35.05%E2%80%AFAM-350x131.png 350w\" sizes=\"(max-width: 1878px) 100vw, 1878px\" \/><\/td>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6308\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173659\/Screenshot-2025-02-21-at-10.36.52%E2%80%AFAM.png\" alt=\"A standard normal distribution with z = 1 labeled and shaded to the right. The probability is listed as 15.87%.\" width=\"1906\" height=\"712\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173659\/Screenshot-2025-02-21-at-10.36.52%E2%80%AFAM.png 1906w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173659\/Screenshot-2025-02-21-at-10.36.52%E2%80%AFAM-300x112.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173659\/Screenshot-2025-02-21-at-10.36.52%E2%80%AFAM-1024x383.png 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173659\/Screenshot-2025-02-21-at-10.36.52%E2%80%AFAM-768x287.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173659\/Screenshot-2025-02-21-at-10.36.52%E2%80%AFAM-1536x574.png 1536w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173659\/Screenshot-2025-02-21-at-10.36.52%E2%80%AFAM-65x24.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173659\/Screenshot-2025-02-21-at-10.36.52%E2%80%AFAM-225x84.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173659\/Screenshot-2025-02-21-at-10.36.52%E2%80%AFAM-350x131.png 350w\" sizes=\"(max-width: 1906px) 100vw, 1906px\" \/><\/td>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6309\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173753\/Screenshot-2025-02-21-at-10.37.46%E2%80%AFAM.png\" alt=\"A standard normal distribution with z = -1 and z = 1 labeled. The graph is shaded to the left of z = -1 and to the right of z = 1. The percentage is shown as 31.73%.\" width=\"1878\" height=\"708\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173753\/Screenshot-2025-02-21-at-10.37.46%E2%80%AFAM.png 1878w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173753\/Screenshot-2025-02-21-at-10.37.46%E2%80%AFAM-300x113.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173753\/Screenshot-2025-02-21-at-10.37.46%E2%80%AFAM-1024x386.png 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173753\/Screenshot-2025-02-21-at-10.37.46%E2%80%AFAM-768x290.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173753\/Screenshot-2025-02-21-at-10.37.46%E2%80%AFAM-1536x579.png 1536w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173753\/Screenshot-2025-02-21-at-10.37.46%E2%80%AFAM-65x25.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173753\/Screenshot-2025-02-21-at-10.37.46%E2%80%AFAM-225x85.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/07\/21173753\/Screenshot-2025-02-21-at-10.37.46%E2%80%AFAM-350x132.png 350w\" sizes=\"(max-width: 1878px) 100vw, 1878px\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">P-value = [latex].8413[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">P-value = [latex].1587[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">P-value = [latex].3173[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The P-value is the probability of the shaded area, typically written as a decimal to the nearest thousandth.<\/p>\n<\/section>\n<p>It is important to remember that a P-value is a probability, which means that it is a number between [latex]0[\/latex] and [latex]1[\/latex].<\/p>\n<section class=\"textbox proTip\">\n<ul>\n<li>The <strong>smaller the P-value<\/strong> is, the more <strong>unlikely<\/strong> it is to observe the sample data given that the null hypothesis is true. Thus, the evidence against the null hypothesis is stronger and is in favor of the alternative hypothesis.<\/li>\n<li>The <strong>larger the P-value<\/strong> is, the more <strong>likely<\/strong> it is to observe the sample data. Thus, the evidence against the null hypothesis is weaker.<\/li>\n<\/ul>\n<\/section>\n<p><strong>Important note:<\/strong><\/p>\n<p>A hypothesis test can be <strong>one-sided <\/strong>or <strong>two-sided<\/strong>. The example about internet access was a one-sided hypothesis test. The P-value was the area of the right (upper) tail because we want to test if the percentage has increased since 2020.<\/p>\n<section class=\"textbox proTip\">If the inequality in the alternative hypothesis is [latex]<[\/latex] or [latex]>[\/latex], the test is one-sided.<\/p>\n<ul>\n<li>If the inequality in the alternative hypothesis is [latex]<[\/latex], you have a lower-tailed test.<\/li>\n<li>If the inequality in the alternative hypothesis is [latex]>[\/latex], you have an upper-tailed test.<\/li>\n<\/ul>\n<p>If the inequality is [latex]\u2260[\/latex], the test is two-sided, or two-tailed. The P-value for a two-tailed test is [latex]2*P(|z|>z^*)[\/latex] or double the corresponding one-tailed test.<\/p>\n<\/section>\n<p>Let&#8217;s calculate our P-value. Use the statistical tool to answer the next question.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/normaldist\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm11074\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=11074&theme=lumen&iframe_resize_id=ohm11074&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":18,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1205,"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\/2060"}],"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\/12"}],"version-history":[{"count":20,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/2060\/revisions"}],"predecessor-version":[{"id":6759,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/2060\/revisions\/6759"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1205"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/2060\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=2060"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=2060"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=2060"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=2060"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}