{"id":2049,"date":"2023-07-26T00:40:38","date_gmt":"2023-07-26T00:40:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=2049"},"modified":"2025-05-16T03:36:38","modified_gmt":"2025-05-16T03:36:38","slug":"one-sample-hypothesis-test-for-proportions-learn-it-1-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/one-sample-hypothesis-test-for-proportions-learn-it-1-2\/","title":{"raw":"One-Sample Hypothesis Test for Proportions: Learn It 2","rendered":"One-Sample Hypothesis Test for Proportions: Learn It 2"},"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>Test Statistic<\/h2>\r\n<p>To draw a conclusion using data, first we have to calculate a test statistic. A\u00a0<strong>test statistic<\/strong>\u00a0is used in a hypothesis test to decide whether the data support or reject the null hypothesis.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>test statistic ([latex]z[\/latex]-statistic)<\/h3>\r\n<p>A <strong>test statistic<\/strong> measures the distance between the sample statistic and the null hypothesis value in terms of the hypothesized standard deviation of the null hypothesis value.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: left; padding-left: 40px;\">[latex] \\text{test statistic}:\u00a0 z = \\dfrac{\\text{sample statistic }-\\text{ null hypothesis value}}{\\text{standard deviation of the null hypothesis value}}[\/latex]<\/p>\r\n<p style=\"text-align: left; padding-left: 40px;\">[latex] \\text{test statistic}:\u00a0 z = \\dfrac{\\stackrel{\u02c6}{p}-p_0}{\\sqrt{\\frac{p_0(1-p_0)}{n}}}[\/latex]<\/p>\r\n<p style=\"padding-left: 40px;\">where [latex]\\stackrel{\u02c6}{p}[\/latex] is the sample statistics and [latex]p_0[\/latex] is the null hypothesis value.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>When the sample statistic is a proportion, the test statistic is also called a<strong> [latex]z[\/latex]-statistic<\/strong>.<\/p>\r\n<\/section>\r\n<section>[reveal-answer q=\"885260\"]z-score formula[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"885260\"]<strong style=\"font-size: 1rem; text-align: initial;\">A standardized value<\/strong><span style=\"font-size: 1rem; text-align: initial;\">, or <\/span><strong style=\"font-size: 1rem; text-align: initial;\">[latex]z[\/latex]-score<\/strong><span style=\"font-size: 1rem; text-align: initial;\">, is the number of standard deviations an observation is away from the mean.<\/span>\r\n<p style=\"text-align: center;\">[latex]z=\\dfrac{\\text{Observed Value}-\\text{mean}}{\\text{standard deviation}} = \\dfrac{x-\\mu}{\\sigma}[\/latex]<\/p>\r\n<p>For the sampling distribution of a sample proportion, the mean is [latex]p[\/latex] and the standard deviation is [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].<\/p>\r\n<p>Thus, the [latex] \\text{test statistic}:\u00a0 z = \\dfrac{\\stackrel{\u02c6}{p}-p}{\\sqrt{\\frac{p(1-p)}{n}}}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1831[\/ohm2_question]<\/section>\r\n<p>The [latex]z[\/latex]-score (the test statistic) tells us how far the sample proportion is from the null hypothesis. The test statistic assesses how consistent the sample data collected are with the null hypothesis in a hypothesis test. We can use this statistic to find the probability of how likely it is that the data would have occurred by random chance.<\/p>\r\n<p>Assuming the sample size is large enough ([latex] np \\geq 10, n(1-p) \\geq 10 [\/latex]), we can then use the normal distribution to calculate such probability,\u00a0<span style=\"text-align: initial; font-size: 1em;\">which we can use to make an inference to draw conclusions about the population parameter.<\/span><\/p>","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>Test Statistic<\/h2>\n<p>To draw a conclusion using data, first we have to calculate a test statistic. A\u00a0<strong>test statistic<\/strong>\u00a0is used in a hypothesis test to decide whether the data support or reject the null hypothesis.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>test statistic ([latex]z[\/latex]-statistic)<\/h3>\n<p>A <strong>test statistic<\/strong> measures the distance between the sample statistic and the null hypothesis value in terms of the hypothesized standard deviation of the null hypothesis value.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left; padding-left: 40px;\">[latex]\\text{test statistic}:\u00a0 z = \\dfrac{\\text{sample statistic }-\\text{ null hypothesis value}}{\\text{standard deviation of the null hypothesis value}}[\/latex]<\/p>\n<p style=\"text-align: left; padding-left: 40px;\">[latex]\\text{test statistic}:\u00a0 z = \\dfrac{\\stackrel{\u02c6}{p}-p_0}{\\sqrt{\\frac{p_0(1-p_0)}{n}}}[\/latex]<\/p>\n<p style=\"padding-left: 40px;\">where [latex]\\stackrel{\u02c6}{p}[\/latex] is the sample statistics and [latex]p_0[\/latex] is the null hypothesis value.<\/p>\n<p>&nbsp;<\/p>\n<p>When the sample statistic is a proportion, the test statistic is also called a<strong> [latex]z[\/latex]-statistic<\/strong>.<\/p>\n<\/section>\n<section>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q885260\">z-score formula<\/button><\/p>\n<div id=\"q885260\" class=\"hidden-answer\" style=\"display: none\"><strong style=\"font-size: 1rem; text-align: initial;\">A standardized value<\/strong><span style=\"font-size: 1rem; text-align: initial;\">, or <\/span><strong style=\"font-size: 1rem; text-align: initial;\">[latex]z[\/latex]-score<\/strong><span style=\"font-size: 1rem; text-align: initial;\">, is the number of standard deviations an observation is away from the mean.<\/span><\/p>\n<p style=\"text-align: center;\">[latex]z=\\dfrac{\\text{Observed Value}-\\text{mean}}{\\text{standard deviation}} = \\dfrac{x-\\mu}{\\sigma}[\/latex]<\/p>\n<p>For the sampling distribution of a sample proportion, the mean is [latex]p[\/latex] and the standard deviation is [latex]\\sqrt{\\frac{p(1-p)}{n}}[\/latex].<\/p>\n<p>Thus, the [latex]\\text{test statistic}:\u00a0 z = \\dfrac{\\stackrel{\u02c6}{p}-p}{\\sqrt{\\frac{p(1-p)}{n}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1831\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1831&theme=lumen&iframe_resize_id=ohm1831&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>The [latex]z[\/latex]-score (the test statistic) tells us how far the sample proportion is from the null hypothesis. The test statistic assesses how consistent the sample data collected are with the null hypothesis in a hypothesis test. We can use this statistic to find the probability of how likely it is that the data would have occurred by random chance.<\/p>\n<p>Assuming the sample size is large enough ([latex]np \\geq 10, n(1-p) \\geq 10[\/latex]), we can then use the normal distribution to calculate such probability,\u00a0<span style=\"text-align: initial; font-size: 1em;\">which we can use to make an inference to draw conclusions about the population parameter.<\/span><\/p>\n","protected":false},"author":12,"menu_order":17,"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\/2049"}],"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":16,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/2049\/revisions"}],"predecessor-version":[{"id":6758,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/2049\/revisions\/6758"}],"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\/2049\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=2049"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=2049"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=2049"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=2049"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}