{"id":1240,"date":"2023-06-22T02:09:30","date_gmt":"2023-06-22T02:09:30","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/one-sample-hypothesis-test-for-proportions-apply-it-2\/"},"modified":"2025-02-20T16:40:35","modified_gmt":"2025-02-20T16:40:35","slug":"one-sample-hypothesis-test-for-proportions-apply-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/one-sample-hypothesis-test-for-proportions-apply-it-2\/","title":{"raw":"One-Sample Hypothesis Test for Proportions: Learn It 5","rendered":"One-Sample Hypothesis Test for Proportions: Learn It 5"},"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<section class=\"textbox keyTakeaway\">\r\n<h3>one-sample [latex]z[\/latex]-test of proportions<\/h3>\r\n<ol>\r\n\t<li>Write out the null and alternative hypotheses.<\/li>\r\n\t<li>Check the conditions for the hypothesis test. For testing a one-sample [latex]z[\/latex]-test for proportions, we require:\r\n\r\n<ul>\r\n\t<li>Large counts: Check that [latex]np\\ge10[\/latex] and [latex]n(1-p)\\ge10[\/latex].<\/li>\r\n\t<li>Random samples\/assignment: Check that the sample is a random sample.<\/li>\r\n\t<li>10% population size: Check that the sample size, [latex]n[\/latex], is less than 10% of the population size, [latex]N[\/latex]: [latex]n&lt;0.10(N)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Calculate a test statistic.\r\n\r\n<ul>\r\n\t<li>\r\n<p style=\"text-align: left;\">[latex]\\text{test statistic}:\u00a0 z = \\dfrac{\\stackrel{\u02c6}{p}-p}{\\sqrt{\\frac{p(1-p)}{n}}}[\/latex] where [latex]\\stackrel{\u02c6}{p}[\/latex] is the sample statistic and [latex]p[\/latex] is the null hypothesis value.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Calculate a P-value.<\/li>\r\n\t<li>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<br \/>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Decision<\/strong><\/td>\r\n<td><strong>Conclusion<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>If P-value [latex]\\le\\alpha[\/latex], there is enough evidence to reject the null hypothesis.<\/td>\r\n<td>At the [latex]\\alpha\\times[\/latex]100% significance level, the data provide convincing evidence in support of the alternative hypothesis.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>If P-value [latex]\\gt\\alpha[\/latex], there is not enough evidence to reject the null hypothesis.<\/td>\r\n<td>At the [latex]\\alpha\\times[\/latex]100% significance level, the data do not provide convincing evidence in support of the alternative hypothesis.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section>Instead of calculating the test statistic by hand, like we did in the previous example about Internet Access, we can also use statistical software to compute the test statistics and the P-value.\u00a0<\/section>\r\n<section>\r\n<section class=\"textbox interact\"><b>Step 1: <\/b>Under <strong>Enter Data<\/strong> select <strong>Number of Successes<\/strong>.<br \/>\r\n<b>Step 2: <\/b>Enter the <strong>Sample Size<\/strong>\u00a0and the <strong># of Successes<\/strong>\u00a0accordingly.<br \/>\r\n<b style=\"text-align: initial; background-color: initial; font-size: 1em;\">Step 3: <\/b><span style=\"text-align: initial; background-color: initial; font-size: 1em;\">Change the <\/span><strong style=\"text-align: initial; background-color: initial; font-size: 1em;\">Type of Inference<\/strong><span style=\"text-align: initial; background-color: initial; font-size: 1em;\"> to <strong>Significance Test<\/strong>.<br \/>\r\n<\/span><b>Step 4: <\/b>Enter the <strong>Null value<\/strong>\u00a0and change the <strong>Alternative<\/strong>\u00a0accordingly.<\/section>\r\n<\/section>\r\n<section><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/inference_prop\/\" width=\"100%\" height=\"850\"><\/iframe>\r\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/inference_prop\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<br \/>\r\nTry it out with the example below.<\/p>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p>[ohm2_question hide_question_numbers=1]1578[\/ohm2_question]<\/p>\r\n<\/section>\r\n<\/section>\r\n<section class=\"textbox example\">[videopicker divId=\"tnh-video-picker\" title=\"One-Sample z-test for Proportions\" label=\"Select Instructor\"]<br \/>\r\n[videooption displayName=\"Dr. Pamela E. Harris\" value=\"https:\/\/www.youtube.com\/watch?v=ULf9GpNmyZc\"][videooption displayName=\"Dr. Aris Winger\" value=\"https:\/\/www.youtube.com\/watch?v=DuNle6Lrk7o\"] [videooption displayName=\"Dr. Lane Fisher\" value=\"https:\/\/www.youtube.com\/watch?v=RX0MuCVTHyk\"]<br \/>\r\n[\/videopicker]<\/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<section class=\"textbox keyTakeaway\">\n<h3>one-sample [latex]z[\/latex]-test of proportions<\/h3>\n<ol>\n<li>Write out the null and alternative hypotheses.<\/li>\n<li>Check the conditions for the hypothesis test. For testing a one-sample [latex]z[\/latex]-test for proportions, we require:\n<ul>\n<li>Large counts: Check that [latex]np\\ge10[\/latex] and [latex]n(1-p)\\ge10[\/latex].<\/li>\n<li>Random samples\/assignment: Check that the sample is a random sample.<\/li>\n<li>10% population size: Check that the sample size, [latex]n[\/latex], is less than 10% of the population size, [latex]N[\/latex]: [latex]n<0.10(N)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Calculate a test statistic.\n<ul>\n<li>\n<p style=\"text-align: left;\">[latex]\\text{test statistic}:\u00a0 z = \\dfrac{\\stackrel{\u02c6}{p}-p}{\\sqrt{\\frac{p(1-p)}{n}}}[\/latex] where [latex]\\stackrel{\u02c6}{p}[\/latex] is the sample statistic and [latex]p[\/latex] is the null hypothesis value.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li>Calculate a P-value.<\/li>\n<li>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<br \/>\n<table>\n<tbody>\n<tr>\n<td><strong>Decision<\/strong><\/td>\n<td><strong>Conclusion<\/strong><\/td>\n<\/tr>\n<tr>\n<td>If P-value [latex]\\le\\alpha[\/latex], there is enough evidence to reject the null hypothesis.<\/td>\n<td>At the [latex]\\alpha\\times[\/latex]100% significance level, the data provide convincing evidence in support of the alternative hypothesis.<\/td>\n<\/tr>\n<tr>\n<td>If P-value [latex]\\gt\\alpha[\/latex], there is not enough evidence to reject the null hypothesis.<\/td>\n<td>At the [latex]\\alpha\\times[\/latex]100% significance level, the data do not provide convincing evidence in support of the alternative hypothesis.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).<\/li>\n<\/ol>\n<\/section>\n<section>Instead of calculating the test statistic by hand, like we did in the previous example about Internet Access, we can also use statistical software to compute the test statistics and the P-value.\u00a0<\/section>\n<section>\n<section class=\"textbox interact\"><b>Step 1: <\/b>Under <strong>Enter Data<\/strong> select <strong>Number of Successes<\/strong>.<br \/>\n<b>Step 2: <\/b>Enter the <strong>Sample Size<\/strong>\u00a0and the <strong># of Successes<\/strong>\u00a0accordingly.<br \/>\n<b style=\"text-align: initial; background-color: initial; font-size: 1em;\">Step 3: <\/b><span style=\"text-align: initial; background-color: initial; font-size: 1em;\">Change the <\/span><strong style=\"text-align: initial; background-color: initial; font-size: 1em;\">Type of Inference<\/strong><span style=\"text-align: initial; background-color: initial; font-size: 1em;\"> to <strong>Significance Test<\/strong>.<br \/>\n<\/span><b>Step 4: <\/b>Enter the <strong>Null value<\/strong>\u00a0and change the <strong>Alternative<\/strong>\u00a0accordingly.<\/section>\n<\/section>\n<section><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/inference_prop\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/inference_prop\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<br \/>\nTry it out with the example below.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm1578\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1578&theme=lumen&iframe_resize_id=ohm1578&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<\/section>\n<section class=\"textbox example\">\n<div id=\"tnh-video-picker\" class=\"videoPicker\">\n<h3>One-Sample z-test for Proportions<\/h3>\n<form><label>Select Instructor:<\/label><select name=\"video\"><option value=\"https:\/\/www.youtube.com\/embed\/ULf9GpNmyZc\">Dr. Pamela E. Harris<\/option><option value=\"https:\/\/www.youtube.com\/embed\/DuNle6Lrk7o\">Dr. Aris Winger<\/option><option value=\"https:\/\/www.youtube.com\/embed\/RX0MuCVTHyk\">Dr. Lane Fisher<\/option><\/select><\/form>\n<div class=\"videoContainer\"><iframe src=\"https:\/\/www.youtube.com\/embed\/ULf9GpNmyZc\" allowfullscreen><\/iframe><\/div>\n<\/section>\n","protected":false},"author":8,"menu_order":20,"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":[{"divId":"tnh-video-picker","title":"One-Sample z-test for Proportions","label":"Select Instructor","video_collection":[{"displayName":"Dr. Pamela E. 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