{"id":1322,"date":"2023-06-22T02:17:32","date_gmt":"2023-06-22T02:17:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/one-sample-hypothesis-test-for-means-learn-it-2\/"},"modified":"2025-05-16T22:25:54","modified_gmt":"2025-05-16T22:25:54","slug":"one-sample-hypothesis-test-for-means-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/one-sample-hypothesis-test-for-means-learn-it-2\/","title":{"raw":"One-Sample Hypothesis Test for Means: Apply It 2","rendered":"One-Sample Hypothesis Test for Means: Apply 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 t-test for means 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]t[\/latex]-test for means from hypotheses to conclusions.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>In the previous page, we looked at the following scenario:<\/p>\r\n<p>Researchers want to examine the effect of diet on cholesterol levels. They select a random sample of [latex]125[\/latex] adult males who are vegetarians and test their cholesterol levels to determine if they are significantly different from [latex]201.5[\/latex] milligrams of cholesterol per deciliter (mg\/dl), which is the average cholesterol level of all adult males without heart disease. We have found that the conditions to conduct a one-sample hypothesis test have been satisfied.<\/p>\r\n<p>Use the following statistical tool to find the test statistic, and let's use it to make an inference about the population.<\/p>\r\n<section class=\"textbox interact\">In the statistical tool, complete the following steps:\r\n\r\n<p style=\"padding-left: 40px;\"><strong>Step 1: <\/strong>Under <strong>Enter Data<\/strong>, select <strong>Summary Statistics<\/strong>.<\/p>\r\n<p style=\"padding-left: 40px;\"><strong>Step 2: <\/strong>The sample results about the cholesterol level are shown in the following table. Enter the sample results accordingly.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 59.7222px;\">\u00a0<\/td>\r\n<td style=\"width: 33.6111px;\"><strong>Size<\/strong><\/td>\r\n<td style=\"width: 42.9167px;\"><strong>Mean<\/strong><\/td>\r\n<td style=\"width: 75.5729px;\"><strong>Standard Deviation<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 59.7222px;\"><strong>Sample<\/strong><\/td>\r\n<td style=\"width: 33.6111px;\">[latex]125[\/latex]<\/td>\r\n<td style=\"width: 42.9167px;\">[latex]183.4[\/latex]<\/td>\r\n<td style=\"width: 75.5729px;\">[latex]15[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"padding-left: 40px;\"><strong>Step 3: <\/strong>Under <strong>Type of Inference<\/strong>, select <strong>Significance Test<\/strong>.<\/p>\r\n<p style=\"padding-left: 40px;\"><strong>Step 4: <\/strong>Enter the null value (in this scenario is [latex]201.5[\/latex]) and select the correct alternative hypothesis.<\/p>\r\n<\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/inference_mean\/ \" width=\"100%\" height=\"1075\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><\/iframe><br \/>\r\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/inference_mean\/\" 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]994[\/ohm2_question]<\/section>\r\n<p>Notice that in the statistical tool above, the test statistic we are using is the [latex]t[\/latex]-statistic. Recall that in practice, we rarely know the population standard deviation. So, we use the sample standard deviation [latex]s[\/latex] as an estimate for [latex]\\sigma[\/latex]. Because of this estimation, we now use the [latex]t[\/latex]-distribution instead of the normal distribution.<\/p>\r\n<section class=\"textbox proTip\">If you draw a simple random sample of size [latex]n[\/latex] from a population that has an approximately a normal distribution with mean [latex]\\mu[\/latex] and unknown population standard deviation [latex]\\sigma[\/latex], then we will use the [latex]t[\/latex]-statistic as the test statistic.\r\n\r\n<p>[latex]t=\\dfrac{\\bar{x}-\\mu}{\\frac{s}{\\sqrt{n}}}[\/latex]<\/p>\r\n<p>For each sample size [latex]n[\/latex], there is a different [latex]t[\/latex]-distribution, which depends on the sample's degree of freedom, [latex]df=n-1[\/latex].<\/p>\r\n<\/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 t-test for means 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]t[\/latex]-test for means from hypotheses to conclusions.<\/span><\/li>\n<\/ul>\n<\/section>\n<p>In the previous page, we looked at the following scenario:<\/p>\n<p>Researchers want to examine the effect of diet on cholesterol levels. They select a random sample of [latex]125[\/latex] adult males who are vegetarians and test their cholesterol levels to determine if they are significantly different from [latex]201.5[\/latex] milligrams of cholesterol per deciliter (mg\/dl), which is the average cholesterol level of all adult males without heart disease. We have found that the conditions to conduct a one-sample hypothesis test have been satisfied.<\/p>\n<p>Use the following statistical tool to find the test statistic, and let&#8217;s use it to make an inference about the population.<\/p>\n<section class=\"textbox interact\">In the statistical tool, complete the following steps:<\/p>\n<p style=\"padding-left: 40px;\"><strong>Step 1: <\/strong>Under <strong>Enter Data<\/strong>, select <strong>Summary Statistics<\/strong>.<\/p>\n<p style=\"padding-left: 40px;\"><strong>Step 2: <\/strong>The sample results about the cholesterol level are shown in the following table. Enter the sample results accordingly.<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"width: 59.7222px;\">\u00a0<\/td>\n<td style=\"width: 33.6111px;\"><strong>Size<\/strong><\/td>\n<td style=\"width: 42.9167px;\"><strong>Mean<\/strong><\/td>\n<td style=\"width: 75.5729px;\"><strong>Standard Deviation<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 59.7222px;\"><strong>Sample<\/strong><\/td>\n<td style=\"width: 33.6111px;\">[latex]125[\/latex]<\/td>\n<td style=\"width: 42.9167px;\">[latex]183.4[\/latex]<\/td>\n<td style=\"width: 75.5729px;\">[latex]15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"padding-left: 40px;\"><strong>Step 3: <\/strong>Under <strong>Type of Inference<\/strong>, select <strong>Significance Test<\/strong>.<\/p>\n<p style=\"padding-left: 40px;\"><strong>Step 4: <\/strong>Enter the null value (in this scenario is [latex]201.5[\/latex]) and select the correct alternative hypothesis.<\/p>\n<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/inference_mean\/\" width=\"100%\" height=\"1075\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><\/iframe><br \/>\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/inference_mean\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm994\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=994&theme=lumen&iframe_resize_id=ohm994&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Notice that in the statistical tool above, the test statistic we are using is the [latex]t[\/latex]-statistic. Recall that in practice, we rarely know the population standard deviation. So, we use the sample standard deviation [latex]s[\/latex] as an estimate for [latex]\\sigma[\/latex]. Because of this estimation, we now use the [latex]t[\/latex]-distribution instead of the normal distribution.<\/p>\n<section class=\"textbox proTip\">If you draw a simple random sample of size [latex]n[\/latex] from a population that has an approximately a normal distribution with mean [latex]\\mu[\/latex] and unknown population standard deviation [latex]\\sigma[\/latex], then we will use the [latex]t[\/latex]-statistic as the test statistic.<\/p>\n<p>[latex]t=\\dfrac{\\bar{x}-\\mu}{\\frac{s}{\\sqrt{n}}}[\/latex]<\/p>\n<p>For each sample size [latex]n[\/latex], there is a different [latex]t[\/latex]-distribution, which depends on the sample&#8217;s degree of freedom, [latex]df=n-1[\/latex].<\/p>\n<\/section>\n","protected":false},"author":8,"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":1309,"module-header":"apply_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\/1322"}],"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":5,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1322\/revisions"}],"predecessor-version":[{"id":6812,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1322\/revisions\/6812"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1309"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1322\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1322"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1322"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1322"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1322"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}