{"id":2842,"date":"2024-07-25T23:04:08","date_gmt":"2024-07-26T03:04:08","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=2842"},"modified":"2024-07-26T04:46:00","modified_gmt":"2024-07-26T08:46:00","slug":"example-of-a-pooled-variance-test","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/example-of-a-pooled-variance-test\/","title":{"raw":"Pooled and UnPooled Variance Tests","rendered":"Pooled and UnPooled Variance Tests"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThis section review the following learning objectives.\r\n<ul>\r\n \t<li>Determine which test to use (Pooled or Unpooled) Variance t-Test<\/li>\r\n \t<li>Use Excel's Data Analysis Toolpak to calculate values in the test<\/li>\r\n \t<li>Draw conclusions based off of the test results<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nLet us first recap what types of scenarios where we used a Pooled Variance t-Test to test for differences in two population means:\r\n<ul>\r\n \t<li>We have two independent samples<\/li>\r\n \t<li>The standard deviation of one of the samples is less than double the other sample<\/li>\r\n \t<li>The variance of one of the samples is less than four times the variance of the other sample<\/li>\r\n<\/ul>\r\n<h1>Determining which Test to Use<\/h1>\r\nLet us revisit the example from the previous section but put a different 'spin' on it. This will, hopefully, help you understand the difference between paired and independent samples.\r\n<h2>Example 64.1<\/h2>\r\n<strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: One hundred suitable individuals who needed to take a statin to lower their cholesterol were selected at random.\r\n<ul>\r\n \t<li>They were given 'Brand X's statin medication<\/li>\r\n \t<li>Their levels were initially recorded (for a 'baseline' measurement).<\/li>\r\n \t<li>Their levels were recorded after six months of taking the medication.<\/li>\r\n<\/ul>\r\nAnother one hundred individuals who needed take a statin were selected at random.\r\n<ul>\r\n \t<li>They were given 'Brand Y's statin medication<\/li>\r\n \t<li>Their levels were initially recorded (for a 'baseline' measurement).<\/li>\r\n \t<li>Their levels were recorded after six months of taking the medication.<\/li>\r\n<\/ul>\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example64-1_Data.xlsx\">Click here<\/a> to download their before, after levels and the difference in levels. The sample standard deviation for the brands' differences between baseline and post-meds levels are given below:\r\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse;width: 65%;height: 88px\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 17px\">\r\n<th style=\"width: 33.3333%;text-align: center;height: 17px\">Statistic<\/th>\r\n<th style=\"width: 33.3333%;text-align: center;height: 17px\">Brand X<\/th>\r\n<th style=\"width: 33.3333%;text-align: center;height: 17px\">Brand Y<\/th>\r\n<\/tr>\r\n<tr style=\"height: 19px\">\r\n<td style=\"width: 33.3333%;height: 19px;text-align: center\">Standard Deviation<\/td>\r\n<td style=\"width: 33.3333%;height: 19px;text-align: center\">78.8395<\/td>\r\n<td style=\"width: 33.3333%;height: 19px;text-align: center\">69.1039<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: Which test should we use if we want to test if there is a difference between the average decrease in levels between the two brands?\r\n\r\n<span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: We need to examine how 'different' the standard deviation or variances are between the two brands' decrease in levels:\r\n<p style=\"padding-left: 40px\">[latex] \\frac{\\text{st dev}_X}{\\text{st dev}_Y} = \\frac{78.8395}{69.1039} = 1.14 &lt; 2 [\/latex]<\/p>\r\nBecause the one brand's standard deviation is less than double the other's, then we can use a Pooled Variance Test. Note: It is easiest to place the larger standard deviation on the top of the fraction and always compare to two:\r\n<ul>\r\n \t<li>If the ratio is smaller than 2 \u2014 use a pooled variance test.<\/li>\r\n \t<li>If the ratio is larger than 2 \u2014 use an unpooled variance test.<\/li>\r\n<\/ul>\r\n<h1>Setting up the Hypotheses<\/h1>\r\nLet us now setup the hypotheses for this problem in the next example.\r\n<h2>Example 64.2<\/h2>\r\n<strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Let us revisit Example 64.1's problem. We want to decide if there is a difference in the effectiveness in dropping cholesterol levels in the two brands.\r\n\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: What are the null and alternate hypotheses for this problem?\r\n\r\n<span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: Let us examine the differences between the 'Decrease in Level' values for the two brands. Let us call the brand X's true average decrease in cholesterol \u03bc<sub>d_x<\/sub>. Let us call the brand Y's true average decrease in cholesterol \u03bc<sub>d_y<\/sub>. This gives:\r\n\r\nH<sub>0<\/sub>: \u03bc<sub>d_x <\/sub>= \u03bc<sub>d_y<\/sub>\u00a0 or\u00a0 \u03bc<sub>d_x <\/sub>\u2212 \u03bc<sub>d_y<\/sub> = 0\r\n\r\nH<sub>A<\/sub>: \u03bc<sub>d_x <\/sub>\u2260 \u03bc<sub>d_y<\/sub>\u00a0 or\u00a0 \u03bc<sub>d_x <\/sub>\u2212 \u03bc<sub>d_y<\/sub> \u2260 0\r\n<h1>Using Excel's Data Analysis Toolpak<\/h1>\r\nLet us continue with Example 64.2. In this section, we will step through how to use Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/use-the-analysis-toolpak-to-perform-complex-data-analysis-6c67ccf0-f4a9-487c-8dec-bdb5a2cefab6\">Data Analysis Toolpak<\/a> to calculate the required metrics for this question.\r\n<h2>Example 64.3<\/h2>\r\n<strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Continue with Example 64.2. We want to decide if there is a difference in the effectiveness in dropping cholesterol levels in the two brands. In this example, pick the correct test within the Data Analysis Toolpak and then run this test to determine the following metrics:\r\n<ul>\r\n \t<li>The test statistic (t<sub>test<\/sub>)<\/li>\r\n \t<li>The p-value for correct tail<\/li>\r\n \t<li>The critical value (t<sub>crit<\/sub>)<\/li>\r\n<\/ul>\r\n<strong>Solutions<\/strong>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example64-1_Solutions.xlsx\">Click here<\/a>\u00a0to download the solutions shown in the video or click to reveal the step-by-step instructions below.\r\n\r\nhttps:\/\/youtu.be\/MUfV9r8ZhF4\r\n<div>\r\n<h1>Step-by-Step Solutions<\/h1>\r\n<ol>\r\n \t<li>Click on the 'Data' tab and select 'Data Analysis'\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/DataAnalysis.jpg\"><img class=\"alignnone wp-image-2854\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/DataAnalysis.jpg\" alt=\"Image of 'Data Analysis' highlighted in the Data Tab in Excel\" width=\"620\" height=\"150\" \/><\/a><\/li>\r\n \t<li>Select 't-test: Two-Sample Assuming Equal Variances and click 'OK'\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled.jpg\"><img class=\"alignnone wp-image-2884 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled.jpg\" alt=\"Image of drop-down menu with t-test: Two-Sample Assuming Equal Variances selected\" width=\"651\" height=\"319\" \/><\/a><\/li>\r\n \t<li>Put the following inputs in the t-test dialogue box:\r\n<ol>\r\n \t<li>Select brand X's 'Decrease in Level' as Variable <span style=\"text-decoration: underline\">1<\/span>.<\/li>\r\n \t<li>Select brand Y's 'Decrease in Level' as Variable <span style=\"text-decoration: underline\">2<\/span>.<\/li>\r\n \t<li>Set the Hypoth<span style=\"text-decoration: underline\">e<\/span>sized Mean Difference to 0.<\/li>\r\n \t<li>Check off <span style=\"text-decoration: underline\">L<\/span>abels.<\/li>\r\n \t<li>Enter 0.01 for <span style=\"text-decoration: underline\">A<\/span>lpha (this is the level of significance)<\/li>\r\n \t<li>Select an '<span style=\"text-decoration: underline\">O<\/span>utput Range' (either somewhere in the worksheet or a new worksheet)<\/li>\r\n \t<li>Click 'OK'\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_dialogue_box.jpg\"><img class=\"alignnone wp-image-2886 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_dialogue_box.jpg\" alt=\"Image with Pooled t-test dialogue box and inputs included. These inputs are also given in the Excel solutions.\" width=\"696\" height=\"449\" \/><\/a><\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nThe following outputs should be given:\r\n\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output.jpg\"><img class=\"alignnone wp-image-2887 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output.jpg\" alt=\"Screenshot of paired t-test outputs that are also given in the excel file provided\" width=\"549\" height=\"420\" \/><\/a>\r\n\r\n<\/div>\r\n<h1>The Decision and Conclusion<\/h1>\r\nSo how do we interpret the output given by the Data Analysis Toolpak? Let us form a conclusion based off the output given in the previous section.\r\n<h2>Example 64.4<\/h2>\r\n<strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Continue with Example 64.3. Interpret the Excel output given in that example (see below).\r\n<p style=\"padding-left: 40px\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output.jpg\"><img class=\"alignnone wp-image-2887 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output.jpg\" alt=\"Screenshot of paired t-test outputs that are also given in the excel file provided\" width=\"549\" height=\"420\" \/><\/a><\/p>\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: What are your decision and conclusion based on the above output?\r\n\r\n<span style=\"color: #003366\"><strong>Solutions<\/strong><\/span>: We are performing a two-tailed test. Therefore, read the P(T&lt;=t) two-tail line to determine the p-value.\r\n<ul>\r\n \t<li><span style=\"color: #003366\">Decision<\/span>: Fail to reject H<sub>0<\/sub> at the 1% level of significance<\/li>\r\n \t<li><span style=\"color: #003366\">Reasoning<\/span>: The p-value = 0.91125 is greater than (&gt;) the level of significance (0.01)<\/li>\r\n \t<li><span style=\"color: #003366\">Conclusion<\/span>: There is not sufficient evidence to conclude one brand's average decrease in cholesterol levels is different from the other brand's decrease in level.<\/li>\r\n<\/ul>\r\n<h1>UnPooled Variance Test Case<\/h1>\r\nLet us continue slightly change up Example 64.1 to help us better understand the difference between pooled an unpooled variance tests.\r\n<h2>Example 64.5<\/h2>\r\n<strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Let us continue with example 64.5. In this, however, let assume the following to be true of the standard deviations for brand X's and brand Y's decrease in cholesterol levels:\r\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse;width: 65%;height: 88px\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 17px\">\r\n<th style=\"width: 33.3333%;text-align: center;height: 17px\">Statistic<\/th>\r\n<th style=\"width: 33.3333%;text-align: center;height: 17px\">Brand X<\/th>\r\n<th style=\"width: 33.3333%;text-align: center;height: 17px\">Brand Y<\/th>\r\n<\/tr>\r\n<tr style=\"height: 19px\">\r\n<td style=\"width: 33.3333%;height: 19px;text-align: center\">Standard Deviation<\/td>\r\n<td style=\"width: 33.3333%;height: 19px;text-align: center\">78.8395<\/td>\r\n<td style=\"width: 33.3333%;height: 19px;text-align: center\">169.1039<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: Which test should we use if we want to test if there is a difference between the average decrease in levels between the two brands?\r\n\r\n<span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: We need to examine how 'different' the standard deviation or variances are between the two brands' decrease in levels. Since Brand Y has a higher standard deviation, let us place its value on the top of the fraction:\r\n<p style=\"padding-left: 40px\">[latex] \\frac{\\text{st dev}_Y}{\\text{st dev}_X} = \\frac{169.1039}{78.8395} = 2.1449 &gt; 2 [\/latex]<\/p>\r\nBecause the one brand's standard deviation is more than double the other's, then we can use an Unpooled Variance Test. Note: It is easiest to place the larger standard deviation on the top of the fraction and always compare to two:\r\n<ul>\r\n \t<li>If the ratio is smaller than 2 \u2014 use a pooled variance test.<\/li>\r\n \t<li>If the ratio is larger than 2 \u2014 use an unpooled variance test.<\/li>\r\n<\/ul>\r\nTo perform this test in Excel, all the steps are identical with the exception that you choose 'Two-Sample Assuming Unequal Variances':\r\n<p style=\"padding-left: 40px\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_UnPooled.jpg\"><img class=\"alignnone size-full wp-image-2898\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_UnPooled.jpg\" alt=\"dialogue box with 't-test: Two-Sample Assuming Unequal Variances' selected\" width=\"653\" height=\"316\" data-wp-editing=\"1\" \/><\/a><\/p>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>This section review the following learning objectives.<\/p>\n<ul>\n<li>Determine which test to use (Pooled or Unpooled) Variance t-Test<\/li>\n<li>Use Excel&#8217;s Data Analysis Toolpak to calculate values in the test<\/li>\n<li>Draw conclusions based off of the test results<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>Let us first recap what types of scenarios where we used a Pooled Variance t-Test to test for differences in two population means:<\/p>\n<ul>\n<li>We have two independent samples<\/li>\n<li>The standard deviation of one of the samples is less than double the other sample<\/li>\n<li>The variance of one of the samples is less than four times the variance of the other sample<\/li>\n<\/ul>\n<h1>Determining which Test to Use<\/h1>\n<p>Let us revisit the example from the previous section but put a different &#8216;spin&#8217; on it. This will, hopefully, help you understand the difference between paired and independent samples.<\/p>\n<h2>Example 64.1<\/h2>\n<p><strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: One hundred suitable individuals who needed to take a statin to lower their cholesterol were selected at random.<\/p>\n<ul>\n<li>They were given &#8216;Brand X&#8217;s statin medication<\/li>\n<li>Their levels were initially recorded (for a &#8216;baseline&#8217; measurement).<\/li>\n<li>Their levels were recorded after six months of taking the medication.<\/li>\n<\/ul>\n<p>Another one hundred individuals who needed take a statin were selected at random.<\/p>\n<ul>\n<li>They were given &#8216;Brand Y&#8217;s statin medication<\/li>\n<li>Their levels were initially recorded (for a &#8216;baseline&#8217; measurement).<\/li>\n<li>Their levels were recorded after six months of taking the medication.<\/li>\n<\/ul>\n<p><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example64-1_Data.xlsx\">Click here<\/a> to download their before, after levels and the difference in levels. The sample standard deviation for the brands&#8217; differences between baseline and post-meds levels are given below:<\/p>\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse;width: 65%;height: 88px\">\n<tbody>\n<tr style=\"height: 17px\">\n<th style=\"width: 33.3333%;text-align: center;height: 17px\">Statistic<\/th>\n<th style=\"width: 33.3333%;text-align: center;height: 17px\">Brand X<\/th>\n<th style=\"width: 33.3333%;text-align: center;height: 17px\">Brand Y<\/th>\n<\/tr>\n<tr style=\"height: 19px\">\n<td style=\"width: 33.3333%;height: 19px;text-align: center\">Standard Deviation<\/td>\n<td style=\"width: 33.3333%;height: 19px;text-align: center\">78.8395<\/td>\n<td style=\"width: 33.3333%;height: 19px;text-align: center\">69.1039<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: Which test should we use if we want to test if there is a difference between the average decrease in levels between the two brands?<\/p>\n<p><span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: We need to examine how &#8216;different&#8217; the standard deviation or variances are between the two brands&#8217; decrease in levels:<\/p>\n<p style=\"padding-left: 40px\">[latex]\\frac{\\text{st dev}_X}{\\text{st dev}_Y} = \\frac{78.8395}{69.1039} = 1.14 < 2[\/latex]<\/p>\n<p>Because the one brand&#8217;s standard deviation is less than double the other&#8217;s, then we can use a Pooled Variance Test. Note: It is easiest to place the larger standard deviation on the top of the fraction and always compare to two:<\/p>\n<ul>\n<li>If the ratio is smaller than 2 \u2014 use a pooled variance test.<\/li>\n<li>If the ratio is larger than 2 \u2014 use an unpooled variance test.<\/li>\n<\/ul>\n<h1>Setting up the Hypotheses<\/h1>\n<p>Let us now setup the hypotheses for this problem in the next example.<\/p>\n<h2>Example 64.2<\/h2>\n<p><strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Let us revisit Example 64.1&#8217;s problem. We want to decide if there is a difference in the effectiveness in dropping cholesterol levels in the two brands.<\/p>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: What are the null and alternate hypotheses for this problem?<\/p>\n<p><span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: Let us examine the differences between the &#8216;Decrease in Level&#8217; values for the two brands. Let us call the brand X&#8217;s true average decrease in cholesterol \u03bc<sub>d_x<\/sub>. Let us call the brand Y&#8217;s true average decrease in cholesterol \u03bc<sub>d_y<\/sub>. This gives:<\/p>\n<p>H<sub>0<\/sub>: \u03bc<sub>d_x <\/sub>= \u03bc<sub>d_y<\/sub>\u00a0 or\u00a0 \u03bc<sub>d_x <\/sub>\u2212 \u03bc<sub>d_y<\/sub> = 0<\/p>\n<p>H<sub>A<\/sub>: \u03bc<sub>d_x <\/sub>\u2260 \u03bc<sub>d_y<\/sub>\u00a0 or\u00a0 \u03bc<sub>d_x <\/sub>\u2212 \u03bc<sub>d_y<\/sub> \u2260 0<\/p>\n<h1>Using Excel&#8217;s Data Analysis Toolpak<\/h1>\n<p>Let us continue with Example 64.2. In this section, we will step through how to use Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/use-the-analysis-toolpak-to-perform-complex-data-analysis-6c67ccf0-f4a9-487c-8dec-bdb5a2cefab6\">Data Analysis Toolpak<\/a> to calculate the required metrics for this question.<\/p>\n<h2>Example 64.3<\/h2>\n<p><strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Continue with Example 64.2. We want to decide if there is a difference in the effectiveness in dropping cholesterol levels in the two brands. In this example, pick the correct test within the Data Analysis Toolpak and then run this test to determine the following metrics:<\/p>\n<ul>\n<li>The test statistic (t<sub>test<\/sub>)<\/li>\n<li>The p-value for correct tail<\/li>\n<li>The critical value (t<sub>crit<\/sub>)<\/li>\n<\/ul>\n<p><strong>Solutions<\/strong>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/Example64-1_Solutions.xlsx\">Click here<\/a>\u00a0to download the solutions shown in the video or click to reveal the step-by-step instructions below.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"How to perform a pooled variance t-test\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/MUfV9r8ZhF4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div>\n<h1>Step-by-Step Solutions<\/h1>\n<ol>\n<li>Click on the &#8216;Data&#8217; tab and select &#8216;Data Analysis&#8217;<br \/>\n<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/DataAnalysis.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2854\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/DataAnalysis.jpg\" alt=\"Image of 'Data Analysis' highlighted in the Data Tab in Excel\" width=\"620\" height=\"150\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/DataAnalysis.jpg 1635w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/DataAnalysis-300x73.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/DataAnalysis-1024x248.jpg 1024w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/DataAnalysis-768x186.jpg 768w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/DataAnalysis-1536x372.jpg 1536w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/DataAnalysis-65x16.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/DataAnalysis-225x54.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/DataAnalysis-350x85.jpg 350w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/a><\/li>\n<li>Select &#8216;t-test: Two-Sample Assuming Equal Variances and click &#8216;OK&#8217;<br \/>\n<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2884 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled.jpg\" alt=\"Image of drop-down menu with t-test: Two-Sample Assuming Equal Variances selected\" width=\"651\" height=\"319\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled.jpg 651w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled-300x147.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled-65x32.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled-225x110.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled-350x172.jpg 350w\" sizes=\"auto, (max-width: 651px) 100vw, 651px\" \/><\/a><\/li>\n<li>Put the following inputs in the t-test dialogue box:\n<ol>\n<li>Select brand X&#8217;s &#8216;Decrease in Level&#8217; as Variable <span style=\"text-decoration: underline\">1<\/span>.<\/li>\n<li>Select brand Y&#8217;s &#8216;Decrease in Level&#8217; as Variable <span style=\"text-decoration: underline\">2<\/span>.<\/li>\n<li>Set the Hypoth<span style=\"text-decoration: underline\">e<\/span>sized Mean Difference to 0.<\/li>\n<li>Check off <span style=\"text-decoration: underline\">L<\/span>abels.<\/li>\n<li>Enter 0.01 for <span style=\"text-decoration: underline\">A<\/span>lpha (this is the level of significance)<\/li>\n<li>Select an &#8216;<span style=\"text-decoration: underline\">O<\/span>utput Range&#8217; (either somewhere in the worksheet or a new worksheet)<\/li>\n<li>Click &#8216;OK&#8217;<br \/>\n<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_dialogue_box.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2886 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_dialogue_box.jpg\" alt=\"Image with Pooled t-test dialogue box and inputs included. These inputs are also given in the Excel solutions.\" width=\"696\" height=\"449\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_dialogue_box.jpg 696w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_dialogue_box-300x194.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_dialogue_box-65x42.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_dialogue_box-225x145.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_dialogue_box-350x226.jpg 350w\" sizes=\"auto, (max-width: 696px) 100vw, 696px\" \/><\/a><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>The following outputs should be given:<\/p>\n<p><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2887 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output.jpg\" alt=\"Screenshot of paired t-test outputs that are also given in the excel file provided\" width=\"549\" height=\"420\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output.jpg 549w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output-300x230.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output-65x50.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output-225x172.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output-350x268.jpg 350w\" sizes=\"auto, (max-width: 549px) 100vw, 549px\" \/><\/a><\/p>\n<\/div>\n<h1>The Decision and Conclusion<\/h1>\n<p>So how do we interpret the output given by the Data Analysis Toolpak? Let us form a conclusion based off the output given in the previous section.<\/p>\n<h2>Example 64.4<\/h2>\n<p><strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Continue with Example 64.3. Interpret the Excel output given in that example (see below).<\/p>\n<p style=\"padding-left: 40px\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2887 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output.jpg\" alt=\"Screenshot of paired t-test outputs that are also given in the excel file provided\" width=\"549\" height=\"420\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output.jpg 549w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output-300x230.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output-65x50.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output-225x172.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_Pooled_output-350x268.jpg 350w\" sizes=\"auto, (max-width: 549px) 100vw, 549px\" \/><\/a><\/p>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: What are your decision and conclusion based on the above output?<\/p>\n<p><span style=\"color: #003366\"><strong>Solutions<\/strong><\/span>: We are performing a two-tailed test. Therefore, read the P(T&lt;=t) two-tail line to determine the p-value.<\/p>\n<ul>\n<li><span style=\"color: #003366\">Decision<\/span>: Fail to reject H<sub>0<\/sub> at the 1% level of significance<\/li>\n<li><span style=\"color: #003366\">Reasoning<\/span>: The p-value = 0.91125 is greater than (&gt;) the level of significance (0.01)<\/li>\n<li><span style=\"color: #003366\">Conclusion<\/span>: There is not sufficient evidence to conclude one brand&#8217;s average decrease in cholesterol levels is different from the other brand&#8217;s decrease in level.<\/li>\n<\/ul>\n<h1>UnPooled Variance Test Case<\/h1>\n<p>Let us continue slightly change up Example 64.1 to help us better understand the difference between pooled an unpooled variance tests.<\/p>\n<h2>Example 64.5<\/h2>\n<p><strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Let us continue with example 64.5. In this, however, let assume the following to be true of the standard deviations for brand X&#8217;s and brand Y&#8217;s decrease in cholesterol levels:<\/p>\n<table class=\"lines aligncenter\" style=\"border-collapse: collapse;width: 65%;height: 88px\">\n<tbody>\n<tr style=\"height: 17px\">\n<th style=\"width: 33.3333%;text-align: center;height: 17px\">Statistic<\/th>\n<th style=\"width: 33.3333%;text-align: center;height: 17px\">Brand X<\/th>\n<th style=\"width: 33.3333%;text-align: center;height: 17px\">Brand Y<\/th>\n<\/tr>\n<tr style=\"height: 19px\">\n<td style=\"width: 33.3333%;height: 19px;text-align: center\">Standard Deviation<\/td>\n<td style=\"width: 33.3333%;height: 19px;text-align: center\">78.8395<\/td>\n<td style=\"width: 33.3333%;height: 19px;text-align: center\">169.1039<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: Which test should we use if we want to test if there is a difference between the average decrease in levels between the two brands?<\/p>\n<p><span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: We need to examine how &#8216;different&#8217; the standard deviation or variances are between the two brands&#8217; decrease in levels. Since Brand Y has a higher standard deviation, let us place its value on the top of the fraction:<\/p>\n<p style=\"padding-left: 40px\">[latex]\\frac{\\text{st dev}_Y}{\\text{st dev}_X} = \\frac{169.1039}{78.8395} = 2.1449 > 2[\/latex]<\/p>\n<p>Because the one brand&#8217;s standard deviation is more than double the other&#8217;s, then we can use an Unpooled Variance Test. Note: It is easiest to place the larger standard deviation on the top of the fraction and always compare to two:<\/p>\n<ul>\n<li>If the ratio is smaller than 2 \u2014 use a pooled variance test.<\/li>\n<li>If the ratio is larger than 2 \u2014 use an unpooled variance test.<\/li>\n<\/ul>\n<p>To perform this test in Excel, all the steps are identical with the exception that you choose &#8216;Two-Sample Assuming Unequal Variances&#8217;:<\/p>\n<p style=\"padding-left: 40px\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_UnPooled.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-2898\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_UnPooled.jpg\" alt=\"dialogue box with 't-test: Two-Sample Assuming Unequal Variances' selected\" width=\"653\" height=\"316\" data-wp-editing=\"1\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_UnPooled.jpg 653w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_UnPooled-300x145.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_UnPooled-65x31.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_UnPooled-225x109.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/07\/t-test_UnPooled-350x169.jpg 350w\" sizes=\"auto, (max-width: 653px) 100vw, 653px\" \/><\/a><\/p>\n","protected":false},"author":865,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2842","chapter","type-chapter","status-publish","hentry"],"part":2690,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2842","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/users\/865"}],"version-history":[{"count":23,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2842\/revisions"}],"predecessor-version":[{"id":2899,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2842\/revisions\/2899"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/2690"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2842\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=2842"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=2842"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=2842"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=2842"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}