{"id":2330,"date":"2024-06-28T21:01:19","date_gmt":"2024-06-29T01:01:19","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=2330"},"modified":"2024-06-30T01:15:20","modified_gmt":"2024-06-30T05:15:20","slug":"confidence-intervals-when-is-%cf%83-unknown","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/confidence-intervals-when-is-%cf%83-unknown\/","title":{"raw":"Confidence Intervals when is \u03c3 Unknown","rendered":"Confidence Intervals when is \u03c3 Unknown"},"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\nIn this section, we will do the following calculations to estimate the true mean when the population standard deviation (\u03c3) is unknown:\r\n<ul>\r\n \t<li>Understand when z-scores and when t-scores are used<\/li>\r\n \t<li>Understand the meaning of t-scores<\/li>\r\n \t<li>Examine several Excel calls to calculate t-scores and the margin of error<\/li>\r\n \t<li>Introduce the confidence intervals formula<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nWhen the population standard deviation (\u03c3) is unknown, we need to use a t-score instead of a z-score. In this case, the confidence interval formulas become:\r\n<ul>\r\n \t<li>[latex] CL_{Lower} = \\bar{x}-t \\cdot \\frac{s}{\\sqrt{n}} [\/latex]<\/li>\r\n \t<li>[latex] CL_{Upper} = \\bar{x}-t \\cdot \\frac{s}{\\sqrt{n}} [\/latex]<\/li>\r\n \t<li>where\u00a0[latex] E = t \\cdot \\frac{s}{\\sqrt{n}} [\/latex]<\/li>\r\n<\/ul>\r\n<h2>Calculating T-scores<\/h2>\r\nWhen the population standard deviation is unknown, we must use a t-score instead of a z-score (that we used previously). We will explore what t-scores and what a t-distribution is in the next section also. For now, we will examine the 3 possible ways of calculating a t-score or the margin of error related to the t-score in Excel:\r\n<ol>\r\n \t<li>[latex] t = \\text{T.INV.2T}(\\alpha, df) [\/latex]<\/li>\r\n \t<li>[latex] t = \\text{T.INV}(\\frac{\\alpha}{2}, df) [\/latex]<\/li>\r\n \t<li>[latex] E = \\text{CONFIDENCE.T}(\\alpha, {2}, s , n) [\/latex]<\/li>\r\n<\/ol>\r\nThere are two new expressions to understand in above formulas (apart from t):\r\n<ul>\r\n \t<li>[latex] df = \\text{degrees of freedom} = n - 1[\/latex]<\/li>\r\n \t<li>[latex] \\alpha = 1 - \\text{confidence level} [\/latex]<\/li>\r\n<\/ul>\r\n<h1>Understanding Excel's T.INV.2T Function<\/h1>\r\nWhen we use Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/t-inv-2t-function-ce72ea19-ec6c-4be7-bed2-b9baf2264f17\">T.INV.2T()<\/a> function, we input the area outside of the confidence interval. This area is also called \u03b1 (alpha). See figure 49.1 below to better understand this area.\r\n\r\n[caption id=\"attachment_2336\" align=\"aligncenter\" width=\"551\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha.jpg\"><img class=\"wp-image-2336 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha.jpg\" alt=\"Bell shaped curve with area between \u2212t and t highlighted. In the inside area is written &quot;Confidence Level.&quot; Above the highlighted area is written alpha divided by 2. Below the shaded area in the lower tail is also written alpha over two.\" width=\"551\" height=\"280\" \/><\/a> Figure 50.1 Confidence interval with t-scores shown on bottom axis.[\/caption]\r\n\r\nWe can see from the above graph that \u03b1 (alpha) makes up the area in the upper and lower tails (split between the two tails). It can be calculated using:\r\n\r\n\\[\\alpha = 100\\% - \\text{Confidence Level} = 1 - \\text{Confidence Level} \\]\r\n\r\nWe can solve for [latex]t[\/latex] using:\r\n\r\n\\[t = \\text{T.INV.2T}(\\alpha, df) \\]\r\n\r\nFinally, remember that [latex] df = n - 1 [\/latex]\r\n<h1>Understanding Excel's T.INV Function<\/h1>\r\nWhen we use Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/t-inv-function-2908272b-4e61-4942-9df9-a25fec9b0e2e\">T.INV()<\/a> Function, we input the area to the left of a [latex]t[\/latex]-score. The easiest is to input the area in the left tail [latex]=\\frac{\\alpha}{2}[\/latex]. This returns the negative (\u2212) [latex]t[\/latex]-score:\r\n\r\n[caption id=\"attachment_2341\" align=\"aligncenter\" width=\"551\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha2.jpg\"><img class=\"wp-image-2341 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha2.jpg\" alt=\"Bell shaped curve with area between \u2212t and t highlighted. In the inside area is written &quot;Confidence Level.&quot; Above the highlighted area is written alpha divided by 2. Below the shaded area in the lower tail is also written alpha over two.\" width=\"551\" height=\"288\" \/><\/a> Figure 50.2 Confidence interval with t-scores and T.INV() formulas indicated.[\/caption]\r\n\r\nWe can see from the above graph that we can solve for the negative (\u2212) or positive [latex]t[\/latex]-scores:\r\n\r\n\\[-t = \\text{T.INV}(\\frac{\\alpha}{2}, df) = \\text{T.INV}(\\frac{\\alpha}{2}, n-1)\\]\r\n\r\n\\[t = \\text{T.INV}(\\frac{\\alpha}{2}+\\text{Conf Level}, df) = \\text{T.INV}(\\frac{\\alpha}{2}+\\text{Conf Level}, n-1)\\]\r\n<h1>Understanding Excel's CONFIDENCE.T Function<\/h1>\r\nWhen we use Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/confidence-t-function-e8eca395-6c3a-4ba9-9003-79ccc61d3c53\">CONFIDENCE.T()<\/a> function, we input the area outside of the confidence interval (\u03b1). This function returns the Margin of Error (E):\r\n\r\n[caption id=\"attachment_2343\" align=\"aligncenter\" width=\"551\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_xbar_E_alpha.jpg\"><img class=\"wp-image-2343 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_xbar_E_alpha.jpg\" alt=\"Bell shaped curve with area between the lower and upper confidence interval limits highlighted. Between the limits and the middle is marked E on either side. \" width=\"551\" height=\"279\" \/><\/a> Figure 50.3 Confidence interval with limits, sample mean and margin of error indicated.[\/caption]\r\n\r\nExcel's CONFIDENCE.T() function is the quickest way to calculate the margin of error (there is no need to calculate the [latex]t[\/latex]-score nor do the margin of error calculation):\r\n\r\n[latex] E = t \\cdot \\frac{s}{\\sqrt{n}} = \\text{CONFIDENCE.T}(\\alpha, {2}, s , n) [\/latex]\r\n\r\nWe can easily calculate the lower and upper limits once the margin of error is known:\r\n\r\n\\[CL_{Lower} = \\bar{x}-E\\]\r\n\r\n\\[CL_{Upper} = \\bar{x}+E\\]","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>In this section, we will do the following calculations to estimate the true mean when the population standard deviation (\u03c3) is unknown:<\/p>\n<ul>\n<li>Understand when z-scores and when t-scores are used<\/li>\n<li>Understand the meaning of t-scores<\/li>\n<li>Examine several Excel calls to calculate t-scores and the margin of error<\/li>\n<li>Introduce the confidence intervals formula<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>When the population standard deviation (\u03c3) is unknown, we need to use a t-score instead of a z-score. In this case, the confidence interval formulas become:<\/p>\n<ul>\n<li>[latex]CL_{Lower} = \\bar{x}-t \\cdot \\frac{s}{\\sqrt{n}}[\/latex]<\/li>\n<li>[latex]CL_{Upper} = \\bar{x}-t \\cdot \\frac{s}{\\sqrt{n}}[\/latex]<\/li>\n<li>where\u00a0[latex]E = t \\cdot \\frac{s}{\\sqrt{n}}[\/latex]<\/li>\n<\/ul>\n<h2>Calculating T-scores<\/h2>\n<p>When the population standard deviation is unknown, we must use a t-score instead of a z-score (that we used previously). We will explore what t-scores and what a t-distribution is in the next section also. For now, we will examine the 3 possible ways of calculating a t-score or the margin of error related to the t-score in Excel:<\/p>\n<ol>\n<li>[latex]t = \\text{T.INV.2T}(\\alpha, df)[\/latex]<\/li>\n<li>[latex]t = \\text{T.INV}(\\frac{\\alpha}{2}, df)[\/latex]<\/li>\n<li>[latex]E = \\text{CONFIDENCE.T}(\\alpha, {2}, s , n)[\/latex]<\/li>\n<\/ol>\n<p>There are two new expressions to understand in above formulas (apart from t):<\/p>\n<ul>\n<li>[latex]df = \\text{degrees of freedom} = n - 1[\/latex]<\/li>\n<li>[latex]\\alpha = 1 - \\text{confidence level}[\/latex]<\/li>\n<\/ul>\n<h1>Understanding Excel&#8217;s T.INV.2T Function<\/h1>\n<p>When we use Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/t-inv-2t-function-ce72ea19-ec6c-4be7-bed2-b9baf2264f17\">T.INV.2T()<\/a> function, we input the area outside of the confidence interval. This area is also called \u03b1 (alpha). See figure 49.1 below to better understand this area.<\/p>\n<figure id=\"attachment_2336\" aria-describedby=\"caption-attachment-2336\" style=\"width: 551px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2336 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha.jpg\" alt=\"Bell shaped curve with area between \u2212t and t highlighted. In the inside area is written &quot;Confidence Level.&quot; Above the highlighted area is written alpha divided by 2. Below the shaded area in the lower tail is also written alpha over two.\" width=\"551\" height=\"280\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha.jpg 551w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha-300x152.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha-65x33.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha-225x114.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha-350x178.jpg 350w\" sizes=\"auto, (max-width: 551px) 100vw, 551px\" \/><\/a><figcaption id=\"caption-attachment-2336\" class=\"wp-caption-text\">Figure 50.1 Confidence interval with t-scores shown on bottom axis.<\/figcaption><\/figure>\n<p>We can see from the above graph that \u03b1 (alpha) makes up the area in the upper and lower tails (split between the two tails). It can be calculated using:<\/p>\n<p>\\[\\alpha = 100\\% &#8211; \\text{Confidence Level} = 1 &#8211; \\text{Confidence Level} \\]<\/p>\n<p>We can solve for [latex]t[\/latex] using:<\/p>\n<p>\\[t = \\text{T.INV.2T}(\\alpha, df) \\]<\/p>\n<p>Finally, remember that [latex]df = n - 1[\/latex]<\/p>\n<h1>Understanding Excel&#8217;s T.INV Function<\/h1>\n<p>When we use Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/t-inv-function-2908272b-4e61-4942-9df9-a25fec9b0e2e\">T.INV()<\/a> Function, we input the area to the left of a [latex]t[\/latex]-score. The easiest is to input the area in the left tail [latex]=\\frac{\\alpha}{2}[\/latex]. This returns the negative (\u2212) [latex]t[\/latex]-score:<\/p>\n<figure id=\"attachment_2341\" aria-describedby=\"caption-attachment-2341\" style=\"width: 551px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha2.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2341 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha2.jpg\" alt=\"Bell shaped curve with area between \u2212t and t highlighted. In the inside area is written &quot;Confidence Level.&quot; Above the highlighted area is written alpha divided by 2. Below the shaded area in the lower tail is also written alpha over two.\" width=\"551\" height=\"288\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha2.jpg 551w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha2-300x157.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha2-65x34.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha2-225x118.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_t_alpha2-350x183.jpg 350w\" sizes=\"auto, (max-width: 551px) 100vw, 551px\" \/><\/a><figcaption id=\"caption-attachment-2341\" class=\"wp-caption-text\">Figure 50.2 Confidence interval with t-scores and T.INV() formulas indicated.<\/figcaption><\/figure>\n<p>We can see from the above graph that we can solve for the negative (\u2212) or positive [latex]t[\/latex]-scores:<\/p>\n<p>\\[-t = \\text{T.INV}(\\frac{\\alpha}{2}, df) = \\text{T.INV}(\\frac{\\alpha}{2}, n-1)\\]<\/p>\n<p>\\[t = \\text{T.INV}(\\frac{\\alpha}{2}+\\text{Conf Level}, df) = \\text{T.INV}(\\frac{\\alpha}{2}+\\text{Conf Level}, n-1)\\]<\/p>\n<h1>Understanding Excel&#8217;s CONFIDENCE.T Function<\/h1>\n<p>When we use Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/confidence-t-function-e8eca395-6c3a-4ba9-9003-79ccc61d3c53\">CONFIDENCE.T()<\/a> function, we input the area outside of the confidence interval (\u03b1). This function returns the Margin of Error (E):<\/p>\n<figure id=\"attachment_2343\" aria-describedby=\"caption-attachment-2343\" style=\"width: 551px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_xbar_E_alpha.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2343 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_xbar_E_alpha.jpg\" alt=\"Bell shaped curve with area between the lower and upper confidence interval limits highlighted. Between the limits and the middle is marked E on either side.\" width=\"551\" height=\"279\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_xbar_E_alpha.jpg 551w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_xbar_E_alpha-300x152.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_xbar_E_alpha-65x33.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_xbar_E_alpha-225x114.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Confidence_w_xbar_E_alpha-350x177.jpg 350w\" sizes=\"auto, (max-width: 551px) 100vw, 551px\" \/><\/a><figcaption id=\"caption-attachment-2343\" class=\"wp-caption-text\">Figure 50.3 Confidence interval with limits, sample mean and margin of error indicated.<\/figcaption><\/figure>\n<p>Excel&#8217;s CONFIDENCE.T() function is the quickest way to calculate the margin of error (there is no need to calculate the [latex]t[\/latex]-score nor do the margin of error calculation):<\/p>\n<p>[latex]E = t \\cdot \\frac{s}{\\sqrt{n}} = \\text{CONFIDENCE.T}(\\alpha, {2}, s , n)[\/latex]<\/p>\n<p>We can easily calculate the lower and upper limits once the margin of error is known:<\/p>\n<p>\\[CL_{Lower} = \\bar{x}-E\\]<\/p>\n<p>\\[CL_{Upper} = \\bar{x}+E\\]<\/p>\n","protected":false},"author":865,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2330","chapter","type-chapter","status-publish","hentry"],"part":487,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2330","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":25,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2330\/revisions"}],"predecessor-version":[{"id":2416,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2330\/revisions\/2416"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/487"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2330\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=2330"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=2330"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=2330"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=2330"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}