{"id":649,"date":"2020-07-20T22:17:49","date_gmt":"2020-07-21T02:17:49","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&#038;p=649"},"modified":"2025-06-24T18:45:12","modified_gmt":"2025-06-24T22:45:12","slug":"5-4-using-the-standard-normal-table","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/5-4-using-the-standard-normal-table\/","title":{"raw":"5.4 Using the Standard Normal Table","rendered":"5.4 Using the Standard Normal Table"},"content":{"raw":"The standard normal table (usually found in the appendix of a Statistics textbook) can be used to solve problems related to normal distributions.\r\n<h2><strong>5.4.1 Find the Area (Probability) for a Given Z-Score<\/strong><\/h2>\r\nIn general, the standard normal table gives the area under the standard normal curve to the left of a specified <em>z<\/em>-score. Using the table, we can calculate the area under the curve to the left of a <em>z<\/em>-score, to the right of a <em>z<\/em>-score, between two <em>z<\/em>-scores, or beyond two <em>z<\/em>-scores. Figure 5.6 shows that the area to the left of 1.96 under the standard normal curve is 0.975.<a id=\"retfig5.6\"><\/a>\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_3063\" align=\"aligncenter\" width=\"5308\"]<img class=\"wp-image-3063 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/standard_normal_table_example.png\" alt=\"An image of the z-table demonstrating how to find the z-score. Image description available.\" width=\"5308\" height=\"4552\" \/> <strong>Figure 5.6<\/strong>: Area Under the Standard Normal Curve (Table II). [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig5.6\">Image Description <\/a><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig5.6\">(See Appendix D Figure 5.6)<\/a>][\/caption]If random variable [latex]Z[\/latex] follows a standard normal distribution, more detailed examples of using the standard normal table can be found in Figure 5.7:\r\n<ul>\r\n \t<li>Left panel: the area to the left of 1.96 is 0.975, i.e., [latex]P(Z&lt;1.96)=0.975.[\/latex]<\/li>\r\n \t<li>Middle panel: the area to the right of 1.96 is 0.025. There are two ways to solve this problem:\r\n<ol>\r\n \t<li>Recall that the total area under any density curve is one, the area to the right of 1.96 equals one minus the area to the left of 1.96, i.e, [latex]\\begin{eqnarray*} P(Z&gt;1.96)&amp;=&amp;\\mbox{area under the standard normal curve to the right of 1.96}\\\\&amp;=&amp;1-\\mbox{area under the curve to the left of 1.96}=1-0.975=0.025.\\end{eqnarray*}[\/latex]<\/li>\r\n \t<li>[latex]P(Z&gt;1.96)=P(Z&lt;-1.96)=0.025.[\/latex] This is because the standard normal curve is symmetric at 0. The area to the right of 1.96 equals the area to the left of -1.96.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Right panel: the area between -1.96 and 1.96 is 0.95, i.e., [latex]\\begin{eqnarray*}P(-1.96 \\lt Z \\lt 1.96)&amp;=&amp;\\mbox{area between -1.96 and 1.96}\\\\&amp;=&amp;(\\mbox{area to the left of 1.96) - (area to the left of -1.96})\\\\&amp;=&amp;0.975-0.025=0.95.\\end{eqnarray*}[\/latex]<a id=\"retfig5.7\"><\/a><\/li>\r\n<\/ul>\r\n[caption id=\"attachment_5541\" align=\"aligncenter\" width=\"901\"]<img class=\"wp-image-5541 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/Figure-5.7-1.png\" alt=\"Three standard normal density curves are presented in a row. Image description available.\" width=\"901\" height=\"269\" \/> <strong>Figure 5.7<\/strong>: Area to the Left (left panel), Right (middle panel) of a Z-score, and Between Two Z-scores (right panel). [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig5.7\">Image Description (See Appendix D Figure 5.7)<\/a>][\/caption]\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example: Finding Areas Under Standard Normal Curve<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSuppose that [latex]Z\\sim N(0, 1)[\/latex], follows a standard normal distribution.\r\n\r\n1. Draw a graph to show and find [latex]P(Z&lt;-2)[\/latex].\r\n<p style=\"padding-left: 40px;\">We can find the area to the left of -2 using the standard normal table directly. [latex]P(Z&lt;-2)=P(Z&lt;-2.00)=0.0228.[\/latex] Graph showing the area can be found in Panel (1) of Figure 5.6.<\/p>\r\n2. Draw a graph to show and find [latex]P(Z&gt;2).[\/latex]\r\n<p style=\"padding-left: 40px;\">This is the area to the right of 2. Recall that the table gives the area to the left of a [latex]z[\/latex]-score. There are two ways to answer this question:<\/p>\r\n\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li>Apply the symmetry property of the standard normal curve. The standard normal curve is symmetric at 0, and the area to the right of 2 equals the area to the left of -2. [latex]P(Z&gt;2)=P(Z&lt;-2)=0.0228.[\/latex]<\/li>\r\n \t<li>Use the property of a density curve: all density curves have an area of one under the curve. The area to the right of 2 equals one minus the area to the left of 2. [latex]P(Z&gt;2)=1-P(Z&lt;2)=1-P(Z&lt;2.00)=1-0.9772=0.0228.[\/latex] Graph showing the area can be found in Panel (2) of Figure 5.8.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n3. Draw a graph to show and find [latex]P(Z&lt;-2 \\mbox{ or } Z&gt;2).[\/latex]\r\n<p style=\"padding-left: 40px;\">Area beyond -2 and 2, i.e., to the left of -2 or to the right of 2. The two events {Z&lt;-2} and {Z&gt;2} don't overlap and, hence, are mutually exclusive; the special addition rule applies. [latex]P(Z&lt;-2 \\mbox{ or } Z&gt;2)=P(Z&lt;-2)+P(Z&gt;2)=0.0228+0.0228=0.0456.[\/latex] Graph showing the area can be found in Panel (3) of Figure 5.8.<\/p>\r\n4. Draw a graph to show and find [latex]P(-4 &lt; Z &lt; 5).[\/latex]\r\n<p style=\"padding-left: 40px;\">The area between -4 and 5 equals the area to the left of 5 minus the area to the left of -4. P(-4&lt;Z&lt;5)=P(Z&lt;5)-P(Z&lt;-4)=1-0=1. Graph showing the area can be found in Panel (4) of Figure 5.8.<\/p>\r\nNote: the standard normal table gives the area (in four decimal places) to the left of the z-score between -3.90 and 3.90. Therefore, the area to the left of any z-score below -3.90 is 0, and the area to the left of any z-score above 3.90 is 1.<a id=\"retfig5.8\"><\/a>\r\n\r\n<\/div>\r\n<div class=\"textbox__content\">\r\n<table class=\"aligncenter no-border\" style=\"border-collapse: collapse; width: 90.1068%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td width=\"25%\"><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_left_negative2.png\"><img class=\"alignnone wp-image-2776 size-medium\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_left_negative2-300x300.png\" alt=\"A standard normal curve showing the area to the right of z = -2. Image description available.\" width=\"300\" height=\"300\" \/><\/a>\r\n<p style=\"text-align: center;\"><strong>(1)<\/strong><\/p>\r\n<\/td>\r\n<td width=\"25%\"><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_right2.png\"><img class=\"alignnone wp-image-2777 size-medium\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_right2-300x300.png\" alt=\"A standard normal curve showing the area to the right of z = 2. Image description available.\" width=\"300\" height=\"300\" \/><\/a>\r\n<p style=\"text-align: center;\"><strong>(2)<\/strong><\/p>\r\n<\/td>\r\n<td width=\"25%\"><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_beyond2.png\"><img class=\"alignnone wp-image-2775 size-medium\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_beyond2-300x300.png\" alt=\"A standard normal curve showing the area below -2 and above 2. Image description available.\" width=\"300\" height=\"300\" \/><\/a>\r\n<p style=\"text-align: center;\"><strong>(3)<\/strong><\/p>\r\n<\/td>\r\n<td width=\"25%\"><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_between_n4and5.png\"><img class=\"alignnone wp-image-2778 size-medium\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_between_n4and5-300x300.png\" alt=\"A standard normal curve showing the area between -2 and 2. Image description available.\" width=\"300\" height=\"300\" \/><\/a>\r\n<p style=\"text-align: center;\"><strong>(4)<\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Figure 5.8<\/strong>: Graphs showing the areas under the standard normal curve corresponding to the probabilities in the example. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig5.8\">Image Description (See Appendix D Figure 5.8)<\/a>] Click on the image to enlarge it.\r\n\r\n<\/div>\r\n<\/div>\r\n<h2><strong>5.4.2 Find the Z-Score for a Given Area (Probability)<\/strong><\/h2>\r\nWe can use the standard normal table in another way: find the [latex]z[\/latex]-score for a specified area or probability (percentage). The steps are as follows:\r\n<ol>\r\n \t<li>Express the given area in terms of a left-tailed probability (or probabilities if there are 2 z-scores).<\/li>\r\n \t<li>Search the main body of the standard normal table for the closest value to the left-tailed probability.<\/li>\r\n \t<li>Obtain the [latex]z[\/latex]-score that corresponds to the given area. If multiple values are equally close to the given left-tailed probability, take the average of their corresponding [latex]z[\/latex]-scores.<\/li>\r\n<\/ol>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example: Given the Area, find the corresponding <em>z<\/em>-score<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"input02\">\r\n<div id=\"answer02\" class=\"hidden\">\r\n\r\nFind the [latex]z[\/latex]-score corresponding to the shaded area in each graph:<a id=\"retex5.1\"><\/a>\r\n<table class=\"no-border\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><img class=\"aligncenter wp-image-627 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image014_0000.png\" alt=\"A standard normal curve with an unknown z-score. To the right, a shaded area has a value of 0.1. Image description available.\" \/><\/td>\r\n<td style=\"width: 33.3333%;\"><img class=\"aligncenter wp-image-625 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image016_0000.png\" alt=\"A standard normal curve with an unknown z-score. To the left, a shaded area has a value of 0.1. Image description available.\" \/><\/td>\r\n<td style=\"width: 33.3333%;\"><img class=\"aligncenter wp-image-652 size-full\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image018_0000.png\" alt=\"A standard normal curve with an unknown z-score. To the left, a shaded area has a value of 0.05. Image description available.\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; vertical-align: top;\">Closest value to [latex]0.1[\/latex] is [latex]0.1003 \\Longrightarrow z= -1.28.[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; vertical-align: top;\"><strong>Method 1<\/strong>: Area on the right is [latex]0.1 \\Longrightarrow[\/latex] area on the left is [latex]0.9[\/latex], the closest value is[latex] 0.8997 \\Longrightarrow z= 1.28.[\/latex]\r\n<strong>Method 2<\/strong>: due to symmetry, [latex]z=1.28.[\/latex]<\/td>\r\n<td style=\"width: 33.3333%; vertical-align: top;\">Area on the right is [latex]0.05 \\Longrightarrow[\/latex] area on the left is [latex]0.95[\/latex], two values [latex]0.9495 (z=1.64)[\/latex] and [latex]0.9505 (z=1.65)[\/latex] are equally close to 0.95 [latex]\\Longrightarrow z = \\frac{1.64 + 1.65}{2} = 1.645.[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#ex5.1\">Image Description (See Appendix D Example 5.1)<\/a>]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"input02\">\r\n<div id=\"answer02\" class=\"hidden\">\r\n<div align=\"left\">\r\n\r\nThe notation [latex]z_{\\alpha}[\/latex]\u00a0(read as z alpha) has a special meaning: the z-score that has an area of [latex]\\alpha[\/latex]\u00a0to its <strong>right<\/strong> under the standard normal curve. The middle and right panels of the example above show that [latex]z_{0.1}=1.28[\/latex]\u00a0and [latex]z_{0.05}=1.645[\/latex].\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise: Finding Z-score [latex]\\color{white}Z_{\\alpha}[\/latex]<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse the standard normal table to find\r\n<ol type=\"a\">\r\n \t<li>[latex]Z_{0.25}[\/latex]: z-score with an area of 0.25 to its right<\/li>\r\n \t<li>[latex]Z_{0.6}[\/latex]: z-score with an area of 0.6 to its right<\/li>\r\n \t<li>[latex]Z_{0.005}[\/latex]: z-score with an area of 0.005 to its right<\/li>\r\n<\/ol>\r\n<details><summary>Show\/Hide Answer<\/summary>\r\n<ol type=\"a\">\r\n \t<li>[latex]Z_{0.25}[\/latex]: z-score with an area of 0.25 to its right, so the area to the left of [latex]Z_{0.25}[\/latex] is 1-0.25=0.75. Search the main body of the table; the closest value to 0.75 is 0.7486, which corresponds to the z-score 0.67; therefore, [latex]Z_{0.25}=0.67[\/latex].<\/li>\r\n \t<li>[latex]Z_{0.6}[\/latex]: z-score with an area of 0.6 to its right, so the area to the left of [latex]Z_{0.6}[\/latex] is 1-0.6=0.4. Search the main body of the table; the closest value to 0.4 is 0.4013, which corresponds to the z-score -0.25; therefore,\u00a0 [latex]Z_{0.6}=-0.25[\/latex].<\/li>\r\n \t<li>[latex]Z_{0.005}[\/latex]: z-score with an area of 0.005 to its right, so the area to the left of [latex]Z_{0.005}[\/latex] is 1-0.005=0.995. Search the main body of the table for 0.995. Two values that are equally close to 0.995 are 0.9949 and 0.9951; the corresponding z-scores are 2.57 and 2.58, respectively. Therefore, [latex]Z_{0.005}=\\frac{2.57+2.58}{2}=2.575[\/latex].<\/li>\r\n<\/ol>\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<p>The standard normal table (usually found in the appendix of a Statistics textbook) can be used to solve problems related to normal distributions.<\/p>\n<h2><strong>5.4.1 Find the Area (Probability) for a Given Z-Score<\/strong><\/h2>\n<p>In general, the standard normal table gives the area under the standard normal curve to the left of a specified <em>z<\/em>-score. Using the table, we can calculate the area under the curve to the left of a <em>z<\/em>-score, to the right of a <em>z<\/em>-score, between two <em>z<\/em>-scores, or beyond two <em>z<\/em>-scores. Figure 5.6 shows that the area to the left of 1.96 under the standard normal curve is 0.975.<a id=\"retfig5.6\"><\/a><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_3063\" aria-describedby=\"caption-attachment-3063\" style=\"width: 5308px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3063 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/standard_normal_table_example.png\" alt=\"An image of the z-table demonstrating how to find the z-score. Image description available.\" width=\"5308\" height=\"4552\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/standard_normal_table_example.png 5308w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/standard_normal_table_example-300x257.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/standard_normal_table_example-1024x878.png 1024w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/standard_normal_table_example-768x659.png 768w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/standard_normal_table_example-1536x1317.png 1536w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/standard_normal_table_example-2048x1756.png 2048w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/standard_normal_table_example-65x56.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/standard_normal_table_example-225x193.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/standard_normal_table_example-350x300.png 350w\" sizes=\"auto, (max-width: 5308px) 100vw, 5308px\" \/><figcaption id=\"caption-attachment-3063\" class=\"wp-caption-text\"><strong>Figure 5.6<\/strong>: Area Under the Standard Normal Curve (Table II). [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig5.6\">Image Description <\/a><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig5.6\">(See Appendix D Figure 5.6)<\/a>]<\/figcaption><\/figure>\n<p>If random variable [latex]Z[\/latex] follows a standard normal distribution, more detailed examples of using the standard normal table can be found in Figure 5.7:<\/p>\n<ul>\n<li>Left panel: the area to the left of 1.96 is 0.975, i.e., [latex]P(Z<1.96)=0.975.[\/latex]<\/li>\n<li>Middle panel: the area to the right of 1.96 is 0.025. There are two ways to solve this problem:\n<ol>\n<li>Recall that the total area under any density curve is one, the area to the right of 1.96 equals one minus the area to the left of 1.96, i.e, [latex]\\begin{eqnarray*} P(Z>1.96)&=&\\mbox{area under the standard normal curve to the right of 1.96}\\\\&=&1-\\mbox{area under the curve to the left of 1.96}=1-0.975=0.025.\\end{eqnarray*}[\/latex]<\/li>\n<li>[latex]P(Z>1.96)=P(Z<-1.96)=0.025.[\/latex] This is because the standard normal curve is symmetric at 0. The area to the right of 1.96 equals the area to the left of -1.96.<\/li>\n<\/ol>\n<\/li>\n<li>Right panel: the area between -1.96 and 1.96 is 0.95, i.e., [latex]\\begin{eqnarray*}P(-1.96 \\lt Z \\lt 1.96)&=&\\mbox{area between -1.96 and 1.96}\\\\&=&(\\mbox{area to the left of 1.96) - (area to the left of -1.96})\\\\&=&0.975-0.025=0.95.\\end{eqnarray*}[\/latex]<a id=\"retfig5.7\"><\/a><\/li>\n<\/ul>\n<figure id=\"attachment_5541\" aria-describedby=\"caption-attachment-5541\" style=\"width: 901px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5541 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/Figure-5.7-1.png\" alt=\"Three standard normal density curves are presented in a row. Image description available.\" width=\"901\" height=\"269\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/Figure-5.7-1.png 901w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/Figure-5.7-1-300x90.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/Figure-5.7-1-768x229.png 768w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/Figure-5.7-1-65x19.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/Figure-5.7-1-225x67.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/Figure-5.7-1-350x104.png 350w\" sizes=\"auto, (max-width: 901px) 100vw, 901px\" \/><figcaption id=\"caption-attachment-5541\" class=\"wp-caption-text\"><strong>Figure 5.7<\/strong>: Area to the Left (left panel), Right (middle panel) of a Z-score, and Between Two Z-scores (right panel). [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig5.7\">Image Description (See Appendix D Figure 5.7)<\/a>]<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example: Finding Areas Under Standard Normal Curve<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Suppose that [latex]Z\\sim N(0, 1)[\/latex], follows a standard normal distribution.<\/p>\n<p>1. Draw a graph to show and find [latex]P(Z<-2)[\/latex].\n\n\n<p style=\"padding-left: 40px;\">We can find the area to the left of -2 using the standard normal table directly. [latex]P(Z<-2)=P(Z<-2.00)=0.0228.[\/latex] Graph showing the area can be found in Panel (1) of Figure 5.6.<\/p>\n<p>2. Draw a graph to show and find [latex]P(Z>2).[\/latex]<\/p>\n<p style=\"padding-left: 40px;\">This is the area to the right of 2. Recall that the table gives the area to the left of a [latex]z[\/latex]-score. There are two ways to answer this question:<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>Apply the symmetry property of the standard normal curve. The standard normal curve is symmetric at 0, and the area to the right of 2 equals the area to the left of -2. [latex]P(Z>2)=P(Z<-2)=0.0228.[\/latex]<\/li>\n<li>Use the property of a density curve: all density curves have an area of one under the curve. The area to the right of 2 equals one minus the area to the left of 2. [latex]P(Z>2)=1-P(Z<2)=1-P(Z<2.00)=1-0.9772=0.0228.[\/latex] Graph showing the area can be found in Panel (2) of Figure 5.8.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>3. Draw a graph to show and find [latex]P(Z<-2 \\mbox{ or } Z>2).[\/latex]<\/p>\n<p style=\"padding-left: 40px;\">Area beyond -2 and 2, i.e., to the left of -2 or to the right of 2. The two events {Z&lt;-2} and {Z&gt;2} don&#8217;t overlap and, hence, are mutually exclusive; the special addition rule applies. [latex]P(Z<-2 \\mbox{ or } Z>2)=P(Z<-2)+P(Z>2)=0.0228+0.0228=0.0456.[\/latex] Graph showing the area can be found in Panel (3) of Figure 5.8.<\/p>\n<p>4. Draw a graph to show and find [latex]P(-4 < Z < 5).[\/latex]\n\n\n<p style=\"padding-left: 40px;\">The area between -4 and 5 equals the area to the left of 5 minus the area to the left of -4. P(-4&lt;Z&lt;5)=P(Z&lt;5)-P(Z&lt;-4)=1-0=1. Graph showing the area can be found in Panel (4) of Figure 5.8.<\/p>\n<p>Note: the standard normal table gives the area (in four decimal places) to the left of the z-score between -3.90 and 3.90. Therefore, the area to the left of any z-score below -3.90 is 0, and the area to the left of any z-score above 3.90 is 1.<a id=\"retfig5.8\"><\/a><\/p>\n<\/div>\n<div class=\"textbox__content\">\n<table class=\"aligncenter no-border\" style=\"border-collapse: collapse; width: 90.1068%;\">\n<tbody>\n<tr>\n<td style=\"width: 25%;\"><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_left_negative2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2776 size-medium\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_left_negative2-300x300.png\" alt=\"A standard normal curve showing the area to the right of z = -2. Image description available.\" width=\"300\" height=\"300\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_left_negative2-300x300.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_left_negative2-150x150.png 150w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_left_negative2-65x65.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_left_negative2-225x225.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_left_negative2-350x350.png 350w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_left_negative2.png 480w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p style=\"text-align: center;\"><strong>(1)<\/strong><\/p>\n<\/td>\n<td style=\"width: 25%;\"><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_right2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2777 size-medium\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_right2-300x300.png\" alt=\"A standard normal curve showing the area to the right of z = 2. Image description available.\" width=\"300\" height=\"300\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_right2-300x300.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_right2-150x150.png 150w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_right2-65x65.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_right2-225x225.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_right2-350x350.png 350w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_right2.png 480w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p style=\"text-align: center;\"><strong>(2)<\/strong><\/p>\n<\/td>\n<td style=\"width: 25%;\"><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_beyond2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2775 size-medium\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_beyond2-300x300.png\" alt=\"A standard normal curve showing the area below -2 and above 2. Image description available.\" width=\"300\" height=\"300\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_beyond2-300x300.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_beyond2-150x150.png 150w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_beyond2-65x65.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_beyond2-225x225.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_beyond2-350x350.png 350w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_beyond2.png 480w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p style=\"text-align: center;\"><strong>(3)<\/strong><\/p>\n<\/td>\n<td style=\"width: 25%;\"><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_between_n4and5.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2778 size-medium\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_between_n4and5-300x300.png\" alt=\"A standard normal curve showing the area between -2 and 2. Image description available.\" width=\"300\" height=\"300\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_between_n4and5-300x300.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_between_n4and5-150x150.png 150w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_between_n4and5-65x65.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_between_n4and5-225x225.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_between_n4and5-350x350.png 350w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/standard_normal_between_n4and5.png 480w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p style=\"text-align: center;\"><strong>(4)<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Figure 5.8<\/strong>: Graphs showing the areas under the standard normal curve corresponding to the probabilities in the example. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig5.8\">Image Description (See Appendix D Figure 5.8)<\/a>] Click on the image to enlarge it.<\/p>\n<\/div>\n<\/div>\n<h2><strong>5.4.2 Find the Z-Score for a Given Area (Probability)<\/strong><\/h2>\n<p>We can use the standard normal table in another way: find the [latex]z[\/latex]-score for a specified area or probability (percentage). The steps are as follows:<\/p>\n<ol>\n<li>Express the given area in terms of a left-tailed probability (or probabilities if there are 2 z-scores).<\/li>\n<li>Search the main body of the standard normal table for the closest value to the left-tailed probability.<\/li>\n<li>Obtain the [latex]z[\/latex]-score that corresponds to the given area. If multiple values are equally close to the given left-tailed probability, take the average of their corresponding [latex]z[\/latex]-scores.<\/li>\n<\/ol>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example: Given the Area, find the corresponding <em>z<\/em>-score<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"input02\">\n<div id=\"answer02\" class=\"hidden\">\n<p>Find the [latex]z[\/latex]-score corresponding to the shaded area in each graph:<a id=\"retex5.1\"><\/a><\/p>\n<table class=\"no-border\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" width=\"270\" height=\"209\" class=\"aligncenter wp-image-627 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image014_0000.png\" alt=\"A standard normal curve with an unknown z-score. To the right, a shaded area has a value of 0.1. Image description available.\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image014_0000.png 270w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image014_0000-65x50.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image014_0000-225x174.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/td>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" width=\"268\" height=\"206\" class=\"aligncenter wp-image-625 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image016_0000.png\" alt=\"A standard normal curve with an unknown z-score. To the left, a shaded area has a value of 0.1. Image description available.\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image016_0000.png 268w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image016_0000-65x50.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image016_0000-225x173.png 225w\" sizes=\"auto, (max-width: 268px) 100vw, 268px\" \/><\/td>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" width=\"270\" height=\"211\" class=\"aligncenter wp-image-652 size-full\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image018_0000.png\" alt=\"A standard normal curve with an unknown z-score. To the left, a shaded area has a value of 0.05. Image description available.\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image018_0000.png 270w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image018_0000-65x51.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/M05-04-StandardNormalDensityCurve_clip_image018_0000-225x176.png 225w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; vertical-align: top;\">Closest value to [latex]0.1[\/latex] is [latex]0.1003 \\Longrightarrow z= -1.28.[\/latex]<\/td>\n<td style=\"width: 33.3333%; vertical-align: top;\"><strong>Method 1<\/strong>: Area on the right is [latex]0.1 \\Longrightarrow[\/latex] area on the left is [latex]0.9[\/latex], the closest value is[latex]0.8997 \\Longrightarrow z= 1.28.[\/latex]<br \/>\n<strong>Method 2<\/strong>: due to symmetry, [latex]z=1.28.[\/latex]<\/td>\n<td style=\"width: 33.3333%; vertical-align: top;\">Area on the right is [latex]0.05 \\Longrightarrow[\/latex] area on the left is [latex]0.95[\/latex], two values [latex]0.9495 (z=1.64)[\/latex] and [latex]0.9505 (z=1.65)[\/latex] are equally close to 0.95 [latex]\\Longrightarrow z = \\frac{1.64 + 1.65}{2} = 1.645.[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>[<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#ex5.1\">Image Description (See Appendix D Example 5.1)<\/a>]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div class=\"hidden\">\n<div style=\"text-align: left;\">\n<p>The notation [latex]z_{\\alpha}[\/latex]\u00a0(read as z alpha) has a special meaning: the z-score that has an area of [latex]\\alpha[\/latex]\u00a0to its <strong>right<\/strong> under the standard normal curve. The middle and right panels of the example above show that [latex]z_{0.1}=1.28[\/latex]\u00a0and [latex]z_{0.05}=1.645[\/latex].<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise: Finding Z-score [latex]\\color{white}Z_{\\alpha}[\/latex]<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use the standard normal table to find<\/p>\n<ol type=\"a\">\n<li>[latex]Z_{0.25}[\/latex]: z-score with an area of 0.25 to its right<\/li>\n<li>[latex]Z_{0.6}[\/latex]: z-score with an area of 0.6 to its right<\/li>\n<li>[latex]Z_{0.005}[\/latex]: z-score with an area of 0.005 to its right<\/li>\n<\/ol>\n<details>\n<summary>Show\/Hide Answer<\/summary>\n<ol type=\"a\">\n<li>[latex]Z_{0.25}[\/latex]: z-score with an area of 0.25 to its right, so the area to the left of [latex]Z_{0.25}[\/latex] is 1-0.25=0.75. Search the main body of the table; the closest value to 0.75 is 0.7486, which corresponds to the z-score 0.67; therefore, [latex]Z_{0.25}=0.67[\/latex].<\/li>\n<li>[latex]Z_{0.6}[\/latex]: z-score with an area of 0.6 to its right, so the area to the left of [latex]Z_{0.6}[\/latex] is 1-0.6=0.4. Search the main body of the table; the closest value to 0.4 is 0.4013, which corresponds to the z-score -0.25; therefore,\u00a0 [latex]Z_{0.6}=-0.25[\/latex].<\/li>\n<li>[latex]Z_{0.005}[\/latex]: z-score with an area of 0.005 to its right, so the area to the left of [latex]Z_{0.005}[\/latex] is 1-0.005=0.995. Search the main body of the table for 0.995. Two values that are equally close to 0.995 are 0.9949 and 0.9951; the corresponding z-scores are 2.57 and 2.58, respectively. Therefore, [latex]Z_{0.005}=\\frac{2.57+2.58}{2}=2.575[\/latex].<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":19,"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-649","chapter","type-chapter","status-publish","hentry"],"part":588,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/649","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":77,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/649\/revisions"}],"predecessor-version":[{"id":5611,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/649\/revisions\/5611"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/588"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/649\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=649"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=649"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=649"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=649"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}