{"id":2079,"date":"2024-06-14T17:56:01","date_gmt":"2024-06-14T21:56:01","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=2079"},"modified":"2024-06-16T13:41:47","modified_gmt":"2024-06-16T17:41:47","slug":"excels-norm-inv-function","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/excels-norm-inv-function\/","title":{"raw":"Excel's NORM.INV Function","rendered":"Excel&#8217;s NORM.INV Function"},"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\nUse Excel's NORM.INV() to calculate x-values related to given areas.\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>Left Area Given<\/h2>\r\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100.426%;height: 203px\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 201px\">\r\n<td style=\"width: 46.869%;height: 203px;vertical-align: middle\">\r\n\r\n[caption id=\"attachment_2085\" align=\"alignnone\" width=\"478\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea.jpg\"><img class=\"wp-image-2085 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea.jpg\" alt=\"Bell shaped curve with area to the left of x-value shaded.\" width=\"478\" height=\"301\" \/><\/a> Figure 40.1 Area to the left of x-value[\/caption]<\/td>\r\n<td style=\"width: 67.6962%;height: 203px;vertical-align: top\">\r\n<ul>\r\n \t<li style=\"text-align: left\">Use NORM.INV(area, \u03bc, \u03c3) = x<\/li>\r\n \t<li style=\"text-align: left\">To calculate the x-value (or percentile)<\/li>\r\n \t<li style=\"text-align: left\">Corresponding to the area to the left of x<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Right Area Given<\/h2>\r\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100.426%;height: 231px\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 231px\">\r\n<td style=\"width: 46.869%;height: 231px;text-align: center;vertical-align: middle\">\r\n\r\n[caption id=\"attachment_2088\" align=\"alignnone\" width=\"328\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea.jpg\"><img class=\"wp-image-2088 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea.jpg\" alt=\"Bell shaped curve with area above (to the right of) x-value shaded.\" width=\"328\" height=\"176\" \/><\/a> Figure 40.2 Area to the right of x-value.[\/caption]<\/td>\r\n<td style=\"width: 67.6962%;height: 231px;vertical-align: top\">\r\n<ul>\r\n \t<li style=\"text-align: left\">Use NORM.INV(1\u2212area, \u03bc, \u03c3) = x<\/li>\r\n \t<li style=\"text-align: left\">To calculate the x-value (or percentile)<\/li>\r\n \t<li style=\"text-align: left\">Corresponding to the area to the right of x<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Middle Area Given<\/h2>\r\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100.426%;height: 203px\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 201px\">\r\n<td style=\"width: 46.869%;height: 203px;vertical-align: middle\">\r\n\r\n[caption id=\"attachment_2089\" align=\"alignnone\" width=\"594\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/MiddleArea.jpg\"><img class=\"wp-image-2089 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/MiddleArea.jpg\" alt=\"Bell shaped curve with middle area shaded between the x-values of x1 and x2. In the middle area is written &quot;Confidence Level&quot;. Below the shaded area is written (1\u2212Confidence Level)\/2. Above the shaded area is written (1\u2212Confidence Level)\/2. \" width=\"594\" height=\"308\" \/><\/a> Figure 40.3 Middle area (or confidence level)[\/caption]<\/td>\r\n<td style=\"width: 67.6962%;height: 203px;vertical-align: top\">\r\n<ul>\r\n \t<li style=\"text-align: left\">'Middle' areas can also be called '<a href=\"https:\/\/en.wikipedia.org\/wiki\/Confidence_interval\">Confidence Levels<\/a>'<\/li>\r\n \t<li style=\"text-align: left\">We will use them in later sections also<\/li>\r\n \t<li style=\"text-align: left\">To calculate the lower and upper limits (x<sub>1<\/sub> and x<sub>2<\/sub>):<\/li>\r\n \t<li style=\"text-align: left\">We need to calculate the area to left of each x-values<\/li>\r\n \t<li style=\"text-align: left\">The left areas are marked on the graph<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h1>Calculating the x-value for a left area (Exercise)<\/h1>\r\nLet us first look at an example where we calculate an [latex]x[\/latex]-value when the left area is given.\r\n<h2>Example 40.1.1<\/h2>\r\n<strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Let us revisit the SAT score problem from the previous section. The average SAT score was 1,010 with a standard deviation of 20.\r\n\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: What is the highest score for the bottom 85% of the students?\r\n\r\n<strong><span style=\"color: #003366\">You try<\/span><\/strong>:\r\n\r\n[h5p id=\"130\"]\r\n\r\n<strong><span style=\"color: #003366\">You try<\/span><\/strong>:\r\n\r\n[h5p id=\"131\"]\r\n\r\n<strong><span style=\"color: #003366\">Conclusion<\/span><\/strong>: 85% of people score at most 1030.729 on their SATs.\r\n\r\n<strong><span style=\"color: #003366\">Need Help<\/span><\/strong>? Go to the last section for a video that reviews all of the content in this section. You can also download a PowerPoint presentation on Normal Distributions.\r\n<h1>Calculating the x-value for a RIGHT area (Exercise)<\/h1>\r\nLet us now look at an example where we calculate an [latex]x[\/latex]-value when the right area is given.\r\n<h2>Example 40.1.2<\/h2>\r\n<strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Let us revisit the SAT score problem from the previous section. The average SAT score was 1,010 with a standard deviation of 20.\r\n\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: Above what score do the top 15% of students score?\r\n\r\n<strong><span style=\"color: #003366\">You try<\/span><\/strong>:\r\n\r\n[h5p id=\"132\"]\r\n\r\n<strong><span style=\"color: #003366\">You try<\/span><\/strong>:\r\n\r\n[h5p id=\"133\"]\r\n\r\n<strong><span style=\"color: #003366\">Conclusion<\/span><\/strong>: 15% of people score at least 1030.729 on their SATs.\r\n\r\n<strong><span style=\"color: #003366\">Need Help<\/span><\/strong>? Click to reveal the solutions below OR go to the last section for a video explaining all content in this section.\r\n<div>\r\n<h1>Solutions To This Problem<\/h1>\r\nWhen we are given the area to the right:\r\n<ul>\r\n \t<li>We need to take a complement to get the area to the left<\/li>\r\n \t<li>This is because Excel's NORM.INV() function works with areas to the left<\/li>\r\n \t<li>So, for the top 15%, this is the same as the bottom 85%:<\/li>\r\n<\/ul>\r\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100%\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%\">\r\n\r\n[caption id=\"attachment_2097\" align=\"alignnone\" width=\"328\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-2_Answer.jpg\"><img class=\"wp-image-2097 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-2_Answer.jpg\" alt=\"Bell curve with right-most 15% of graph highlighted and &quot;x=?&quot; at the start of this area. One thousand and ten is written at the bottom in the middle of the graph.\" width=\"328\" height=\"176\" \/><\/a> Figure 40.4 Top 15% of bell curve.[\/caption]<\/td>\r\n<td style=\"width: 50%\">\r\n\r\n[caption id=\"attachment_2098\" align=\"alignnone\" width=\"326\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-1_Answer.jpg\"><img class=\"size-full wp-image-2098\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-1_Answer.jpg\" alt=\"Bell curve with area to left highlighted and 85% written within this area. In the middle below the bell curve 1010 is written. Below, at the end of the highlighted area is written 'x=?'.\" width=\"326\" height=\"179\" \/><\/a> Figure 40.5 Lower 85% area in bell curve[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo calculate the [latex]x[\/latex]-value associated with the above graphs, we use NORM.INV():\r\n\r\n\\[ x = \\text{NORM.INV}(1-0.15,1010,20) = \\text{NORM.INV}(0.85, 1010, 20) = 1030.729\\]\r\n\r\n<\/div>\r\n<h1>Calculating the x-values for a MIddle area (Exercise)<\/h1>\r\nLet us finally look at an example where we calculate an [latex]x[\/latex]-values when a middle area is given.\r\n<h2>Example 40.1.3<\/h2>\r\n<strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Let us revisit the SAT score problem from the previous section. The average SAT score was 1,010 with a standard deviation of 20.\r\n\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: What is the range of SAT scores for the middle 85% of students?\r\n\r\n<strong><span style=\"color: #003366\">You try<\/span><\/strong>:\r\n\r\n[h5p id=\"134\"]\r\n\r\n<strong><span style=\"color: #003366\">You try<\/span><\/strong>:\r\n\r\n[h5p id=\"135\"]\r\n\r\n<strong><span style=\"color: #003366\">You try<\/span><\/strong>:\r\n\r\n[h5p id=\"136\"]\r\n\r\n<strong><span style=\"color: #003366\">You try<\/span><\/strong>:\r\n\r\n[h5p id=\"137\"]\r\n\r\n<strong><span style=\"color: #003366\">You try<\/span><\/strong>:\r\n\r\n[h5p id=\"138\"]\r\n\r\n<strong><span style=\"color: #003366\">Conclusion<\/span><\/strong>: 85% of students score between 981 and 1,038\u00a0 on their SATs.\r\n\r\n<strong><span style=\"color: #003366\">Need Help<\/span><\/strong>? Click to reveal the solutions below OR go to the last section for a video explaining all content in this section.\r\n<div>\r\n<h1>Solutions To This Problem<\/h1>\r\nWhen we are given the middle area:\r\n<ul>\r\n \t<li>We need to calculate the two [latex]x[\/latex]-values separately<\/li>\r\n \t<li>Input the area to the left of [latex]x_1[\/latex]\u00a0and [latex]x_2[\/latex]\u00a0into NORM.INV<\/li>\r\n \t<li>The area to the left of [latex]x_1[\/latex]: [latex]\\frac{100\\%-85\\%}{2}=7.5\\%[\/latex]<\/li>\r\n \t<li>The area to the left of [latex]x_2[\/latex]: [latex]\\frac{100\\%-85\\%}{2}+85\\%=92.5\\%[\/latex]<\/li>\r\n<\/ul>\r\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100%\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%\">\r\n\r\n[caption id=\"attachment_2103\" align=\"alignnone\" width=\"592\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x1_Area.jpg\"><img class=\"wp-image-2103 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x1_Area.jpg\" alt=\"Bell curve with left-most 7.5% of graph highlighted and &quot;x1=?&quot; at the bottom far right of this area. One thousand and ten is written at the bottom in the middle of the graph.\" width=\"592\" height=\"278\" \/><\/a> Figure 40.6 Left 7.5% of bell curve selected.[\/caption]<\/td>\r\n<td style=\"width: 50%\">\r\n\r\n[caption id=\"attachment_2104\" align=\"alignnone\" width=\"577\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x2_Area.jpg\"><img class=\"wp-image-2104 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x2_Area.jpg\" alt=\"Bell curve with left-most 92.5% of graph highlighted and &quot;x2=?&quot; at the bottom far right of this area. One thousand and ten is written at the bottom in the middle of the graph.\" width=\"577\" height=\"313\" \/><\/a> Figure 40.7 Left 92.5% of bell curve selected.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo calculate the [latex]x[\/latex]-value associated with the above graphs, we use NORM.INV:\r\n\r\n\\[ x_1 = \\text{NORM.INV}((1-0.85)\/2,1010,20) = \\text{NORM.INV}(0.075, 1010, 20) = 989.2713\\]\r\n\r\n\\[ x_2 = \\text{NORM.INV}((1-0.85)\/2+0.85,1010,20) = \\text{NORM.INV}(0.925, 1010, 20) = 1038.791\\]\r\n\r\n<\/div>\r\n<h1>Video Explaining ALl topics in this section<\/h1>\r\nhttps:\/\/youtu.be\/h0wWLcmNtZM\r\n<span style=\"color: #003366\"><strong>Additional Resources<\/strong><\/span>:\r\n<ul>\r\n \t<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions.pptx\">Click here to download the Powerpoint slides<\/a> that accompany the video.<\/li>\r\n \t<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributionSolutions.xlsx\">Click here to download the Excel solutions<\/a> for the Normal Distribution section.<\/li>\r\n<\/ul>\r\n<h1>Key Takeaways (EXERCISE)<\/h1>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways: Excel's NORM.INV Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDrag the words into the correct boxes for each section below:\r\n\r\n[h5p id=\"139\"]\r\n\r\nClick the sections below to reveal the solutions to the above exercises\r\n\r\n[h5p id=\"140\"]\r\n\r\n<\/div>\r\n<\/div>\r\n<h1>Your Own Notes (EXERCISE)<\/h1>\r\n<ul>\r\n \t<li>Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t<li>These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t<li>Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"16\"]","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>Use Excel&#8217;s NORM.INV() to calculate x-values related to given areas.<\/p>\n<\/div>\n<\/div>\n<h2>Left Area Given<\/h2>\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100.426%;height: 203px\">\n<tbody>\n<tr style=\"height: 201px\">\n<td style=\"width: 46.869%;height: 203px;vertical-align: middle\">\n<figure id=\"attachment_2085\" aria-describedby=\"caption-attachment-2085\" style=\"width: 478px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2085 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea.jpg\" alt=\"Bell shaped curve with area to the left of x-value shaded.\" width=\"478\" height=\"301\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea.jpg 478w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea-300x189.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea-65x41.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea-225x142.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea-350x220.jpg 350w\" sizes=\"auto, (max-width: 478px) 100vw, 478px\" \/><\/a><figcaption id=\"caption-attachment-2085\" class=\"wp-caption-text\">Figure 40.1 Area to the left of x-value<\/figcaption><\/figure>\n<\/td>\n<td style=\"width: 67.6962%;height: 203px;vertical-align: top\">\n<ul>\n<li style=\"text-align: left\">Use NORM.INV(area, \u03bc, \u03c3) = x<\/li>\n<li style=\"text-align: left\">To calculate the x-value (or percentile)<\/li>\n<li style=\"text-align: left\">Corresponding to the area to the left of x<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Right Area Given<\/h2>\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100.426%;height: 231px\">\n<tbody>\n<tr style=\"height: 231px\">\n<td style=\"width: 46.869%;height: 231px;text-align: center;vertical-align: middle\">\n<figure id=\"attachment_2088\" aria-describedby=\"caption-attachment-2088\" style=\"width: 328px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2088 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea.jpg\" alt=\"Bell shaped curve with area above (to the right of) x-value shaded.\" width=\"328\" height=\"176\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea.jpg 328w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea-300x161.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea-65x35.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea-225x121.jpg 225w\" sizes=\"auto, (max-width: 328px) 100vw, 328px\" \/><\/a><figcaption id=\"caption-attachment-2088\" class=\"wp-caption-text\">Figure 40.2 Area to the right of x-value.<\/figcaption><\/figure>\n<\/td>\n<td style=\"width: 67.6962%;height: 231px;vertical-align: top\">\n<ul>\n<li style=\"text-align: left\">Use NORM.INV(1\u2212area, \u03bc, \u03c3) = x<\/li>\n<li style=\"text-align: left\">To calculate the x-value (or percentile)<\/li>\n<li style=\"text-align: left\">Corresponding to the area to the right of x<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Middle Area Given<\/h2>\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100.426%;height: 203px\">\n<tbody>\n<tr style=\"height: 201px\">\n<td style=\"width: 46.869%;height: 203px;vertical-align: middle\">\n<figure id=\"attachment_2089\" aria-describedby=\"caption-attachment-2089\" style=\"width: 594px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/MiddleArea.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2089 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/MiddleArea.jpg\" alt=\"Bell shaped curve with middle area shaded between the x-values of x1 and x2. In the middle area is written &quot;Confidence Level&quot;. Below the shaded area is written (1\u2212Confidence Level)\/2. Above the shaded area is written (1\u2212Confidence Level)\/2.\" width=\"594\" height=\"308\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/MiddleArea.jpg 594w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/MiddleArea-300x156.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/MiddleArea-65x34.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/MiddleArea-225x117.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/MiddleArea-350x181.jpg 350w\" sizes=\"auto, (max-width: 594px) 100vw, 594px\" \/><\/a><figcaption id=\"caption-attachment-2089\" class=\"wp-caption-text\">Figure 40.3 Middle area (or confidence level)<\/figcaption><\/figure>\n<\/td>\n<td style=\"width: 67.6962%;height: 203px;vertical-align: top\">\n<ul>\n<li style=\"text-align: left\">&#8216;Middle&#8217; areas can also be called &#8216;<a href=\"https:\/\/en.wikipedia.org\/wiki\/Confidence_interval\">Confidence Levels<\/a>&#8216;<\/li>\n<li style=\"text-align: left\">We will use them in later sections also<\/li>\n<li style=\"text-align: left\">To calculate the lower and upper limits (x<sub>1<\/sub> and x<sub>2<\/sub>):<\/li>\n<li style=\"text-align: left\">We need to calculate the area to left of each x-values<\/li>\n<li style=\"text-align: left\">The left areas are marked on the graph<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Calculating the x-value for a left area (Exercise)<\/h1>\n<p>Let us first look at an example where we calculate an [latex]x[\/latex]-value when the left area is given.<\/p>\n<h2>Example 40.1.1<\/h2>\n<p><strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Let us revisit the SAT score problem from the previous section. The average SAT score was 1,010 with a standard deviation of 20.<\/p>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: What is the highest score for the bottom 85% of the students?<\/p>\n<p><strong><span style=\"color: #003366\">You try<\/span><\/strong>:<\/p>\n<div id=\"h5p-130\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-130\" class=\"h5p-iframe\" data-content-id=\"130\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 40.1.1 - Using Norm.inv() - Left area given\"><\/iframe><\/div>\n<\/div>\n<p><strong><span style=\"color: #003366\">You try<\/span><\/strong>:<\/p>\n<div id=\"h5p-131\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-131\" class=\"h5p-iframe\" data-content-id=\"131\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 40.1.1 - Using Norm.inv() - Left area given - Formula\"><\/iframe><\/div>\n<\/div>\n<p><strong><span style=\"color: #003366\">Conclusion<\/span><\/strong>: 85% of people score at most 1030.729 on their SATs.<\/p>\n<p><strong><span style=\"color: #003366\">Need Help<\/span><\/strong>? Go to the last section for a video that reviews all of the content in this section. You can also download a PowerPoint presentation on Normal Distributions.<\/p>\n<h1>Calculating the x-value for a RIGHT area (Exercise)<\/h1>\n<p>Let us now look at an example where we calculate an [latex]x[\/latex]-value when the right area is given.<\/p>\n<h2>Example 40.1.2<\/h2>\n<p><strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Let us revisit the SAT score problem from the previous section. The average SAT score was 1,010 with a standard deviation of 20.<\/p>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: Above what score do the top 15% of students score?<\/p>\n<p><strong><span style=\"color: #003366\">You try<\/span><\/strong>:<\/p>\n<div id=\"h5p-132\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-132\" class=\"h5p-iframe\" data-content-id=\"132\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 40.1.2 - Using Norm.inv() - Right area given\"><\/iframe><\/div>\n<\/div>\n<p><strong><span style=\"color: #003366\">You try<\/span><\/strong>:<\/p>\n<div id=\"h5p-133\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-133\" class=\"h5p-iframe\" data-content-id=\"133\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 40.1.2 - Using Norm.inv() - Right area given - Formula\"><\/iframe><\/div>\n<\/div>\n<p><strong><span style=\"color: #003366\">Conclusion<\/span><\/strong>: 15% of people score at least 1030.729 on their SATs.<\/p>\n<p><strong><span style=\"color: #003366\">Need Help<\/span><\/strong>? Click to reveal the solutions below OR go to the last section for a video explaining all content in this section.<\/p>\n<div>\n<h1>Solutions To This Problem<\/h1>\n<p>When we are given the area to the right:<\/p>\n<ul>\n<li>We need to take a complement to get the area to the left<\/li>\n<li>This is because Excel&#8217;s NORM.INV() function works with areas to the left<\/li>\n<li>So, for the top 15%, this is the same as the bottom 85%:<\/li>\n<\/ul>\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100%\">\n<tbody>\n<tr>\n<td style=\"width: 50%\">\n<figure id=\"attachment_2097\" aria-describedby=\"caption-attachment-2097\" style=\"width: 328px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-2_Answer.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2097 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-2_Answer.jpg\" alt=\"Bell curve with right-most 15% of graph highlighted and &quot;x=?&quot; at the start of this area. One thousand and ten is written at the bottom in the middle of the graph.\" width=\"328\" height=\"176\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-2_Answer.jpg 328w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-2_Answer-300x161.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-2_Answer-65x35.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-2_Answer-225x121.jpg 225w\" sizes=\"auto, (max-width: 328px) 100vw, 328px\" \/><\/a><figcaption id=\"caption-attachment-2097\" class=\"wp-caption-text\">Figure 40.4 Top 15% of bell curve.<\/figcaption><\/figure>\n<\/td>\n<td style=\"width: 50%\">\n<figure id=\"attachment_2098\" aria-describedby=\"caption-attachment-2098\" style=\"width: 326px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-1_Answer.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2098\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-1_Answer.jpg\" alt=\"Bell curve with area to left highlighted and 85% written within this area. In the middle below the bell curve 1010 is written. Below, at the end of the highlighted area is written 'x=?'.\" width=\"326\" height=\"179\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-1_Answer.jpg 326w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-1_Answer-300x165.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-1_Answer-65x36.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-1_Answer-225x124.jpg 225w\" sizes=\"auto, (max-width: 326px) 100vw, 326px\" \/><\/a><figcaption id=\"caption-attachment-2098\" class=\"wp-caption-text\">Figure 40.5 Lower 85% area in bell curve<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To calculate the [latex]x[\/latex]-value associated with the above graphs, we use NORM.INV():<\/p>\n<p>\\[ x = \\text{NORM.INV}(1-0.15,1010,20) = \\text{NORM.INV}(0.85, 1010, 20) = 1030.729\\]<\/p>\n<\/div>\n<h1>Calculating the x-values for a MIddle area (Exercise)<\/h1>\n<p>Let us finally look at an example where we calculate an [latex]x[\/latex]-values when a middle area is given.<\/p>\n<h2>Example 40.1.3<\/h2>\n<p><strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Let us revisit the SAT score problem from the previous section. The average SAT score was 1,010 with a standard deviation of 20.<\/p>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: What is the range of SAT scores for the middle 85% of students?<\/p>\n<p><strong><span style=\"color: #003366\">You try<\/span><\/strong>:<\/p>\n<div id=\"h5p-134\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-134\" class=\"h5p-iframe\" data-content-id=\"134\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 40.1.3 - Using Norm.inv() - Middle area given\"><\/iframe><\/div>\n<\/div>\n<p><strong><span style=\"color: #003366\">You try<\/span><\/strong>:<\/p>\n<div id=\"h5p-135\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-135\" class=\"h5p-iframe\" data-content-id=\"135\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 40.1.3 - Using Norm.inv() - Middle area given (x1)\"><\/iframe><\/div>\n<\/div>\n<p><strong><span style=\"color: #003366\">You try<\/span><\/strong>:<\/p>\n<div id=\"h5p-136\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-136\" class=\"h5p-iframe\" data-content-id=\"136\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 40.1.3 - Using Norm.inv() - Middle area given (x2)\"><\/iframe><\/div>\n<\/div>\n<p><strong><span style=\"color: #003366\">You try<\/span><\/strong>:<\/p>\n<div id=\"h5p-137\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-137\" class=\"h5p-iframe\" data-content-id=\"137\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 40.1.3 - Using Norm.inv() - Right area given - x1 Formula\"><\/iframe><\/div>\n<\/div>\n<p><strong><span style=\"color: #003366\">You try<\/span><\/strong>:<\/p>\n<div id=\"h5p-138\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-138\" class=\"h5p-iframe\" data-content-id=\"138\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 40.1.3 - Using Norm.inv() - Right area given - x2 Formula\"><\/iframe><\/div>\n<\/div>\n<p><strong><span style=\"color: #003366\">Conclusion<\/span><\/strong>: 85% of students score between 981 and 1,038\u00a0 on their SATs.<\/p>\n<p><strong><span style=\"color: #003366\">Need Help<\/span><\/strong>? Click to reveal the solutions below OR go to the last section for a video explaining all content in this section.<\/p>\n<div>\n<h1>Solutions To This Problem<\/h1>\n<p>When we are given the middle area:<\/p>\n<ul>\n<li>We need to calculate the two [latex]x[\/latex]-values separately<\/li>\n<li>Input the area to the left of [latex]x_1[\/latex]\u00a0and [latex]x_2[\/latex]\u00a0into NORM.INV<\/li>\n<li>The area to the left of [latex]x_1[\/latex]: [latex]\\frac{100\\%-85\\%}{2}=7.5\\%[\/latex]<\/li>\n<li>The area to the left of [latex]x_2[\/latex]: [latex]\\frac{100\\%-85\\%}{2}+85\\%=92.5\\%[\/latex]<\/li>\n<\/ul>\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100%\">\n<tbody>\n<tr>\n<td style=\"width: 50%\">\n<figure id=\"attachment_2103\" aria-describedby=\"caption-attachment-2103\" style=\"width: 592px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x1_Area.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2103 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x1_Area.jpg\" alt=\"Bell curve with left-most 7.5% of graph highlighted and &quot;x1=?&quot; at the bottom far right of this area. One thousand and ten is written at the bottom in the middle of the graph.\" width=\"592\" height=\"278\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x1_Area.jpg 592w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x1_Area-300x141.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x1_Area-65x31.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x1_Area-225x106.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x1_Area-350x164.jpg 350w\" sizes=\"auto, (max-width: 592px) 100vw, 592px\" \/><\/a><figcaption id=\"caption-attachment-2103\" class=\"wp-caption-text\">Figure 40.6 Left 7.5% of bell curve selected.<\/figcaption><\/figure>\n<\/td>\n<td style=\"width: 50%\">\n<figure id=\"attachment_2104\" aria-describedby=\"caption-attachment-2104\" style=\"width: 577px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x2_Area.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2104 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x2_Area.jpg\" alt=\"Bell curve with left-most 92.5% of graph highlighted and &quot;x2=?&quot; at the bottom far right of this area. One thousand and ten is written at the bottom in the middle of the graph.\" width=\"577\" height=\"313\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x2_Area.jpg 577w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x2_Area-300x163.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x2_Area-65x35.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x2_Area-225x122.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example40-1-3_x2_Area-350x190.jpg 350w\" sizes=\"auto, (max-width: 577px) 100vw, 577px\" \/><\/a><figcaption id=\"caption-attachment-2104\" class=\"wp-caption-text\">Figure 40.7 Left 92.5% of bell curve selected.<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To calculate the [latex]x[\/latex]-value associated with the above graphs, we use NORM.INV:<\/p>\n<p>\\[ x_1 = \\text{NORM.INV}((1-0.85)\/2,1010,20) = \\text{NORM.INV}(0.075, 1010, 20) = 989.2713\\]<\/p>\n<p>\\[ x_2 = \\text{NORM.INV}((1-0.85)\/2+0.85,1010,20) = \\text{NORM.INV}(0.925, 1010, 20) = 1038.791\\]<\/p>\n<\/div>\n<h1>Video Explaining ALl topics in this section<\/h1>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Using Excel&#39;s NORM.INV() function\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/h0wWLcmNtZM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<span style=\"color: #003366\"><strong>Additional Resources<\/strong><\/span>:<\/p>\n<ul>\n<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributions.pptx\">Click here to download the Powerpoint slides<\/a> that accompany the video.<\/li>\n<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/02\/NormalDistributionSolutions.xlsx\">Click here to download the Excel solutions<\/a> for the Normal Distribution section.<\/li>\n<\/ul>\n<h1>Key Takeaways (EXERCISE)<\/h1>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways: Excel&#8217;s NORM.INV Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Drag the words into the correct boxes for each section below:<\/p>\n<div id=\"h5p-139\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-139\" class=\"h5p-iframe\" data-content-id=\"139\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Excel&#039;s NORM.INV Function Key Takeaways\"><\/iframe><\/div>\n<\/div>\n<p>Click the sections below to reveal the solutions to the above exercises<\/p>\n<div id=\"h5p-140\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-140\" class=\"h5p-iframe\" data-content-id=\"140\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Excel&#039;s NORM.INV Function Key Takeaway Solutions\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Your Own Notes (EXERCISE)<\/h1>\n<ul>\n<li>Are there any notes you want to take from this section? Is there anything you&#8217;d like to copy and paste below?<\/li>\n<li>These notes are for you only (they will not be stored anywhere)<\/li>\n<li>Make sure to download them at the end to use as a reference<\/li>\n<\/ul>\n<div id=\"h5p-16\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-16\" class=\"h5p-iframe\" data-content-id=\"16\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key takeaways, notes and comments from this section document tool.\"><\/iframe><\/div>\n<\/div>\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-2079","chapter","type-chapter","status-publish","hentry"],"part":263,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2079","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":21,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2079\/revisions"}],"predecessor-version":[{"id":2116,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2079\/revisions\/2116"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/263"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2079\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=2079"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=2079"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=2079"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=2079"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}