{"id":1970,"date":"2024-06-12T20:01:27","date_gmt":"2024-06-13T00:01:27","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=1970"},"modified":"2024-06-16T13:39:42","modified_gmt":"2024-06-16T17:39:42","slug":"1970","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/1970\/","title":{"raw":"The Empirical Rule, Outliers and Excel's NORM.DIST()","rendered":"The Empirical Rule, Outliers and Excel&#8217;s NORM.DIST()"},"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 the Empirical Rule and Excel's NORM.DIST() function to calculate probabilities.\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>Empirical Rule<\/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_2073\" align=\"alignnone\" width=\"654\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/EmpericalRule2.jpg\"><img class=\"wp-image-2073 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/EmpericalRule2.jpg\" alt=\"Graph showing arrows and the 68%, 95% and 99.7% regions highlighted between 1, 2 and 3 standard deviations from the mean.\" width=\"654\" height=\"381\" \/><\/a> Figure 39.1: % of data within 1, 2 and, 3 standard deviations of mean[\/caption]<\/td>\r\n<td style=\"width: 67.6962%;height: 203px;vertical-align: top\">\r\n<p style=\"text-align: left\">When data are Normally Distributed:<\/p>\r\n\r\n<ul>\r\n \t<li style=\"text-align: left\">Roughly 68% of all data are within 1 standard deviation of the mean<\/li>\r\n \t<li style=\"text-align: left\">Roughly 95% of all data are within 2 standard deviations of the mean<\/li>\r\n \t<li style=\"text-align: left\">Roughly 99.7% of all data are within 3 standard deviations of the mean<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Outliers<\/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_2075\" align=\"alignnone\" width=\"713\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Outliers_new2.jpg\"><img class=\"wp-image-2075 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Outliers_new2.jpg\" alt=\"Image of bell curve with the word 'outliers' and arrows pointing to regions 3 standard deviations above and below the mean. Below each of these regions is marked 0.15% (because 0.15% of the data is located in these regions).\" width=\"713\" height=\"385\" \/><\/a> Figure 39.2: Outliers above\/below 3 standard deviations[\/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\">Data beyond 3 standard deviations are called '<a href=\"https:\/\/en.wikipedia.org\/wiki\/Outlier\">outliers<\/a>'<\/li>\r\n \t<li style=\"text-align: left\">Roughly 0.15% of the data are 3 standard deviations above or below the mean<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h1>Applying the Empirical Rule (Exercises)<\/h1>\r\nLet us first try some examples\/exercises where we will seek to understand the Empirical Rule percentages. This understanding will also help when calculating more complicated probabilities using Excel's NORM.DIST() function in later sections.\r\n<h2>Example 39.1<\/h2>\r\n<strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: In this problem, we will 'divide up' the sections of the normal curve into 'slices'.\r\n\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: Can you figure out the percent of data that lie in each slice of the curve?\r\n\r\n<strong><span style=\"color: #003366\">You try<\/span><\/strong>: Drag the appropriate percentage into each slice in the exercise below:\r\n\r\n[h5p id=\"118\"]\r\n\r\n<strong><span style=\"color: #003366\">Need Help<\/span><\/strong>? Try the exercise below first.\r\n\r\n[h5p id=\"119\"]\r\n<h2>Example 39.2<\/h2>\r\n<strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Demand for the 12-pack of extra plush toilet paper follows a normal distribution at a local drug store. On average, they sell 50 packs per week with a standard deviation of 10 packs. They get deliveries once per week and stock 80 packs of toilet paper per week.\r\n\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: Can you solve for the probabilities below and match them to their answers?\r\n\r\n[h5p id=\"120\"]\r\n\r\n<strong><span style=\"color: #003366\">Need Help<\/span><\/strong>? Click on the problems below to reveal their answers:\r\n<div>\r\n<h1>Probability of 40 to 60 packs sold<\/h1>\r\nWe are given the following values in the problem:\r\n<ul>\r\n \t<li>\u03bc = 50, \u03c3 = 10,\u00a0x<sub>1<\/sub> = 40, x<sub>2<\/sub> = 60<\/li>\r\n \t<li>z<sub>1<\/sub> = <sup>(40\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>\u221210<\/sup>\/<sub>10 <\/sub>= \u22121<\/li>\r\n \t<li>z<sub>2<\/sub> = <sup>(60\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>10<\/sup>\/<sub>10 <\/sub>= 1<\/li>\r\n<\/ul>\r\nThis gives P(40 &lt; x &lt; 60) = 34% + 34% = 68% (see below):\r\n\r\n[caption id=\"attachment_2039\" align=\"aligncenter\" width=\"413\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2a_Solution.jpg\"><img class=\"wp-image-2039\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2a_Solution.jpg\" alt=\"Picture with area from z equals minus one to one highlighted. The area above each section is 34%.\" width=\"413\" height=\"210\" \/><\/a> Figure 39.4: Areas from \u22121 to +1 standard deviations from the mean.[\/caption]\r\n<h1>probability of 30 and 70 packs sold<\/h1>\r\nWe are given the following values in the problem:\r\n<ul>\r\n \t<li>\u03bc = 50, \u03c3 = 10,\u00a0x<sub>1<\/sub> = 30, x<sub>2<\/sub> = 70<\/li>\r\n \t<li>z<sub>1<\/sub> = <sup>(30\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>\u221220<\/sup>\/<sub>10 <\/sub>= \u22122<\/li>\r\n \t<li>z<sub>2<\/sub> = <sup>(70\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>20<\/sup>\/<sub>10 <\/sub>= 2<\/li>\r\n<\/ul>\r\nThis gives P(30 &lt; x &lt; 70) = 13.5% + 34% + 34% + 13.5% = 95% (see below):\r\n\r\n[caption id=\"attachment_2043\" align=\"aligncenter\" width=\"414\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2b_Solution.jpg\"><img class=\"wp-image-2043 \" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2b_Solution.jpg\" alt=\"Image with areas between -2 to -1, -1 to 0, 0 to 1 and, 1 to 2 standard deviations highlighted. Above them are the values: 13.5%, 34%, 34% and, 13.5%.\" width=\"414\" height=\"217\" \/><\/a> Figure 39.5: Areas from \u22122 to +2 standard deviations from the mean.[\/caption]\r\n<h1>probability of 30 and 80 packs sold<\/h1>\r\nWe are given the following values in the problem:\r\n<ul>\r\n \t<li>\u03bc = 50, \u03c3 = 10,\u00a0x<sub>1<\/sub> = 30, x<sub>2<\/sub> = 80<\/li>\r\n \t<li>z<sub>1<\/sub> = <sup>(30\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>\u221220<\/sup>\/<sub>10 <\/sub>= \u22122<\/li>\r\n \t<li>z<sub>2<\/sub> = <sup>(80\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>30<\/sup>\/<sub>10 <\/sub>= 3<\/li>\r\n<\/ul>\r\nThis gives P(30 &lt; x &lt; 80) = 13.5% + 34% + 34% + 13.5% + 2.35% = 97.35% (see below):\r\n\r\n[caption id=\"attachment_2047\" align=\"aligncenter\" width=\"413\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2c_Solution.jpg\"><img class=\"wp-image-2047 \" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2c_Solution.jpg\" alt=\"Image with areas between -2 to -1, -1 to 0, 0 to 1, 1 to 2 and, 2 to 3 standard deviations highlighted. Above them are the values: 13.5%, 34%, 34%, 13.5% and, 2.35%.\" width=\"413\" height=\"216\" \/><\/a> Figure 39.6: Areas from \u22122 to +3 standard deviations from the mean.[\/caption]\r\n<h1>probability of Less than 20 packs sold<\/h1>\r\nWe are given the following values in the problem:\r\n<ul>\r\n \t<li>\u03bc = 50, \u03c3 = 10, x = 20<\/li>\r\n \t<li>z = <sup>(20\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>\u221230<\/sup>\/<sub>10 <\/sub>= \u22123<\/li>\r\n<\/ul>\r\nThis gives P(x &lt; 20) = 0.15% (see below):\r\n\r\n[caption id=\"attachment_2049\" align=\"aligncenter\" width=\"413\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2d_Solution.jpg\"><img class=\"wp-image-2049 \" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2d_Solution.jpg\" alt=\"Image with area below -3 standard deviations highlighted. Above it is the value: 0.15%.\" width=\"413\" height=\"222\" \/><\/a> Figure 39.7: Area below 3 standard deviations from the mean.[\/caption]\r\n<h1>probability of Greater than 50 packs sold<\/h1>\r\nWe are given the following values in the problem:\r\n<ul>\r\n \t<li>\u03bc = 50, \u03c3 = 10,\u00a0x<sub>1<\/sub> = 30, x = 50<\/li>\r\n \t<li>z = <sup>(50\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>0<\/sup>\/<sub>10 <\/sub>= 0<\/li>\r\n<\/ul>\r\nThis gives P(x &gt; 50) = 34% + 13.5% + 0.15% = 50% (see below):\r\n\r\n[caption id=\"attachment_2050\" align=\"aligncenter\" width=\"413\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2e_Solution.jpg\"><img class=\"wp-image-2050\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2e_Solution.jpg\" alt=\"Image with all areas above 0 standard deviations from the mean highlighted. Above them are the values: 34%, 13.5%, 2.35% and, 0.15%.\" width=\"413\" height=\"215\" \/><\/a> Figure 39.8: Area above 0 standard deviations (top half of graph).[\/caption]\r\n\r\n<span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: We see that the area above the mean (z=0) makes up half (50%) of the graph.\r\n<h1>probability of 40 and 70 packs sold<\/h1>\r\nWe are given the following values in the problem:\r\n<ul>\r\n \t<li>\u03bc = 50, \u03c3 = 10,\u00a0x<sub>1<\/sub> = 40, x<sub>2<\/sub> = 70<\/li>\r\n \t<li>z<sub>1<\/sub> = <sup>(40\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>\u221210<\/sup>\/<sub>10 <\/sub>= \u22121<\/li>\r\n \t<li>z<sub>2<\/sub> = <sup>(70\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>20<\/sup>\/<sub>10 <\/sub>= 2<\/li>\r\n<\/ul>\r\nThis gives P(40 &lt; x &lt; 70) = 34% + 34% + 13.5% = 81.5% (see below):\r\n\r\n[caption id=\"attachment_2052\" align=\"aligncenter\" width=\"413\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2f_Solution.jpg\"><img class=\"wp-image-2052\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2f_Solution.jpg\" alt=\"Image with all areas between -1 and 2 standard deviations from the mean highlighted. Above them are the values: 34%, 34%,13.5%.\" width=\"413\" height=\"208\" \/><\/a> Figure 39.9: Areas from \u22121 to +2 standard deviations from the mean.[\/caption]\r\n<h1>probability Of stocking out<\/h1>\r\nIf we stock out, demand is higher than supply. We stock 80 packs per week. This means that demand is higher than 80 packs that week:\r\n<ul>\r\n \t<li>\u03bc = 50, \u03c3 = 10, x = 80<\/li>\r\n \t<li>z = <sup>(80\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>30<\/sup>\/<sub>10 <\/sub>= 3<\/li>\r\n<\/ul>\r\nThis gives P(x &gt; 80) = 0.15% (see below):\r\n\r\n[caption id=\"attachment_2053\" align=\"aligncenter\" width=\"413\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2g_Solution.jpg\"><img class=\"wp-image-2053 \" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2g_Solution.jpg\" alt=\"Image with area above 3 standard deviations highlighted. Above it is the value 0.15%.\" width=\"413\" height=\"218\" \/><\/a> Figure 39.10: Area highlighted above 3 standard deviations from mean.[\/caption]\r\n\r\n<\/div>\r\n<h1>Calculating the Area Below an X-Value (Norm.DIST Exercise)<\/h1>\r\nLet us now practice using Excel's NORM.DIST() function to solve for probabilities. Remember the following is true when calculating the area below an [latex]x[\/latex]-value:\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_2004\" align=\"alignnone\" width=\"585\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea_x-value_large.jpg\"><img class=\"wp-image-2004 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea_x-value_large.jpg\" alt=\"Bell shaped curve with area to the left of x-value shaded.\" width=\"585\" height=\"367\" \/><\/a> Figure 39.3: 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\">For more precise area calculations or<\/li>\r\n \t<li style=\"text-align: left\">Areas that aren't precisely 1, 2 or 3 standard deviations above or below the mean,<\/li>\r\n \t<li style=\"text-align: left\">Use Excel's <a href=\"https:\/\/support.microsoft.com\/en-gb\/office\/norm-dist-function-edb1cc14-a21c-4e53-839d-8082074c9f8d\">NORM.DIST()<\/a> function<\/li>\r\n \t<li style=\"text-align: left\">P(X \u2264 x) = NORM.DIST(x, \u03bc, \u03c3, 1)<\/li>\r\n \t<li style=\"text-align: left\">This gives 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>Example 39.3.1<\/h2>\r\n<strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: A school's SAT scores are normally distributed with a mean of 1,010 and standard deviation of 20.\r\n\r\n<span style=\"color: #003366\"><strong>Question:<\/strong><\/span> What percent of students have scores below 1,040 points?\r\n\r\n<span style=\"color: #003366\"><strong>You try 1<\/strong><\/span>: Drag the region we would highlight for this question onto the graph below:\r\n\r\n[h5p id=\"122\"]\r\n\r\n<span style=\"color: #003366\"><strong>You try 2<\/strong><\/span>: Select the correct Excel formula and resulting solution for this question:\r\n\r\n[h5p id=\"124\"]\r\n<h1>Calculating the Area Above an X-Value (Norm.DIST Exercise)<\/h1>\r\nRemember, the following is true when calculating the area above an [latex]x[\/latex]-value:\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: top\">\r\n\r\n[caption id=\"attachment_2006\" align=\"alignnone\" width=\"585\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea_x-value_large.jpg\"><img class=\"wp-image-2006 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea_x-value_large.jpg\" alt=\"Bell shaped curve with area above (to the right of) x-value shaded.\" width=\"585\" height=\"314\" \/><\/a> Figure 39.11: Area to the right 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\">Excel's NORM.DIST function returns the area to the left of x<\/li>\r\n \t<li style=\"text-align: left\">To calculate the area to the right, we take a complement<\/li>\r\n \t<li style=\"text-align: left\">P(X&gt;x) = 1\u2212NORM.DIST(x, \u03bc, \u03c3,1)<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Example 39.3.2<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup:<\/strong><\/span> Let us keep going with our previous example of average SAT scores of 1,010 with a standard deviation of 20.\r\n\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: What percent of students score above 1,040 on their SATs?\r\n\r\n<span style=\"color: #003366\"><strong>You try 1<\/strong><\/span>: Drag the region we would highlight for this question onto the graph below:\r\n\r\n[h5p id=\"123\"]\r\n\r\n<span style=\"color: #003366\"><strong>You try 2<\/strong><\/span>: Select the correct Excel formula and resulting solution for this question:\r\n\r\n[h5p id=\"125\"]\r\n<h1>Calculating the Area Between Two X-Values (Norm.DIST Exercise)<\/h1>\r\nRemember, the following is true when calculating the area between two [latex]x[\/latex]-values:\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: top\">\r\n\r\n[caption id=\"attachment_2005\" align=\"alignnone\" width=\"585\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/BtwArea_x-value_large.jpg\"><img class=\"wp-image-2005 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/BtwArea_x-value_large.jpg\" alt=\"Bell shaped curve with area shaded between two values, x1 and x2.\" width=\"585\" height=\"316\" \/><\/a> Figure 5: Area between x-values[\/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\">To calculate the area between two x-values<\/li>\r\n \t<li style=\"text-align: left\">We need to deduct the area to the left of each x-value<\/li>\r\n \t<li style=\"text-align: left\">The remaining, middle area, will be the answer.<\/li>\r\n \t<li style=\"text-align: left\">P(x<sub>1<\/sub>\u2264x\u2264x<sub>2<\/sub>) = NORM.DIST(x<sub>2<\/sub>, \u03bc, \u03c3,1)\u2212NORM.DIST(x<sub>1<\/sub>, \u03bc, \u03c3,1)<\/li>\r\n<\/ul>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Example 39.3.3<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup:<\/strong><\/span> Let us, again, keep going with our previous example with average SAT scores of 1,010 with a standard deviation of 20.\r\n\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: What percent of students score between 955 and 1,035 on their SATs?\r\n\r\n<span style=\"color: #003366\"><strong>You try 1<\/strong><\/span>: Drag the region we would highlight for this question onto the graph below:\r\n\r\n[h5p id=\"127\"]\r\n\r\n<span style=\"color: #003366\"><strong>You try 2<\/strong><\/span>: Select the correct Excel formula and resulting solution for this question:\r\n\r\n[h5p id=\"126\"]\r\n\r\n<span style=\"color: #003366\"><strong>Need more help<\/strong><\/span>? See the video in the section below for a full walk-through of these examples\r\n<h1>Video &amp; Additional Resources Explaining this section<\/h1>\r\nhttps:\/\/youtu.be\/fN5RQ8Gtlsc\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: The Empirical Rule, Outliers and Using Excel's NORM.DIST<\/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=\"128\"]\r\n\r\nClick the sections below to reveal the solutions to the above exercises\r\n\r\n[h5p id=\"129\"]\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 the Empirical Rule and Excel&#8217;s NORM.DIST() function to calculate probabilities.<\/p>\n<\/div>\n<\/div>\n<h2>Empirical Rule<\/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_2073\" aria-describedby=\"caption-attachment-2073\" style=\"width: 654px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/EmpericalRule2.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2073 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/EmpericalRule2.jpg\" alt=\"Graph showing arrows and the 68%, 95% and 99.7% regions highlighted between 1, 2 and 3 standard deviations from the mean.\" width=\"654\" height=\"381\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/EmpericalRule2.jpg 654w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/EmpericalRule2-300x175.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/EmpericalRule2-65x38.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/EmpericalRule2-225x131.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/EmpericalRule2-350x204.jpg 350w\" sizes=\"auto, (max-width: 654px) 100vw, 654px\" \/><\/a><figcaption id=\"caption-attachment-2073\" class=\"wp-caption-text\">Figure 39.1: % of data within 1, 2 and, 3 standard deviations of mean<\/figcaption><\/figure>\n<\/td>\n<td style=\"width: 67.6962%;height: 203px;vertical-align: top\">\n<p style=\"text-align: left\">When data are Normally Distributed:<\/p>\n<ul>\n<li style=\"text-align: left\">Roughly 68% of all data are within 1 standard deviation of the mean<\/li>\n<li style=\"text-align: left\">Roughly 95% of all data are within 2 standard deviations of the mean<\/li>\n<li style=\"text-align: left\">Roughly 99.7% of all data are within 3 standard deviations of the mean<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Outliers<\/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_2075\" aria-describedby=\"caption-attachment-2075\" style=\"width: 713px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Outliers_new2.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2075 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Outliers_new2.jpg\" alt=\"Image of bell curve with the word 'outliers' and arrows pointing to regions 3 standard deviations above and below the mean. Below each of these regions is marked 0.15% (because 0.15% of the data is located in these regions).\" width=\"713\" height=\"385\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Outliers_new2.jpg 713w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Outliers_new2-300x162.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Outliers_new2-65x35.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Outliers_new2-225x121.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Outliers_new2-350x189.jpg 350w\" sizes=\"auto, (max-width: 713px) 100vw, 713px\" \/><\/a><figcaption id=\"caption-attachment-2075\" class=\"wp-caption-text\">Figure 39.2: Outliers above\/below 3 standard deviations<\/figcaption><\/figure>\n<\/td>\n<td style=\"width: 67.6962%;height: 231px;vertical-align: top\">\n<ul>\n<li style=\"text-align: left\">Data beyond 3 standard deviations are called &#8216;<a href=\"https:\/\/en.wikipedia.org\/wiki\/Outlier\">outliers<\/a>&#8216;<\/li>\n<li style=\"text-align: left\">Roughly 0.15% of the data are 3 standard deviations above or below the mean<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Applying the Empirical Rule (Exercises)<\/h1>\n<p>Let us first try some examples\/exercises where we will seek to understand the Empirical Rule percentages. This understanding will also help when calculating more complicated probabilities using Excel&#8217;s NORM.DIST() function in later sections.<\/p>\n<h2>Example 39.1<\/h2>\n<p><strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: In this problem, we will &#8216;divide up&#8217; the sections of the normal curve into &#8216;slices&#8217;.<\/p>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: Can you figure out the percent of data that lie in each slice of the curve?<\/p>\n<p><strong><span style=\"color: #003366\">You try<\/span><\/strong>: Drag the appropriate percentage into each slice in the exercise below:<\/p>\n<div id=\"h5p-118\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-118\" class=\"h5p-iframe\" data-content-id=\"118\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 39.1a - Normal Distributions - Using the Emperical Rule\"><\/iframe><\/div>\n<\/div>\n<p><strong><span style=\"color: #003366\">Need Help<\/span><\/strong>? Try the exercise below first.<\/p>\n<div id=\"h5p-119\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-119\" class=\"h5p-iframe\" data-content-id=\"119\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 39.1b - Normal Distributions - Using the Emperical Rule\"><\/iframe><\/div>\n<\/div>\n<h2>Example 39.2<\/h2>\n<p><strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: Demand for the 12-pack of extra plush toilet paper follows a normal distribution at a local drug store. On average, they sell 50 packs per week with a standard deviation of 10 packs. They get deliveries once per week and stock 80 packs of toilet paper per week.<\/p>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: Can you solve for the probabilities below and match them to their answers?<\/p>\n<div id=\"h5p-120\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-120\" class=\"h5p-iframe\" data-content-id=\"120\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 39.2 - Normal Distributions - Empirical Rule Exercise\"><\/iframe><\/div>\n<\/div>\n<p><strong><span style=\"color: #003366\">Need Help<\/span><\/strong>? Click on the problems below to reveal their answers:<\/p>\n<div>\n<h1>Probability of 40 to 60 packs sold<\/h1>\n<p>We are given the following values in the problem:<\/p>\n<ul>\n<li>\u03bc = 50, \u03c3 = 10,\u00a0x<sub>1<\/sub> = 40, x<sub>2<\/sub> = 60<\/li>\n<li>z<sub>1<\/sub> = <sup>(40\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>\u221210<\/sup>\/<sub>10 <\/sub>= \u22121<\/li>\n<li>z<sub>2<\/sub> = <sup>(60\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>10<\/sup>\/<sub>10 <\/sub>= 1<\/li>\n<\/ul>\n<p>This gives P(40 &lt; x &lt; 60) = 34% + 34% = 68% (see below):<\/p>\n<figure id=\"attachment_2039\" aria-describedby=\"caption-attachment-2039\" style=\"width: 413px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2a_Solution.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2039\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2a_Solution.jpg\" alt=\"Picture with area from z equals minus one to one highlighted. The area above each section is 34%.\" width=\"413\" height=\"210\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2a_Solution.jpg 849w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2a_Solution-300x153.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2a_Solution-768x391.jpg 768w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2a_Solution-65x33.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2a_Solution-225x114.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2a_Solution-350x178.jpg 350w\" sizes=\"auto, (max-width: 413px) 100vw, 413px\" \/><\/a><figcaption id=\"caption-attachment-2039\" class=\"wp-caption-text\">Figure 39.4: Areas from \u22121 to +1 standard deviations from the mean.<\/figcaption><\/figure>\n<h1>probability of 30 and 70 packs sold<\/h1>\n<p>We are given the following values in the problem:<\/p>\n<ul>\n<li>\u03bc = 50, \u03c3 = 10,\u00a0x<sub>1<\/sub> = 30, x<sub>2<\/sub> = 70<\/li>\n<li>z<sub>1<\/sub> = <sup>(30\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>\u221220<\/sup>\/<sub>10 <\/sub>= \u22122<\/li>\n<li>z<sub>2<\/sub> = <sup>(70\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>20<\/sup>\/<sub>10 <\/sub>= 2<\/li>\n<\/ul>\n<p>This gives P(30 &lt; x &lt; 70) = 13.5% + 34% + 34% + 13.5% = 95% (see below):<\/p>\n<figure id=\"attachment_2043\" aria-describedby=\"caption-attachment-2043\" style=\"width: 414px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2b_Solution.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2043\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2b_Solution.jpg\" alt=\"Image with areas between -2 to -1, -1 to 0, 0 to 1 and, 1 to 2 standard deviations highlighted. Above them are the values: 13.5%, 34%, 34% and, 13.5%.\" width=\"414\" height=\"217\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2b_Solution.jpg 549w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2b_Solution-300x157.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2b_Solution-65x34.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2b_Solution-225x118.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2b_Solution-350x184.jpg 350w\" sizes=\"auto, (max-width: 414px) 100vw, 414px\" \/><\/a><figcaption id=\"caption-attachment-2043\" class=\"wp-caption-text\">Figure 39.5: Areas from \u22122 to +2 standard deviations from the mean.<\/figcaption><\/figure>\n<h1>probability of 30 and 80 packs sold<\/h1>\n<p>We are given the following values in the problem:<\/p>\n<ul>\n<li>\u03bc = 50, \u03c3 = 10,\u00a0x<sub>1<\/sub> = 30, x<sub>2<\/sub> = 80<\/li>\n<li>z<sub>1<\/sub> = <sup>(30\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>\u221220<\/sup>\/<sub>10 <\/sub>= \u22122<\/li>\n<li>z<sub>2<\/sub> = <sup>(80\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>30<\/sup>\/<sub>10 <\/sub>= 3<\/li>\n<\/ul>\n<p>This gives P(30 &lt; x &lt; 80) = 13.5% + 34% + 34% + 13.5% + 2.35% = 97.35% (see below):<\/p>\n<figure id=\"attachment_2047\" aria-describedby=\"caption-attachment-2047\" style=\"width: 413px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2c_Solution.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2047\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2c_Solution.jpg\" alt=\"Image with areas between -2 to -1, -1 to 0, 0 to 1, 1 to 2 and, 2 to 3 standard deviations highlighted. Above them are the values: 13.5%, 34%, 34%, 13.5% and, 2.35%.\" width=\"413\" height=\"216\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2c_Solution.jpg 522w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2c_Solution-300x157.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2c_Solution-65x34.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2c_Solution-225x118.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2c_Solution-350x183.jpg 350w\" sizes=\"auto, (max-width: 413px) 100vw, 413px\" \/><\/a><figcaption id=\"caption-attachment-2047\" class=\"wp-caption-text\">Figure 39.6: Areas from \u22122 to +3 standard deviations from the mean.<\/figcaption><\/figure>\n<h1>probability of Less than 20 packs sold<\/h1>\n<p>We are given the following values in the problem:<\/p>\n<ul>\n<li>\u03bc = 50, \u03c3 = 10, x = 20<\/li>\n<li>z = <sup>(20\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>\u221230<\/sup>\/<sub>10 <\/sub>= \u22123<\/li>\n<\/ul>\n<p>This gives P(x &lt; 20) = 0.15% (see below):<\/p>\n<figure id=\"attachment_2049\" aria-describedby=\"caption-attachment-2049\" style=\"width: 413px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2d_Solution.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2049\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2d_Solution.jpg\" alt=\"Image with area below -3 standard deviations highlighted. Above it is the value: 0.15%.\" width=\"413\" height=\"222\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2d_Solution.jpg 532w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2d_Solution-300x161.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2d_Solution-65x35.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2d_Solution-225x121.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2d_Solution-350x188.jpg 350w\" sizes=\"auto, (max-width: 413px) 100vw, 413px\" \/><\/a><figcaption id=\"caption-attachment-2049\" class=\"wp-caption-text\">Figure 39.7: Area below 3 standard deviations from the mean.<\/figcaption><\/figure>\n<h1>probability of Greater than 50 packs sold<\/h1>\n<p>We are given the following values in the problem:<\/p>\n<ul>\n<li>\u03bc = 50, \u03c3 = 10,\u00a0x<sub>1<\/sub> = 30, x = 50<\/li>\n<li>z = <sup>(50\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>0<\/sup>\/<sub>10 <\/sub>= 0<\/li>\n<\/ul>\n<p>This gives P(x &gt; 50) = 34% + 13.5% + 0.15% = 50% (see below):<\/p>\n<figure id=\"attachment_2050\" aria-describedby=\"caption-attachment-2050\" style=\"width: 413px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2e_Solution.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2050\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2e_Solution.jpg\" alt=\"Image with all areas above 0 standard deviations from the mean highlighted. Above them are the values: 34%, 13.5%, 2.35% and, 0.15%.\" width=\"413\" height=\"215\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2e_Solution.jpg 551w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2e_Solution-300x156.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2e_Solution-65x34.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2e_Solution-225x117.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2e_Solution-350x182.jpg 350w\" sizes=\"auto, (max-width: 413px) 100vw, 413px\" \/><\/a><figcaption id=\"caption-attachment-2050\" class=\"wp-caption-text\">Figure 39.8: Area above 0 standard deviations (top half of graph).<\/figcaption><\/figure>\n<p><span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: We see that the area above the mean (z=0) makes up half (50%) of the graph.<\/p>\n<h1>probability of 40 and 70 packs sold<\/h1>\n<p>We are given the following values in the problem:<\/p>\n<ul>\n<li>\u03bc = 50, \u03c3 = 10,\u00a0x<sub>1<\/sub> = 40, x<sub>2<\/sub> = 70<\/li>\n<li>z<sub>1<\/sub> = <sup>(40\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>\u221210<\/sup>\/<sub>10 <\/sub>= \u22121<\/li>\n<li>z<sub>2<\/sub> = <sup>(70\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>20<\/sup>\/<sub>10 <\/sub>= 2<\/li>\n<\/ul>\n<p>This gives P(40 &lt; x &lt; 70) = 34% + 34% + 13.5% = 81.5% (see below):<\/p>\n<figure id=\"attachment_2052\" aria-describedby=\"caption-attachment-2052\" style=\"width: 413px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2f_Solution.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2052\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2f_Solution.jpg\" alt=\"Image with all areas between -1 and 2 standard deviations from the mean highlighted. Above them are the values: 34%, 34%,13.5%.\" width=\"413\" height=\"208\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2f_Solution.jpg 512w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2f_Solution-300x151.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2f_Solution-65x33.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2f_Solution-225x113.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2f_Solution-350x176.jpg 350w\" sizes=\"auto, (max-width: 413px) 100vw, 413px\" \/><\/a><figcaption id=\"caption-attachment-2052\" class=\"wp-caption-text\">Figure 39.9: Areas from \u22121 to +2 standard deviations from the mean.<\/figcaption><\/figure>\n<h1>probability Of stocking out<\/h1>\n<p>If we stock out, demand is higher than supply. We stock 80 packs per week. This means that demand is higher than 80 packs that week:<\/p>\n<ul>\n<li>\u03bc = 50, \u03c3 = 10, x = 80<\/li>\n<li>z = <sup>(80\u221250)<\/sup>\/<sub>10 <\/sub>= <sup>30<\/sup>\/<sub>10 <\/sub>= 3<\/li>\n<\/ul>\n<p>This gives P(x &gt; 80) = 0.15% (see below):<\/p>\n<figure id=\"attachment_2053\" aria-describedby=\"caption-attachment-2053\" style=\"width: 413px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2g_Solution.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2053\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2g_Solution.jpg\" alt=\"Image with area above 3 standard deviations highlighted. Above it is the value 0.15%.\" width=\"413\" height=\"218\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2g_Solution.jpg 487w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2g_Solution-300x158.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2g_Solution-65x34.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2g_Solution-225x119.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example39-2g_Solution-350x185.jpg 350w\" sizes=\"auto, (max-width: 413px) 100vw, 413px\" \/><\/a><figcaption id=\"caption-attachment-2053\" class=\"wp-caption-text\">Figure 39.10: Area highlighted above 3 standard deviations from mean.<\/figcaption><\/figure>\n<\/div>\n<h1>Calculating the Area Below an X-Value (Norm.DIST Exercise)<\/h1>\n<p>Let us now practice using Excel&#8217;s NORM.DIST() function to solve for probabilities. Remember the following is true when calculating the area below an [latex]x[\/latex]-value:<\/p>\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_2004\" aria-describedby=\"caption-attachment-2004\" style=\"width: 585px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea_x-value_large.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2004 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea_x-value_large.jpg\" alt=\"Bell shaped curve with area to the left of x-value shaded.\" width=\"585\" height=\"367\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea_x-value_large.jpg 585w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea_x-value_large-300x188.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea_x-value_large-65x41.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea_x-value_large-225x141.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/LeftArea_x-value_large-350x220.jpg 350w\" sizes=\"auto, (max-width: 585px) 100vw, 585px\" \/><\/a><figcaption id=\"caption-attachment-2004\" class=\"wp-caption-text\">Figure 39.3: 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\">For more precise area calculations or<\/li>\n<li style=\"text-align: left\">Areas that aren&#8217;t precisely 1, 2 or 3 standard deviations above or below the mean,<\/li>\n<li style=\"text-align: left\">Use Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-gb\/office\/norm-dist-function-edb1cc14-a21c-4e53-839d-8082074c9f8d\">NORM.DIST()<\/a> function<\/li>\n<li style=\"text-align: left\">P(X \u2264 x) = NORM.DIST(x, \u03bc, \u03c3, 1)<\/li>\n<li style=\"text-align: left\">This gives the area to the left of x<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Example 39.3.1<\/h2>\n<p><strong><span style=\"color: #003366\">Problem Setup<\/span><\/strong>: A school&#8217;s SAT scores are normally distributed with a mean of 1,010 and standard deviation of 20.<\/p>\n<p><span style=\"color: #003366\"><strong>Question:<\/strong><\/span> What percent of students have scores below 1,040 points?<\/p>\n<p><span style=\"color: #003366\"><strong>You try 1<\/strong><\/span>: Drag the region we would highlight for this question onto the graph below:<\/p>\n<div id=\"h5p-122\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-122\" class=\"h5p-iframe\" data-content-id=\"122\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 39.3a - Normal Distributions - Using NORM.DIST\"><\/iframe><\/div>\n<\/div>\n<p><span style=\"color: #003366\"><strong>You try 2<\/strong><\/span>: Select the correct Excel formula and resulting solution for this question:<\/p>\n<div id=\"h5p-124\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-124\" class=\"h5p-iframe\" data-content-id=\"124\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 39.3a - Normal Distributions - Using NORM.DIST formula and solution\"><\/iframe><\/div>\n<\/div>\n<h1>Calculating the Area Above an X-Value (Norm.DIST Exercise)<\/h1>\n<p>Remember, the following is true when calculating the area above an [latex]x[\/latex]-value:<\/p>\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: top\">\n<figure id=\"attachment_2006\" aria-describedby=\"caption-attachment-2006\" style=\"width: 585px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea_x-value_large.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2006 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea_x-value_large.jpg\" alt=\"Bell shaped curve with area above (to the right of) x-value shaded.\" width=\"585\" height=\"314\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea_x-value_large.jpg 585w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea_x-value_large-300x161.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea_x-value_large-65x35.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea_x-value_large-225x121.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/RightArea_x-value_large-350x188.jpg 350w\" sizes=\"auto, (max-width: 585px) 100vw, 585px\" \/><\/a><figcaption id=\"caption-attachment-2006\" class=\"wp-caption-text\">Figure 39.11: Area to the right 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\">Excel&#8217;s NORM.DIST function returns the area to the left of x<\/li>\n<li style=\"text-align: left\">To calculate the area to the right, we take a complement<\/li>\n<li style=\"text-align: left\">P(X&gt;x) = 1\u2212NORM.DIST(x, \u03bc, \u03c3,1)<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Example 39.3.2<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup:<\/strong><\/span> Let us keep going with our previous example of average SAT scores of 1,010 with a standard deviation of 20.<\/p>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: What percent of students score above 1,040 on their SATs?<\/p>\n<p><span style=\"color: #003366\"><strong>You try 1<\/strong><\/span>: Drag the region we would highlight for this question onto the graph below:<\/p>\n<div id=\"h5p-123\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-123\" class=\"h5p-iframe\" data-content-id=\"123\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 39.3b - Normal Distributions - Using NORM.DIST\"><\/iframe><\/div>\n<\/div>\n<p><span style=\"color: #003366\"><strong>You try 2<\/strong><\/span>: Select the correct Excel formula and resulting solution for this question:<\/p>\n<div id=\"h5p-125\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-125\" class=\"h5p-iframe\" data-content-id=\"125\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 39.3b - Normal Distributions - Using NORM.DIST formula and solution\"><\/iframe><\/div>\n<\/div>\n<h1>Calculating the Area Between Two X-Values (Norm.DIST Exercise)<\/h1>\n<p>Remember, the following is true when calculating the area between two [latex]x[\/latex]-values:<\/p>\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: top\">\n<figure id=\"attachment_2005\" aria-describedby=\"caption-attachment-2005\" style=\"width: 585px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/BtwArea_x-value_large.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2005 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/BtwArea_x-value_large.jpg\" alt=\"Bell shaped curve with area shaded between two values, x1 and x2.\" width=\"585\" height=\"316\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/BtwArea_x-value_large.jpg 585w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/BtwArea_x-value_large-300x162.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/BtwArea_x-value_large-65x35.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/BtwArea_x-value_large-225x122.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/BtwArea_x-value_large-350x189.jpg 350w\" sizes=\"auto, (max-width: 585px) 100vw, 585px\" \/><\/a><figcaption id=\"caption-attachment-2005\" class=\"wp-caption-text\">Figure 5: Area between x-values<\/figcaption><\/figure>\n<\/td>\n<td style=\"width: 67.6962%;height: 203px;vertical-align: top\">\n<ul>\n<li style=\"text-align: left\">To calculate the area between two x-values<\/li>\n<li style=\"text-align: left\">We need to deduct the area to the left of each x-value<\/li>\n<li style=\"text-align: left\">The remaining, middle area, will be the answer.<\/li>\n<li style=\"text-align: left\">P(x<sub>1<\/sub>\u2264x\u2264x<sub>2<\/sub>) = NORM.DIST(x<sub>2<\/sub>, \u03bc, \u03c3,1)\u2212NORM.DIST(x<sub>1<\/sub>, \u03bc, \u03c3,1)<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Example 39.3.3<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup:<\/strong><\/span> Let us, again, keep going with our previous example with average SAT scores of 1,010 with a standard deviation of 20.<\/p>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: What percent of students score between 955 and 1,035 on their SATs?<\/p>\n<p><span style=\"color: #003366\"><strong>You try 1<\/strong><\/span>: Drag the region we would highlight for this question onto the graph below:<\/p>\n<div id=\"h5p-127\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-127\" class=\"h5p-iframe\" data-content-id=\"127\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 39.3c - Normal Distributions - Using NORM.DIST\"><\/iframe><\/div>\n<\/div>\n<p><span style=\"color: #003366\"><strong>You try 2<\/strong><\/span>: Select the correct Excel formula and resulting solution for this question:<\/p>\n<div id=\"h5p-126\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-126\" class=\"h5p-iframe\" data-content-id=\"126\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 39.3c - Normal Distributions - Using NORM.DIST formula and solution\"><\/iframe><\/div>\n<\/div>\n<p><span style=\"color: #003366\"><strong>Need more help<\/strong><\/span>? See the video in the section below for a full walk-through of these examples<\/p>\n<h1>Video &amp; Additional Resources Explaining this section<\/h1>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"The Empirical Rule, Outliers and Using Excel\u2019s NORM.DIST()\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/fN5RQ8Gtlsc?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: The Empirical Rule, Outliers and Using Excel&#8217;s NORM.DIST<\/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-128\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-128\" class=\"h5p-iframe\" data-content-id=\"128\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"The Emperical Rule, Outliers and Using Excel&#039;s NORM.DIST 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-129\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-129\" class=\"h5p-iframe\" data-content-id=\"129\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"The Emperical Rule, Outliers and Using Excel&#039;s NORM.DIST Key Takeaways 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":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1970","chapter","type-chapter","status-publish","hentry"],"part":263,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1970","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\/1970\/revisions"}],"predecessor-version":[{"id":2114,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1970\/revisions\/2114"}],"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\/1970\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=1970"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1970"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=1970"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=1970"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}