{"id":280,"date":"2017-04-15T03:20:35","date_gmt":"2017-04-15T03:20:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/conceptstest1\/chapter\/discrete-random-variables-2-of-4\/"},"modified":"2017-06-06T01:42:00","modified_gmt":"2017-06-06T01:42:00","slug":"discrete-random-variables-3-of-5","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/chapter\/discrete-random-variables-3-of-5\/","title":{"raw":"Discrete Random Variables (3 of 5)","rendered":"Discrete Random Variables (3 of 5)"},"content":{"raw":"&nbsp;\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use probability distributions for discrete and continuous random variables to estimate probabilities and identify unusual events.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<strong>The Mean and Standard Deviation of a Discrete Random Variable<\/strong>\r\n\r\nWe now focus on the mean and standard deviation of a discrete random variable. We discuss how to calculate these measures of center and spread for this type of probability distribution, but in general we will use technology to do these calculations.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\n<h2>The Mean of a Discrete Random Variable<\/h2>\r\nAt Rushmore Community College, there have been complaints about how long it takes to get food from the college cafeteria. In response, a study was conducted to record the total amount of time students had to wait to get their food. The following table gives the total times (rounded to the nearest 5 minutes) to get food for 200 randomly selected students.\r\n\r\nHere is the <strong>frequency table<\/strong>.\r\n<table>\r\n<tbody>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Time (minutes)<\/td>\r\n<td align=\"center\">5<\/td>\r\n<td align=\"center\">10<\/td>\r\n<td align=\"center\">15<\/td>\r\n<td align=\"center\">20<\/td>\r\n<td align=\"center\">25<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Number of students<\/td>\r\n<td align=\"center\">30<\/td>\r\n<td align=\"center\">52<\/td>\r\n<td align=\"center\">62<\/td>\r\n<td align=\"center\">40<\/td>\r\n<td align=\"center\">16<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUsing this data, we can create a <strong>probability distribution <\/strong>for the random variable <em>X<\/em> = \u201ctime to get food.\u201d As we have done before, we divide each frequency (count) by the total number of observations. For example, to calculate the probability that a student will have to wait 10 minutes to get their food we divide: (the number of students in the sample that waited 10 minutes) by (the total number of students in the sample) = 52 \/ 200 = 0.26.\r\n<table>\r\n<tbody>\r\n<tr class=\"oli_table\">\r\n<td><em>X<\/em> = Time (minutes)<\/td>\r\n<td align=\"center\">5<\/td>\r\n<td align=\"center\">10<\/td>\r\n<td align=\"center\">15<\/td>\r\n<td align=\"center\">20<\/td>\r\n<td align=\"center\">25<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\"><em>P<\/em>(<em>X<\/em>)<\/td>\r\n<td>30 \/ 200 = <strong>0.15<\/strong><\/td>\r\n<td>52 \/ 200 = <strong>0.26<\/strong><\/td>\r\n<td>62 \/ 200 = <strong>0.31<\/strong><\/td>\r\n<td>40 \/ 200 = <strong>0.20<\/strong><\/td>\r\n<td>16 \/ 200 = <strong>0.08<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nHere is the corresponding probability histogram:\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032030\/m6_probability_topic_6_1_6_1DiscreteRandomVariables2of4_image1.png\" alt=\"Histogram of waiting times for food. The gray bars are in increments of 5 minutes, and the highest bar indicates a .31 wait time in minutes.\" width=\"491\" height=\"303\" \/>\r\n\r\n<strong>A comment on probability histograms<\/strong>\r\n\r\nIn this probability histogram, the area, instead of the height, is the probability. In general, when we work with probability histograms, the area will represent the probability, so we will not worry about the units on the y-axis. Since the area represents the probabilities, the total area is 1.\r\n\r\nBecause in this case we have the actual data in the first table, we start by using that table of actual counts to calculate the mean. However, usually all we have is the probability distribution, so we will also consider how to calculate the mean directly from this information alone.\r\n\r\n<strong>Calculating the Mean from the Frequency Table<\/strong>\r\n<table>\r\n<tbody>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Time (minutes)<\/td>\r\n<td align=\"center\">5<\/td>\r\n<td align=\"center\">10<\/td>\r\n<td align=\"center\">15<\/td>\r\n<td align=\"center\">20<\/td>\r\n<td align=\"center\">25<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Number of students<\/td>\r\n<td align=\"center\">30<\/td>\r\n<td align=\"center\">52<\/td>\r\n<td align=\"center\">62<\/td>\r\n<td align=\"center\">40<\/td>\r\n<td align=\"center\">16<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe have 200 observations that are summarized in this table. We have 30 students with a time of 5 minutes, 52 students with a time of 10 minutes, 62 students with a time of 15 minutes, and so on.\r\n\r\nTo calculate the mean (that is the average), we have to add 30 fives + 52 tens + 62 fifteens + 40 twenties + 16 twenty-fives and then divide by 200. Here is that calculation:\r\n<p style=\"text-align: center\">[latex]\\frac{\\text{5}(\\text{30})+\\text{10}(\\text{52})+\\text{15}(\\text{62})+\\text{20}(\\text{40})+\\text{25}(\\text{16})}{\\text{200}}=\\text{14}[\/latex]<\/p>\r\nSo the mean time for students to get their food in the cafeteria is 14 minutes.\r\n\r\n<strong>Calculating the Mean from the Probability Distribution<\/strong>\r\n\r\nNow let's take a closer look at the calculation we just did.\r\n\r\nNotice that the large fraction on the left could be broken up into a sum of five smaller fractions all with the denominator 200:\r\n<p style=\"text-align: center\">[latex]\\frac{\\text{5}(\\text{30})}{\\text{200}}+\\frac{\\text{10}(\\text{52})}{\\text{200}}+\\frac{\\text{15}(\\text{62})}{\\text{200}}+\\frac{\\text{20}(\\text{40})}{\\text{200}}+\\frac{\\text{25}(\\text{16})}{\\text{200}}=\\text{14}[\/latex]<\/p>\r\nOkay, we are almost there. The last thing to do is rewrite each of these fractions like this:\r\n<p style=\"text-align: center\">[latex]\\text{5}\\left(\\frac{30}{200}\\right)+\\text{10}\\left(\\frac{52}{200}\\right)+\\text{15}\\left(\\frac{62}{200}\\right)+\\text{20}\\left(\\frac{40}{200}\\right)+\\text{25}\\left(\\frac{16}{200}\\right)=\\text{14}[\/latex]<\/p>\r\nHere is the same equation with the fractions expressed as decimals:\r\n<p style=\"text-align: center\">[latex]\\text{5}\\left(0.15\\right)+\\text{10}\\left(0.26\\right)+\\text{15}\\left(0.31\\right)+\\text{20}\\left(0.20\\right)+\\text{25}\\left(0.08\\right)=\\text{14}[\/latex]<\/p>\r\nLook closely at the terms we are adding. In each case, we have the product of one of the possible values of <em>X<\/em> and its corresponding probability:\r\n<table>\r\n<tbody>\r\n<tr class=\"oli_table\">\r\n<td><em>X<\/em> = Time (minutes)<\/td>\r\n<td align=\"center\">5<\/td>\r\n<td align=\"center\">10<\/td>\r\n<td align=\"center\">15<\/td>\r\n<td align=\"center\">20<\/td>\r\n<td align=\"center\">25<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\"><em>P<\/em>(<em>X<\/em>)<\/td>\r\n<td>30 \/ 200 = <strong>0.15<\/strong><\/td>\r\n<td>52 \/ 200 = <strong>0.26<\/strong><\/td>\r\n<td>62 \/ 200 = <strong>0.31<\/strong><\/td>\r\n<td>40 \/ 200 = <strong>0.20<\/strong><\/td>\r\n<td>16 \/ 200 = <strong>0.08<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAs we can see, the mean is just a <strong>weighted average<\/strong>. That is, the mean is the weighted sum of all the possible values of the random variable <em>X<\/em>, where each value is weighted by its probability.\r\n\r\n<\/div>\r\n<h3>Comment<\/h3>\r\n<strong>Why Is the Mean a Weighted Average?<\/strong>\r\n\r\nThe mean of a discrete random variable <em>X<\/em> should give us a measure of the long-run average value for <em>X<\/em>. It therefore makes sense to count more heavily those values of <em>X<\/em> that have a high probability, because they are more likely to occur and will consequently influence the long-run average. On the other hand, those values of <em>X<\/em> with low probability will not occur very often, so they will have little effect on the long-run average. It therefore makes sense to not give them much weight in our calculation.\r\n<h3>The Formula for the Mean of a Discrete Random Variable<\/h3>\r\nEarlier in the course, when we calculated the mean of a data set, we used the symbol [latex]\\stackrel{\u00af}{x}[\/latex] (x-bar) to represent that value. We do not use [latex]\\stackrel{\u00af}{x}[\/latex] to represent the mean of a random variable; instead we use [latex]{\\mathrm{\u03bc}}_{x}[\/latex] (pronounced \u201cmu-sub-x\u201d).\r\n\r\nHere is the formula that we have come up with for the mean of a discrete random variable. Note that [latex]P(x)[\/latex] represents the probability of <em>x<\/em>, where <em>x<\/em> is a value of the random variable <em>X<\/em>.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032032\/m6_probability_topic_6_1_6_1DiscreteRandomVariables2of4_image10.png\" alt=\"Equation: The mean equals the sum of all the values of x times their probabilities.\" width=\"486\" height=\"156\" \/>\r\n\r\nAnother term often used to describe the mean is <strong>expected value<\/strong>. It is a useful term because it reminds us that the mean of a random variable is not calculated on a fixed data set. Rather, the mean (expected value) is a measure of the expected long-term behavior of the random variable.\r\n<div class=\"textbox exercises\">\r\n<h3>Learn By Doing<\/h3>\r\nDrivers entering the short-term parking facility at an airport are given the option to purchase a parking permit for one of four possible time periods: \u00bd hour, 1 hour, 1\u00bd hours, or 2 hours. Thus, for each driver who enters the parking facility, we can consider their choice of parking time as a discrete random variable. In this case, the random variable <em>X<\/em> has four possible values: 0.5, 1, 1.5, and 2.\r\n\r\nAssume that the probability distribution for <em>X<\/em> is given by the following table.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032034\/m6_probability_topic_6_1_6_1DiscreteRandomVariables2of4_image11.png\" alt=\"table and histogram that show x=parking time in hours, with four possible times in hours (.5, 1, 1.5, and 2 hours) they might choose to park.\" width=\"640\" height=\"207\" \/>\r\n\r\nFor example, reading from this table, it appears that there is a 15% chance that the next driver entering the parking facility will opt for a \u00bd-hour permit. In the probability histogram, the area of each rectangle (not the height) is the probability of the corresponding <em>x<\/em>-value occurring.\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3558\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3559\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<p>&nbsp;<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use probability distributions for discrete and continuous random variables to estimate probabilities and identify unusual events.<\/li>\n<\/ul>\n<\/div>\n<p><strong>The Mean and Standard Deviation of a Discrete Random Variable<\/strong><\/p>\n<p>We now focus on the mean and standard deviation of a discrete random variable. We discuss how to calculate these measures of center and spread for this type of probability distribution, but in general we will use technology to do these calculations.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<h2>The Mean of a Discrete Random Variable<\/h2>\n<p>At Rushmore Community College, there have been complaints about how long it takes to get food from the college cafeteria. In response, a study was conducted to record the total amount of time students had to wait to get their food. The following table gives the total times (rounded to the nearest 5 minutes) to get food for 200 randomly selected students.<\/p>\n<p>Here is the <strong>frequency table<\/strong>.<\/p>\n<table>\n<tbody>\n<tr class=\"oli_table\">\n<td align=\"center\">Time (minutes)<\/td>\n<td align=\"center\">5<\/td>\n<td align=\"center\">10<\/td>\n<td align=\"center\">15<\/td>\n<td align=\"center\">20<\/td>\n<td align=\"center\">25<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\">Number of students<\/td>\n<td align=\"center\">30<\/td>\n<td align=\"center\">52<\/td>\n<td align=\"center\">62<\/td>\n<td align=\"center\">40<\/td>\n<td align=\"center\">16<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Using this data, we can create a <strong>probability distribution <\/strong>for the random variable <em>X<\/em> = \u201ctime to get food.\u201d As we have done before, we divide each frequency (count) by the total number of observations. For example, to calculate the probability that a student will have to wait 10 minutes to get their food we divide: (the number of students in the sample that waited 10 minutes) by (the total number of students in the sample) = 52 \/ 200 = 0.26.<\/p>\n<table>\n<tbody>\n<tr class=\"oli_table\">\n<td><em>X<\/em> = Time (minutes)<\/td>\n<td align=\"center\">5<\/td>\n<td align=\"center\">10<\/td>\n<td align=\"center\">15<\/td>\n<td align=\"center\">20<\/td>\n<td align=\"center\">25<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\"><em>P<\/em>(<em>X<\/em>)<\/td>\n<td>30 \/ 200 = <strong>0.15<\/strong><\/td>\n<td>52 \/ 200 = <strong>0.26<\/strong><\/td>\n<td>62 \/ 200 = <strong>0.31<\/strong><\/td>\n<td>40 \/ 200 = <strong>0.20<\/strong><\/td>\n<td>16 \/ 200 = <strong>0.08<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Here is the corresponding probability histogram:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032030\/m6_probability_topic_6_1_6_1DiscreteRandomVariables2of4_image1.png\" alt=\"Histogram of waiting times for food. The gray bars are in increments of 5 minutes, and the highest bar indicates a .31 wait time in minutes.\" width=\"491\" height=\"303\" \/><\/p>\n<p><strong>A comment on probability histograms<\/strong><\/p>\n<p>In this probability histogram, the area, instead of the height, is the probability. In general, when we work with probability histograms, the area will represent the probability, so we will not worry about the units on the y-axis. Since the area represents the probabilities, the total area is 1.<\/p>\n<p>Because in this case we have the actual data in the first table, we start by using that table of actual counts to calculate the mean. However, usually all we have is the probability distribution, so we will also consider how to calculate the mean directly from this information alone.<\/p>\n<p><strong>Calculating the Mean from the Frequency Table<\/strong><\/p>\n<table>\n<tbody>\n<tr class=\"oli_table\">\n<td align=\"center\">Time (minutes)<\/td>\n<td align=\"center\">5<\/td>\n<td align=\"center\">10<\/td>\n<td align=\"center\">15<\/td>\n<td align=\"center\">20<\/td>\n<td align=\"center\">25<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\">Number of students<\/td>\n<td align=\"center\">30<\/td>\n<td align=\"center\">52<\/td>\n<td align=\"center\">62<\/td>\n<td align=\"center\">40<\/td>\n<td align=\"center\">16<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We have 200 observations that are summarized in this table. We have 30 students with a time of 5 minutes, 52 students with a time of 10 minutes, 62 students with a time of 15 minutes, and so on.<\/p>\n<p>To calculate the mean (that is the average), we have to add 30 fives + 52 tens + 62 fifteens + 40 twenties + 16 twenty-fives and then divide by 200. Here is that calculation:<\/p>\n<p style=\"text-align: center\">[latex]\\frac{\\text{5}(\\text{30})+\\text{10}(\\text{52})+\\text{15}(\\text{62})+\\text{20}(\\text{40})+\\text{25}(\\text{16})}{\\text{200}}=\\text{14}[\/latex]<\/p>\n<p>So the mean time for students to get their food in the cafeteria is 14 minutes.<\/p>\n<p><strong>Calculating the Mean from the Probability Distribution<\/strong><\/p>\n<p>Now let&#8217;s take a closer look at the calculation we just did.<\/p>\n<p>Notice that the large fraction on the left could be broken up into a sum of five smaller fractions all with the denominator 200:<\/p>\n<p style=\"text-align: center\">[latex]\\frac{\\text{5}(\\text{30})}{\\text{200}}+\\frac{\\text{10}(\\text{52})}{\\text{200}}+\\frac{\\text{15}(\\text{62})}{\\text{200}}+\\frac{\\text{20}(\\text{40})}{\\text{200}}+\\frac{\\text{25}(\\text{16})}{\\text{200}}=\\text{14}[\/latex]<\/p>\n<p>Okay, we are almost there. The last thing to do is rewrite each of these fractions like this:<\/p>\n<p style=\"text-align: center\">[latex]\\text{5}\\left(\\frac{30}{200}\\right)+\\text{10}\\left(\\frac{52}{200}\\right)+\\text{15}\\left(\\frac{62}{200}\\right)+\\text{20}\\left(\\frac{40}{200}\\right)+\\text{25}\\left(\\frac{16}{200}\\right)=\\text{14}[\/latex]<\/p>\n<p>Here is the same equation with the fractions expressed as decimals:<\/p>\n<p style=\"text-align: center\">[latex]\\text{5}\\left(0.15\\right)+\\text{10}\\left(0.26\\right)+\\text{15}\\left(0.31\\right)+\\text{20}\\left(0.20\\right)+\\text{25}\\left(0.08\\right)=\\text{14}[\/latex]<\/p>\n<p>Look closely at the terms we are adding. In each case, we have the product of one of the possible values of <em>X<\/em> and its corresponding probability:<\/p>\n<table>\n<tbody>\n<tr class=\"oli_table\">\n<td><em>X<\/em> = Time (minutes)<\/td>\n<td align=\"center\">5<\/td>\n<td align=\"center\">10<\/td>\n<td align=\"center\">15<\/td>\n<td align=\"center\">20<\/td>\n<td align=\"center\">25<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\"><em>P<\/em>(<em>X<\/em>)<\/td>\n<td>30 \/ 200 = <strong>0.15<\/strong><\/td>\n<td>52 \/ 200 = <strong>0.26<\/strong><\/td>\n<td>62 \/ 200 = <strong>0.31<\/strong><\/td>\n<td>40 \/ 200 = <strong>0.20<\/strong><\/td>\n<td>16 \/ 200 = <strong>0.08<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As we can see, the mean is just a <strong>weighted average<\/strong>. That is, the mean is the weighted sum of all the possible values of the random variable <em>X<\/em>, where each value is weighted by its probability.<\/p>\n<\/div>\n<h3>Comment<\/h3>\n<p><strong>Why Is the Mean a Weighted Average?<\/strong><\/p>\n<p>The mean of a discrete random variable <em>X<\/em> should give us a measure of the long-run average value for <em>X<\/em>. It therefore makes sense to count more heavily those values of <em>X<\/em> that have a high probability, because they are more likely to occur and will consequently influence the long-run average. On the other hand, those values of <em>X<\/em> with low probability will not occur very often, so they will have little effect on the long-run average. It therefore makes sense to not give them much weight in our calculation.<\/p>\n<h3>The Formula for the Mean of a Discrete Random Variable<\/h3>\n<p>Earlier in the course, when we calculated the mean of a data set, we used the symbol [latex]\\stackrel{\u00af}{x}[\/latex] (x-bar) to represent that value. We do not use [latex]\\stackrel{\u00af}{x}[\/latex] to represent the mean of a random variable; instead we use [latex]{\\mathrm{\u03bc}}_{x}[\/latex] (pronounced \u201cmu-sub-x\u201d).<\/p>\n<p>Here is the formula that we have come up with for the mean of a discrete random variable. Note that [latex]P(x)[\/latex] represents the probability of <em>x<\/em>, where <em>x<\/em> is a value of the random variable <em>X<\/em>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032032\/m6_probability_topic_6_1_6_1DiscreteRandomVariables2of4_image10.png\" alt=\"Equation: The mean equals the sum of all the values of x times their probabilities.\" width=\"486\" height=\"156\" \/><\/p>\n<p>Another term often used to describe the mean is <strong>expected value<\/strong>. It is a useful term because it reminds us that the mean of a random variable is not calculated on a fixed data set. Rather, the mean (expected value) is a measure of the expected long-term behavior of the random variable.<\/p>\n<div class=\"textbox exercises\">\n<h3>Learn By Doing<\/h3>\n<p>Drivers entering the short-term parking facility at an airport are given the option to purchase a parking permit for one of four possible time periods: \u00bd hour, 1 hour, 1\u00bd hours, or 2 hours. Thus, for each driver who enters the parking facility, we can consider their choice of parking time as a discrete random variable. In this case, the random variable <em>X<\/em> has four possible values: 0.5, 1, 1.5, and 2.<\/p>\n<p>Assume that the probability distribution for <em>X<\/em> is given by the following table.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15032034\/m6_probability_topic_6_1_6_1DiscreteRandomVariables2of4_image11.png\" alt=\"table and histogram that show x=parking time in hours, with four possible times in hours (.5, 1, 1.5, and 2 hours) they might choose to park.\" width=\"640\" height=\"207\" \/><\/p>\n<p>For example, reading from this table, it appears that there is a 15% chance that the next driver entering the parking facility will opt for a \u00bd-hour permit. In the probability histogram, the area of each rectangle (not the height) is the probability of the corresponding <em>x<\/em>-value occurring.<\/p>\n<p>\t<iframe id=\"lumen_assessment_3558\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3558&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3558\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3559\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3559&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3559\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-280\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Concepts in Statistics. <strong>Provided by<\/strong>: Open Learning Initiative. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/oli.cmu.edu\">http:\/\/oli.cmu.edu<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":163,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Concepts in Statistics\",\"author\":\"\",\"organization\":\"Open Learning Initiative\",\"url\":\"http:\/\/oli.cmu.edu\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"8f648af7-5196-4bb9-ad2c-dfac537bff39","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-280","chapter","type-chapter","status-web-only","hentry"],"part":258,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/280","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/users\/163"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/280\/revisions"}],"predecessor-version":[{"id":1539,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/280\/revisions\/1539"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/parts\/258"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/280\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/media?parent=280"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapter-type?post=280"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/contributor?post=280"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/license?post=280"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}