{"id":307,"date":"2016-10-10T19:56:27","date_gmt":"2016-10-10T19:56:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/math4libarts\/?post_type=chapter&#038;p=307"},"modified":"2019-05-30T16:58:13","modified_gmt":"2019-05-30T16:58:13","slug":"sampling-methods","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/sampling-methods\/","title":{"raw":"Sampling Methods and Bias","rendered":"Sampling Methods and Bias"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify\u00a0methods for obtaining a random sample of the intended population of a study<\/li>\r\n \t<li>Identify ineffective ways of obtaining a random sample from a population<\/li>\r\n \t<li>Identify types of sample bias<\/li>\r\n \t<li>Identify the differences between observational study and an experiment<\/li>\r\n \t<li>Identify the treatment in an experiment<\/li>\r\n \t<li>Determine whether an experiment may have been influenced by confounding<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Selecting a Population<\/h2>\r\nSuppose we are hired by a politician to determine the amount of support he has among the electorate should he decide to run for another term.\u00a0 What population should we study? Every person in the district? Not every person is eligible to vote, and regardless of how strongly someone likes or dislikes the candidate, they don't have much to do with him being re-elected if they are not able to vote.\r\n\r\nWhat about eligible voters in the district? That might be better, but if someone is eligible to vote but does not\u00a0register by the deadline, they won't have any say in the election either. What about registered voters? Many people are registered but choose not to vote. What about \"likely voters?\"\r\n\r\nThis is the criteria used in much political polling, but it is sometimes difficult to define a \"likely voter.\" Is it someone who voted in the last election? In the last general election? In the last presidential election? Should we consider someone who just turned 18 a \"likely voter?\" They weren't eligible to vote in the past, so how do we judge the likelihood that they will vote in the next election?\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/29220533\/crowd-1699137_640.jpg\"><img class=\"aligncenter size-full wp-image-1026\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/29220533\/crowd-1699137_640.jpg\" alt=\"rows of Lego figurines\" width=\"640\" height=\"360\" \/><\/a>\r\n\r\n&nbsp;\r\n\r\nIn November 1998, former professional wrestler Jesse \"The Body\" Ventura was elected governor of Minnesota. Up until right before the election, most polls showed he had little chance of winning. There were several contributing factors to the polls not reflecting the actual intent of the electorate:\r\n<ul>\r\n \t<li>Ventura was running on a third-party ticket and most polling methods are better suited to a two-candidate race.<\/li>\r\n \t<li>Many respondents to polls may have been embarrassed to tell pollsters that they were planning to vote for a professional wrestler.<\/li>\r\n \t<li>The mere fact that the polls showed Ventura had little chance of winning might have prompted some people to vote for him in protest to send a message to the major-party candidates.<\/li>\r\n<\/ul>\r\nBut one of the major contributing factors was that Ventura recruited a substantial amount of support from young people, particularly college students, who had never voted before and who registered specifically to vote in the gubernatorial election. The polls did not deem these young people likely voters (since in most cases young people have a lower rate of voter registration and a turnout rate for elections) and so the polling samples were subject to <strong>sampling bias<\/strong>: they omitted a portion of the electorate that was weighted in favor of the winning candidate.\r\n<div>\r\n<div class=\"textbox\">\r\n<div>\r\n<h3>Sampling bias<\/h3>\r\n<\/div>\r\n<div>\r\n\r\nA sampling method is biased if every member of the population doesn\u2019t have equal likelihood of being in the sample.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nSo even identifying the population can be a difficult job, but once we have identified the population, how do we choose an appropriate sample? Remember, although we would prefer to survey all members of the population, this is usually impractical unless the population is very small, so we choose a sample. There are many ways to sample a population, but there is one goal we need to keep in mind: we would like the sample to be <em>representative of the population<\/em>.\r\n\r\nReturning to our hypothetical job as a political pollster, we would not anticipate very accurate results if we drew all of our samples from among the customers at a Starbucks, nor would we expect that a sample drawn entirely from the membership list of the local Elks club would provide a useful picture of district-wide support for our candidate.\r\n\r\nOne way to ensure that the sample has a reasonable chance of mirroring the population is to employ <em>randomness<\/em>. The most basic random method is simple random sampling.\r\n<div>\r\n<div class=\"textbox\">\r\n<div>\r\n<h3>Simple random sample<\/h3>\r\n<\/div>\r\n<div>\r\n\r\nA <strong>random sample<\/strong> is one in which each member of the population has an equal probability of being chosen. A <strong>simple random sample<\/strong> is one in which every member of the population and any group of members has an equal probability of being chosen.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n<div>\r\n\r\nIf we could somehow identify all likely voters in the state, put each of their names on a piece of paper, toss the slips into a (very large) hat and draw 1000 slips out of the hat, we would have a simple random sample.\r\n\r\n<\/div>\r\nIn practice, computers are better suited for this sort of endeavor than millions of slips of paper and extremely large headgear.\r\n\r\n<\/div>\r\n<\/div>\r\nIt is always possible, however, that even a random sample might end up not being totally representative of the population.\u00a0 If we repeatedly take samples of 1000 people from among the population of likely voters in the state of Washington, some of these samples might tend to have a slightly higher percentage of Democrats (or Republicans) than does the general population; some samples might include more older people and some samples might include more younger people; etc. In most cases, this <strong>sampling variability<\/strong> is not significant.\r\n<div>\r\n<div class=\"textbox\">\r\n<div>\r\n<h3>Sampling variability<\/h3>\r\n<\/div>\r\n<div>\r\n\r\nThe natural variation of samples is called <strong>sampling variability<\/strong>.\r\n\r\nThis is unavoidable and expected in random sampling, and in most cases is not an issue.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nTo help account for variability, pollsters might instead use a <strong>stratified sample<\/strong>.\r\n<div>\r\n<div class=\"textbox\">\r\n<div>\r\n<h3>Stratified sampling<\/h3>\r\n<\/div>\r\n<div>\r\n\r\nIn <strong>stratified sampling<\/strong>, a population is divided into a number of subgroups (or strata). Random samples are then taken from each subgroup with sample sizes proportional to the size of the subgroup in the population.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n<div>\r\n\r\nSuppose in a particular state that previous data indicated that the electorate was comprised of 39% Democrats, 37% Republicans and 24% independents.\u00a0 In a sample of 1000 people, they would then expect to get about 390 Democrats, 370 Republicans and 240 independents.\u00a0 To accomplish this, they could randomly select 390 people from among those voters known to be Democrats, 370 from those known to be Republicans, and 240 from those with no party affiliation.\r\n\r\n<\/div>\r\n<\/div>\r\nStratified sampling can also be used to select a sample with people in desired age groups, a specified mix ratio of males and females, etc. A variation on this technique is called <strong>quota sampling<\/strong>.\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox\">\r\n<div>\r\n<h3>Quota sampling<\/h3>\r\n<\/div>\r\n<div>\r\n\r\n<strong>Quota sampling<\/strong> is a variation on stratified sampling, wherein samples are collected in each subgroup until the desired quota is met.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n<div>\r\n\r\nSuppose the pollsters call people at random, but once they have met their quota of 390 Democrats, they only gather people who do not identify themselves as a Democrat.\r\n\r\n<\/div>\r\nYou may have had the experience of being called by a telephone pollster who started by asking you your age, income, etc. and then thanked you for your time and hung up before asking any \"real\" questions. Most likely, they already had contacted enough people in your demographic group and were looking for people who were older or younger, richer or poorer, etc. Quota sampling is usually a bit easier than stratified sampling, but also does not ensure the same level of randomness.\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n\r\nAnother sampling method is <strong>cluster sampling<\/strong>, in which the population is divided into groups, and one or more groups are randomly selected to be in the sample.\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox\">\r\n<div>\r\n<h3>Cluster sampling<\/h3>\r\n<\/div>\r\n<div>\r\n\r\nIn <strong>cluster sampling<\/strong>, the population is divided into subgroups (clusters), and a set of subgroups are selected to be in the sample.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nIf the college wanted to survey students, since students are already divided into classes, they could randomly select 10 classes and give the survey to all the students in those classes. This would be cluster sampling.\r\n\r\n<\/div>\r\n<\/div>\r\nOther sampling methods include <strong>systematic sampling<\/strong>.\r\n<div>\r\n<div class=\"textbox\">\r\n<div>\r\n<h3>Systematic sampling<\/h3>\r\n<\/div>\r\n<div>\r\n\r\nIn <strong>systematic sampling<\/strong>, every <em>n<sup>th<\/sup><\/em> member of the population is selected to be in the sample.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nTo select a sample using systematic sampling, a pollster calls every 100th name in the phone book.\r\n\r\nSystematic sampling is not as random as a simple random sample (if your name is Albert Aardvark and your sister Alexis Aardvark is right after you in the phone book, there is no way you could both end up in the sample) but it can yield acceptable samples.\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>The Worst Way to Sample<\/h2>\r\nPerhaps the worst types of sampling methods are <strong>convenience samples<\/strong> and <strong>voluntary response samples<\/strong>.\r\n<div>\r\n<div class=\"textbox\">\r\n<div>\r\n<h3>Convenience sampling and voluntary response sampling<\/h3>\r\n<\/div>\r\n<div>\r\n\r\n<strong>Convenience sampling<\/strong> is the practice of samples chosen by selecting whoever is convenient.\r\n\r\n<strong>Voluntary response sampling<\/strong> is allowing the sample to volunteer.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nA pollster stands on a street corner and interviews the first 100 people who agree to speak to him. Which sampling method is represented by this scenario?\r\n[reveal-answer q=\"386299\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"386299\"]This is a convenience sample.[\/hidden-answer]\r\n<div>\r\n\r\n<hr \/>\r\n\r\nA website has a survey asking readers to give their opinion on a tax proposal. Which sampling method is represented?\r\n[reveal-answer q=\"503402\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"503402\"]\r\n<div>\r\n\r\nThis is a self-selected sample, or voluntary response sample, in which respondents volunteer to participate.\r\n\r\n<\/div>\r\nUsually voluntary response samples are skewed towards people who have a particularly strong opinion about the subject of the survey or who just have way too much time on their hands and enjoy taking surveys.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nWatch the following video for an overview of all the sampling methods discussed so far.\r\n\r\nhttps:\/\/youtu.be\/sk4TDJU7QbY\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nIn each case, indicate what sampling method was used\r\n\r\na. Every 4th person in the class was selected\r\n\r\nb. A sample was selected to contain 25 men and 35 women\r\n\r\nc. Viewers of a new show are asked to vote on the show\u2019s website\r\n\r\nd. A website randomly selects 50 of their customers to send a satisfaction survey to\r\n\r\ne. To survey voters in a town, a polling company randomly selects 10 city blocks, and interviews everyone who lives on those blocks.\r\n\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6733&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Problematic Sampling and Surveying<\/h2>\r\nThere are number of ways that a study can be ruined before you even start collecting data. The first we have already explored \u2013 <strong>sampling <\/strong>or<strong> selection bias<\/strong>, which is when the sample is not representative of the population. One example of this is <strong>voluntary response bias<\/strong>, which is bias introduced by only collecting data from those who volunteer to participate. This is not the only potential source of bias.\r\n<div class=\"textbox\">\r\n<h3>Sources of bias<\/h3>\r\n<ul>\r\n \t<li><strong>Sampling bias<\/strong> \u2013 when the sample is not representative of the population<\/li>\r\n \t<li><strong>Voluntary response bias<\/strong> \u2013 the sampling bias that often occurs when the sample is volunteers<\/li>\r\n \t<li><strong>Self-interest study<\/strong> \u2013 bias that can occur when the researchers have an interest in the outcome<\/li>\r\n \t<li><strong>Response bias<\/strong> \u2013 when the responder gives inaccurate responses for any reason<\/li>\r\n \t<li><strong>Perceived lack of anonymity<\/strong> \u2013 when the responder fears giving an honest answer might negatively affect them<\/li>\r\n \t<li><strong>Loaded questions<\/strong> \u2013 when the question wording influences the responses<\/li>\r\n \t<li><strong>Non-response bias<\/strong> \u2013 when people refusing to participate in the study can influence the validity of the outcome<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>examples<\/h3>\r\nConsider a recent study which found that chewing gum may raise math grades in teenagers[footnote]Reuters. <a href=\"http:\/\/news.yahoo.com\/s\/nm\/20090423\/od_uk_nm\/oukoe_uk_gum_learning\" target=\"_blank\" rel=\"noopener\">http:\/\/news.yahoo.com\/s\/nm\/20090423\/od_uk_nm\/oukoe_uk_gum_learning<\/a>. Retrieved 4\/27\/09[\/footnote]. This study was conducted by the Wrigley Science Institute, a branch of the Wrigley chewing gum company. Identify the type of sampling bias found in this example.\r\n[reveal-answer q=\"561139\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"561139\"]This is an example of a <strong>self-interest study<\/strong>; one in which the researchers have a vested interest in the outcome of the study. While this does not necessarily ensure that the study was biased, it certainly suggests that we should subject the study to extra scrutiny.[\/hidden-answer]\r\n<div>\r\n\r\n<hr \/>\r\n\r\nA survey asks people \u201cwhen was the last time you visited your doctor?\u201d What type of sampling bias might this lead to?\r\n[reveal-answer q=\"965739\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"965739\"]\r\n<div>\r\n\r\nThis might suffer from <strong>response bias<\/strong>, since many people might not remember exactly when they last saw a doctor and give inaccurate responses.\r\n\r\n<\/div>\r\nSources of response bias may be innocent, such as bad memory, or as intentional as pressuring by the pollster. Consider, for example, how many voting initiative petitions people sign without even reading them.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n<hr \/>\r\n\r\nA survey asks participants a question about their interactions with members of other races. Which sampling bias might occur for this survey strategy?\r\n[reveal-answer q=\"610809\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"610809\"]Here, a <strong>perceived lack of anonymity<\/strong> could influence the outcome. The respondent might not want to be perceived as racist even if they are, and give an untruthful answer.[\/hidden-answer]\r\n\r\n<hr \/>\r\n\r\nAn employer puts out a survey asking their employees if they have a drug abuse problem and need treatment help. Which sampling bias may occur in this scenario?\r\n[reveal-answer q=\"801450\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"801450\"]Here, answering truthfully might have consequences; responses might not be accurate if the employees do not feel their responses are anonymous or fear retribution from their employer. This survey has the\u00a0potential for\u00a0<strong>perceived lack of anonymity<\/strong>.[\/hidden-answer]\r\n<div>\r\n\r\n<hr \/>\r\n\r\nA survey asks \u201cdo you support funding research of alternative energy sources to reduce our reliance on high-polluting fossil fuels?\u201d Which sampling bias may result from this survey?\r\n[reveal-answer q=\"464953\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"464953\"]\r\n<div>\r\n\r\nThis is an example of a <strong>loaded <\/strong>or<strong> leading question<\/strong> \u2013 questions whose wording leads the respondent towards an answer.\r\n\r\n<\/div>\r\nLoaded questions can occur intentionally by pollsters with an agenda, or accidentally through poor question wording. Also a concern is <strong>question order<\/strong>, where the order of questions changes the results. A psychology researcher provides an example[footnote]Swartz, Norbert. <a href=\"http:\/\/www.umich.edu\/~newsinfo\/MT\/01\/Fal01\/mt6f01.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.umich.edu\/~newsinfo\/MT\/01\/Fal01\/mt6f01.html<\/a>. Retrieved 3\/31\/2009[\/footnote]:\r\n<blockquote>\u201cMy favorite finding is this: we did a study where we asked students, 'How satisfied are you with your life? How often do you have a date?' The two answers were not statistically related - you would conclude that there is no relationship between dating frequency and life satisfaction. But when we reversed the order and asked, 'How often do you have a date? How satisfied are you with your life?' the statistical relationship was a strong one. You would now conclude that there is nothing as important in a student's life as dating frequency.\u201d<\/blockquote>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n<hr \/>\r\n\r\nA telephone poll asks the question \u201cDo you often have time to relax and read a book?\u201d, and 50% of the people called refused to answer the survey. Which sampling bias is represented by this survey?\r\n[reveal-answer q=\"782411\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"782411\"]It is unlikely that the results will be representative of the entire population. This is an example of <strong>non-response bias<\/strong>, introduced by people refusing to participate in a study or dropping out of an experiment. When people refuse to participate, we can no longer be so certain that our sample is representative of the population.[\/hidden-answer]\r\n\r\n<\/div>\r\nThese problematic scenarios for statistics gathering are discussed further in the following video.\r\n\r\nhttps:\/\/youtu.be\/GXAi06PiVsk\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nIn each situation, identify a potential source of bias\r\n\r\na. A survey asks how many sexual partners a person has had in the last year\r\n\r\nb. A radio station asks readers to phone in their choice in a daily poll.\r\n\r\nc. A substitute teacher wants to know how students in the class did on their last test. The teacher asks the 10 students sitting in the front row to state their latest test score.\r\n\r\nd. High school students are asked if they have consumed alcohol in the last two weeks.\r\n\r\ne. The Beef Council releases a study stating that consuming red meat poses little cardiovascular risk.\r\n\r\nf. A poll asks \u201cDo you support a new transportation tax, or would you prefer to see our public transportation system fall apart?\u201d\r\n\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6727&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify\u00a0methods for obtaining a random sample of the intended population of a study<\/li>\n<li>Identify ineffective ways of obtaining a random sample from a population<\/li>\n<li>Identify types of sample bias<\/li>\n<li>Identify the differences between observational study and an experiment<\/li>\n<li>Identify the treatment in an experiment<\/li>\n<li>Determine whether an experiment may have been influenced by confounding<\/li>\n<\/ul>\n<\/div>\n<h2>Selecting a Population<\/h2>\n<p>Suppose we are hired by a politician to determine the amount of support he has among the electorate should he decide to run for another term.\u00a0 What population should we study? Every person in the district? Not every person is eligible to vote, and regardless of how strongly someone likes or dislikes the candidate, they don&#8217;t have much to do with him being re-elected if they are not able to vote.<\/p>\n<p>What about eligible voters in the district? That might be better, but if someone is eligible to vote but does not\u00a0register by the deadline, they won&#8217;t have any say in the election either. What about registered voters? Many people are registered but choose not to vote. What about &#8220;likely voters?&#8221;<\/p>\n<p>This is the criteria used in much political polling, but it is sometimes difficult to define a &#8220;likely voter.&#8221; Is it someone who voted in the last election? In the last general election? In the last presidential election? Should we consider someone who just turned 18 a &#8220;likely voter?&#8221; They weren&#8217;t eligible to vote in the past, so how do we judge the likelihood that they will vote in the next election?<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/29220533\/crowd-1699137_640.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1026\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/29220533\/crowd-1699137_640.jpg\" alt=\"rows of Lego figurines\" width=\"640\" height=\"360\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>In November 1998, former professional wrestler Jesse &#8220;The Body&#8221; Ventura was elected governor of Minnesota. Up until right before the election, most polls showed he had little chance of winning. There were several contributing factors to the polls not reflecting the actual intent of the electorate:<\/p>\n<ul>\n<li>Ventura was running on a third-party ticket and most polling methods are better suited to a two-candidate race.<\/li>\n<li>Many respondents to polls may have been embarrassed to tell pollsters that they were planning to vote for a professional wrestler.<\/li>\n<li>The mere fact that the polls showed Ventura had little chance of winning might have prompted some people to vote for him in protest to send a message to the major-party candidates.<\/li>\n<\/ul>\n<p>But one of the major contributing factors was that Ventura recruited a substantial amount of support from young people, particularly college students, who had never voted before and who registered specifically to vote in the gubernatorial election. The polls did not deem these young people likely voters (since in most cases young people have a lower rate of voter registration and a turnout rate for elections) and so the polling samples were subject to <strong>sampling bias<\/strong>: they omitted a portion of the electorate that was weighted in favor of the winning candidate.<\/p>\n<div>\n<div class=\"textbox\">\n<div>\n<h3>Sampling bias<\/h3>\n<\/div>\n<div>\n<p>A sampling method is biased if every member of the population doesn\u2019t have equal likelihood of being in the sample.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>So even identifying the population can be a difficult job, but once we have identified the population, how do we choose an appropriate sample? Remember, although we would prefer to survey all members of the population, this is usually impractical unless the population is very small, so we choose a sample. There are many ways to sample a population, but there is one goal we need to keep in mind: we would like the sample to be <em>representative of the population<\/em>.<\/p>\n<p>Returning to our hypothetical job as a political pollster, we would not anticipate very accurate results if we drew all of our samples from among the customers at a Starbucks, nor would we expect that a sample drawn entirely from the membership list of the local Elks club would provide a useful picture of district-wide support for our candidate.<\/p>\n<p>One way to ensure that the sample has a reasonable chance of mirroring the population is to employ <em>randomness<\/em>. The most basic random method is simple random sampling.<\/p>\n<div>\n<div class=\"textbox\">\n<div>\n<h3>Simple random sample<\/h3>\n<\/div>\n<div>\n<p>A <strong>random sample<\/strong> is one in which each member of the population has an equal probability of being chosen. A <strong>simple random sample<\/strong> is one in which every member of the population and any group of members has an equal probability of being chosen.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<div>\n<p>If we could somehow identify all likely voters in the state, put each of their names on a piece of paper, toss the slips into a (very large) hat and draw 1000 slips out of the hat, we would have a simple random sample.<\/p>\n<\/div>\n<p>In practice, computers are better suited for this sort of endeavor than millions of slips of paper and extremely large headgear.<\/p>\n<\/div>\n<\/div>\n<p>It is always possible, however, that even a random sample might end up not being totally representative of the population.\u00a0 If we repeatedly take samples of 1000 people from among the population of likely voters in the state of Washington, some of these samples might tend to have a slightly higher percentage of Democrats (or Republicans) than does the general population; some samples might include more older people and some samples might include more younger people; etc. In most cases, this <strong>sampling variability<\/strong> is not significant.<\/p>\n<div>\n<div class=\"textbox\">\n<div>\n<h3>Sampling variability<\/h3>\n<\/div>\n<div>\n<p>The natural variation of samples is called <strong>sampling variability<\/strong>.<\/p>\n<p>This is unavoidable and expected in random sampling, and in most cases is not an issue.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To help account for variability, pollsters might instead use a <strong>stratified sample<\/strong>.<\/p>\n<div>\n<div class=\"textbox\">\n<div>\n<h3>Stratified sampling<\/h3>\n<\/div>\n<div>\n<p>In <strong>stratified sampling<\/strong>, a population is divided into a number of subgroups (or strata). Random samples are then taken from each subgroup with sample sizes proportional to the size of the subgroup in the population.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<div>\n<p>Suppose in a particular state that previous data indicated that the electorate was comprised of 39% Democrats, 37% Republicans and 24% independents.\u00a0 In a sample of 1000 people, they would then expect to get about 390 Democrats, 370 Republicans and 240 independents.\u00a0 To accomplish this, they could randomly select 390 people from among those voters known to be Democrats, 370 from those known to be Republicans, and 240 from those with no party affiliation.<\/p>\n<\/div>\n<\/div>\n<p>Stratified sampling can also be used to select a sample with people in desired age groups, a specified mix ratio of males and females, etc. A variation on this technique is called <strong>quota sampling<\/strong>.<\/p>\n<\/div>\n<div>\n<div class=\"textbox\">\n<div>\n<h3>Quota sampling<\/h3>\n<\/div>\n<div>\n<p><strong>Quota sampling<\/strong> is a variation on stratified sampling, wherein samples are collected in each subgroup until the desired quota is met.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<div>\n<p>Suppose the pollsters call people at random, but once they have met their quota of 390 Democrats, they only gather people who do not identify themselves as a Democrat.<\/p>\n<\/div>\n<p>You may have had the experience of being called by a telephone pollster who started by asking you your age, income, etc. and then thanked you for your time and hung up before asking any &#8220;real&#8221; questions. Most likely, they already had contacted enough people in your demographic group and were looking for people who were older or younger, richer or poorer, etc. Quota sampling is usually a bit easier than stratified sampling, but also does not ensure the same level of randomness.<\/p>\n<\/div>\n<\/div>\n<div>\n<p>Another sampling method is <strong>cluster sampling<\/strong>, in which the population is divided into groups, and one or more groups are randomly selected to be in the sample.<\/p>\n<\/div>\n<div>\n<div class=\"textbox\">\n<div>\n<h3>Cluster sampling<\/h3>\n<\/div>\n<div>\n<p>In <strong>cluster sampling<\/strong>, the population is divided into subgroups (clusters), and a set of subgroups are selected to be in the sample.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>If the college wanted to survey students, since students are already divided into classes, they could randomly select 10 classes and give the survey to all the students in those classes. This would be cluster sampling.<\/p>\n<\/div>\n<\/div>\n<p>Other sampling methods include <strong>systematic sampling<\/strong>.<\/p>\n<div>\n<div class=\"textbox\">\n<div>\n<h3>Systematic sampling<\/h3>\n<\/div>\n<div>\n<p>In <strong>systematic sampling<\/strong>, every <em>n<sup>th<\/sup><\/em> member of the population is selected to be in the sample.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>To select a sample using systematic sampling, a pollster calls every 100th name in the phone book.<\/p>\n<p>Systematic sampling is not as random as a simple random sample (if your name is Albert Aardvark and your sister Alexis Aardvark is right after you in the phone book, there is no way you could both end up in the sample) but it can yield acceptable samples.<\/p>\n<\/div>\n<\/div>\n<h2>The Worst Way to Sample<\/h2>\n<p>Perhaps the worst types of sampling methods are <strong>convenience samples<\/strong> and <strong>voluntary response samples<\/strong>.<\/p>\n<div>\n<div class=\"textbox\">\n<div>\n<h3>Convenience sampling and voluntary response sampling<\/h3>\n<\/div>\n<div>\n<p><strong>Convenience sampling<\/strong> is the practice of samples chosen by selecting whoever is convenient.<\/p>\n<p><strong>Voluntary response sampling<\/strong> is allowing the sample to volunteer.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>A pollster stands on a street corner and interviews the first 100 people who agree to speak to him. Which sampling method is represented by this scenario?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q386299\">Show Solution<\/span><\/p>\n<div id=\"q386299\" class=\"hidden-answer\" style=\"display: none\">This is a convenience sample.<\/div>\n<\/div>\n<div>\n<hr \/>\n<p>A website has a survey asking readers to give their opinion on a tax proposal. Which sampling method is represented?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q503402\">Show Solution<\/span><\/p>\n<div id=\"q503402\" class=\"hidden-answer\" style=\"display: none\">\n<div>\n<p>This is a self-selected sample, or voluntary response sample, in which respondents volunteer to participate.<\/p>\n<\/div>\n<p>Usually voluntary response samples are skewed towards people who have a particularly strong opinion about the subject of the survey or who just have way too much time on their hands and enjoy taking surveys.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video for an overview of all the sampling methods discussed so far.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Sampling methods\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/sk4TDJU7QbY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>In each case, indicate what sampling method was used<\/p>\n<p>a. Every 4th person in the class was selected<\/p>\n<p>b. A sample was selected to contain 25 men and 35 women<\/p>\n<p>c. Viewers of a new show are asked to vote on the show\u2019s website<\/p>\n<p>d. A website randomly selects 50 of their customers to send a satisfaction survey to<\/p>\n<p>e. To survey voters in a town, a polling company randomly selects 10 city blocks, and interviews everyone who lives on those blocks.<\/p>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6733&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Problematic Sampling and Surveying<\/h2>\n<p>There are number of ways that a study can be ruined before you even start collecting data. The first we have already explored \u2013 <strong>sampling <\/strong>or<strong> selection bias<\/strong>, which is when the sample is not representative of the population. One example of this is <strong>voluntary response bias<\/strong>, which is bias introduced by only collecting data from those who volunteer to participate. This is not the only potential source of bias.<\/p>\n<div class=\"textbox\">\n<h3>Sources of bias<\/h3>\n<ul>\n<li><strong>Sampling bias<\/strong> \u2013 when the sample is not representative of the population<\/li>\n<li><strong>Voluntary response bias<\/strong> \u2013 the sampling bias that often occurs when the sample is volunteers<\/li>\n<li><strong>Self-interest study<\/strong> \u2013 bias that can occur when the researchers have an interest in the outcome<\/li>\n<li><strong>Response bias<\/strong> \u2013 when the responder gives inaccurate responses for any reason<\/li>\n<li><strong>Perceived lack of anonymity<\/strong> \u2013 when the responder fears giving an honest answer might negatively affect them<\/li>\n<li><strong>Loaded questions<\/strong> \u2013 when the question wording influences the responses<\/li>\n<li><strong>Non-response bias<\/strong> \u2013 when people refusing to participate in the study can influence the validity of the outcome<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>examples<\/h3>\n<p>Consider a recent study which found that chewing gum may raise math grades in teenagers<a class=\"footnote\" title=\"Reuters. http:\/\/news.yahoo.com\/s\/nm\/20090423\/od_uk_nm\/oukoe_uk_gum_learning. Retrieved 4\/27\/09\" id=\"return-footnote-307-1\" href=\"#footnote-307-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>. This study was conducted by the Wrigley Science Institute, a branch of the Wrigley chewing gum company. Identify the type of sampling bias found in this example.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q561139\">Show Solution<\/span><\/p>\n<div id=\"q561139\" class=\"hidden-answer\" style=\"display: none\">This is an example of a <strong>self-interest study<\/strong>; one in which the researchers have a vested interest in the outcome of the study. While this does not necessarily ensure that the study was biased, it certainly suggests that we should subject the study to extra scrutiny.<\/div>\n<\/div>\n<div>\n<hr \/>\n<p>A survey asks people \u201cwhen was the last time you visited your doctor?\u201d What type of sampling bias might this lead to?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q965739\">Show Solution<\/span><\/p>\n<div id=\"q965739\" class=\"hidden-answer\" style=\"display: none\">\n<div>\n<p>This might suffer from <strong>response bias<\/strong>, since many people might not remember exactly when they last saw a doctor and give inaccurate responses.<\/p>\n<\/div>\n<p>Sources of response bias may be innocent, such as bad memory, or as intentional as pressuring by the pollster. Consider, for example, how many voting initiative petitions people sign without even reading them.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<hr \/>\n<p>A survey asks participants a question about their interactions with members of other races. Which sampling bias might occur for this survey strategy?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q610809\">Show Solution<\/span><\/p>\n<div id=\"q610809\" class=\"hidden-answer\" style=\"display: none\">Here, a <strong>perceived lack of anonymity<\/strong> could influence the outcome. The respondent might not want to be perceived as racist even if they are, and give an untruthful answer.<\/div>\n<\/div>\n<hr \/>\n<p>An employer puts out a survey asking their employees if they have a drug abuse problem and need treatment help. Which sampling bias may occur in this scenario?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q801450\">Show Solution<\/span><\/p>\n<div id=\"q801450\" class=\"hidden-answer\" style=\"display: none\">Here, answering truthfully might have consequences; responses might not be accurate if the employees do not feel their responses are anonymous or fear retribution from their employer. This survey has the\u00a0potential for\u00a0<strong>perceived lack of anonymity<\/strong>.<\/div>\n<\/div>\n<div>\n<hr \/>\n<p>A survey asks \u201cdo you support funding research of alternative energy sources to reduce our reliance on high-polluting fossil fuels?\u201d Which sampling bias may result from this survey?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q464953\">Show Solution<\/span><\/p>\n<div id=\"q464953\" class=\"hidden-answer\" style=\"display: none\">\n<div>\n<p>This is an example of a <strong>loaded <\/strong>or<strong> leading question<\/strong> \u2013 questions whose wording leads the respondent towards an answer.<\/p>\n<\/div>\n<p>Loaded questions can occur intentionally by pollsters with an agenda, or accidentally through poor question wording. Also a concern is <strong>question order<\/strong>, where the order of questions changes the results. A psychology researcher provides an example<a class=\"footnote\" title=\"Swartz, Norbert. http:\/\/www.umich.edu\/~newsinfo\/MT\/01\/Fal01\/mt6f01.html. Retrieved 3\/31\/2009\" id=\"return-footnote-307-2\" href=\"#footnote-307-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>:<\/p>\n<blockquote><p>\u201cMy favorite finding is this: we did a study where we asked students, &#8216;How satisfied are you with your life? How often do you have a date?&#8217; The two answers were not statistically related &#8211; you would conclude that there is no relationship between dating frequency and life satisfaction. But when we reversed the order and asked, &#8216;How often do you have a date? How satisfied are you with your life?&#8217; the statistical relationship was a strong one. You would now conclude that there is nothing as important in a student&#8217;s life as dating frequency.\u201d<\/p><\/blockquote>\n<\/div>\n<\/div>\n<\/div>\n<hr \/>\n<p>A telephone poll asks the question \u201cDo you often have time to relax and read a book?\u201d, and 50% of the people called refused to answer the survey. Which sampling bias is represented by this survey?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q782411\">Show Solution<\/span><\/p>\n<div id=\"q782411\" class=\"hidden-answer\" style=\"display: none\">It is unlikely that the results will be representative of the entire population. This is an example of <strong>non-response bias<\/strong>, introduced by people refusing to participate in a study or dropping out of an experiment. When people refuse to participate, we can no longer be so certain that our sample is representative of the population.<\/div>\n<\/div>\n<\/div>\n<p>These problematic scenarios for statistics gathering are discussed further in the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Bias\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GXAi06PiVsk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>In each situation, identify a potential source of bias<\/p>\n<p>a. A survey asks how many sexual partners a person has had in the last year<\/p>\n<p>b. A radio station asks readers to phone in their choice in a daily poll.<\/p>\n<p>c. A substitute teacher wants to know how students in the class did on their last test. The teacher asks the 10 students sitting in the front row to state their latest test score.<\/p>\n<p>d. High school students are asked if they have consumed alcohol in the last two weeks.<\/p>\n<p>e. The Beef Council releases a study stating that consuming red meat poses little cardiovascular risk.<\/p>\n<p>f. A poll asks \u201cDo you support a new transportation tax, or would you prefer to see our public transportation system fall apart?\u201d<\/p>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=6727&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\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-307\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Sampling Methods. <strong>Authored by<\/strong>: David Lippman. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <strong>Project<\/strong>: Math in Society. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>crowd-lego-staff-choice-selector. <strong>Authored by<\/strong>: mwewering. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/pixabay.com\/en\/crowd-lego-staff-choice-selector-1699137\/\">https:\/\/pixabay.com\/en\/crowd-lego-staff-choice-selector-1699137\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/cc0\">CC0: No Rights Reserved<\/a><\/em><\/li><li>Sampling methods. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/sk4TDJU7QbY\">https:\/\/youtu.be\/sk4TDJU7QbY<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 6733, 6727. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY +GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-307-1\">Reuters. <a href=\"http:\/\/news.yahoo.com\/s\/nm\/20090423\/od_uk_nm\/oukoe_uk_gum_learning\" target=\"_blank\" rel=\"noopener\">http:\/\/news.yahoo.com\/s\/nm\/20090423\/od_uk_nm\/oukoe_uk_gum_learning<\/a>. Retrieved 4\/27\/09 <a href=\"#return-footnote-307-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-307-2\">Swartz, Norbert. <a href=\"http:\/\/www.umich.edu\/~newsinfo\/MT\/01\/Fal01\/mt6f01.html\" target=\"_blank\" rel=\"noopener\">http:\/\/www.umich.edu\/~newsinfo\/MT\/01\/Fal01\/mt6f01.html<\/a>. Retrieved 3\/31\/2009 <a href=\"#return-footnote-307-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":20,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Sampling Methods\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"Math in Society\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"crowd-lego-staff-choice-selector\",\"author\":\"mwewering\",\"organization\":\"\",\"url\":\"https:\/\/pixabay.com\/en\/crowd-lego-staff-choice-selector-1699137\/\",\"project\":\"\",\"license\":\"cc0\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Sampling methods\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/sk4TDJU7QbY\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 6733, 6727\",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY +GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"81b9ffbd-0bca-4913-af9b-6272f00fdb29","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-307","chapter","type-chapter","status-publish","hentry"],"part":299,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/307","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/users\/20"}],"version-history":[{"count":21,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/307\/revisions"}],"predecessor-version":[{"id":3026,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/307\/revisions\/3026"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/parts\/299"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/307\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/media?parent=307"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapter-type?post=307"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/contributor?post=307"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/license?post=307"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}