{"id":241,"date":"2023-12-21T11:05:31","date_gmt":"2023-12-21T16:05:31","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=241"},"modified":"2024-06-17T23:55:36","modified_gmt":"2024-06-18T03:55:36","slug":"point-and-interval-estimates","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/point-and-interval-estimates\/","title":{"raw":"Sample Versus Population Metrics","rendered":"Sample Versus Population Metrics"},"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\nIn this section, you will learn:\r\n<ul>\r\n \t<li>Why it matters to differentiate between samples and populations<\/li>\r\n \t<li>Some population and sample parameters of interest<\/li>\r\n \t<li>When sampling should be used<\/li>\r\n \t<li>The difference between a sample and a census<\/li>\r\n \t<li>How to measure population characteristics using sampling<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<h2>Sample vs Population<\/h2>\r\nUp to this point in time, it really has not mattered if we were dealing with a sample or a population. The only difference in this course so far has been:\r\n<ul>\r\n \t<li>when calculating the standard deviation<\/li>\r\n \t<li>there were different formulas for calculating the standard deviation of a population<\/li>\r\n \t<li>compared to calculating the standard deviation of a sample.<\/li>\r\n<\/ul>\r\n<h1>Why it Matters to Differentiate Samples and Populations<\/h1>\r\nIt matters that we differentiate between sample and populations because of three important reasons:\r\n<ol>\r\n \t<li>You will be working with problems that contain information about the sample <em>and<\/em> the population.<\/li>\r\n \t<li>If you don't clearly distinguish between the two, you can't possibly be successful solving the problems.<\/li>\r\n \t<li>If you don't distinguish the two elements, you probably don't really understand why we're solving the problem in the first place!<\/li>\r\n<\/ol>\r\n<h1>Population and Sample Parameters<\/h1>\r\nFigure 43.1 below illustrates conceptually what we're trying to do with sampling:\r\n<ul>\r\n \t<li>The large circle on the left depicts the population (these are things we really want to know).<\/li>\r\n \t<li>The symbols inside represent population characteristics or <strong><em>population <\/em><\/strong><strong><em>parameters <\/em><\/strong>(these represent characteristics of the population and are considered to be the truth of the population)<\/li>\r\n \t<li>For example, what do BCIT students really think about something? Do they really believe that Global Warming is a problem or not?<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_2176\" align=\"aligncenter\" width=\"390\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/SamplevsPopulation.jpg\"><img class=\"wp-image-2176 \" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/SamplevsPopulation.jpg\" alt=\"Figure comparing sample versus population characteristics, and metrics.\" width=\"390\" height=\"160\" \/><\/a> Figure 43.1 Population versus Sample Characteristics[\/caption]\r\n<h2>Population Characteristics of Interest<\/h2>\r\nSome population characteristics that we are interested in include:\r\n<ul>\r\n \t<li>the population mean: [latex]\\mu[\/latex] (stated as \"mu\")<\/li>\r\n \t<li>the population standard deviation: [latex]\\sigma[\/latex] (stated as \"sigma\")<\/li>\r\n \t<li>the population slope: [latex]\\beta[\/latex] (stated as \"beta\")<\/li>\r\n \t<li>the population percentage: [latex]\\pi[\/latex] (stated as \"pie\")<\/li>\r\n<\/ul>\r\nIn the situation where the population is rather small:\r\n<ul>\r\n \t<li>It may be possible to measure such characteristics directly.<\/li>\r\n \t<li>But this is not where sampling is used.<\/li>\r\n<\/ul>\r\n<h2>Sample Characteristics Recorded<\/h2>\r\nThe most common sample metrics or (statistics) that we will record are:\r\n<ul>\r\n \t<li>the sample mean: [latex]\\overline{x}[\/latex] (stated as \"x-bar\")<\/li>\r\n \t<li>the sample standard deviation: [latex]s[\/latex]<\/li>\r\n \t<li>the sample slope: [latex]b_1[\/latex]<\/li>\r\n \t<li>the sample percentage: [latex]\\overline{p}[\/latex] (stated as \"p-bar\")<\/li>\r\n<\/ul>\r\n<h1>When is Sampling Used?<\/h1>\r\n<h2>For large populations<\/h2>\r\n<ul>\r\n \t<li>We might not have the resources to collect data from the entire population.<\/li>\r\n \t<li>It is especially problematic if the population is also dynamic in nature.<\/li>\r\n \t<li><span style=\"text-align: initial\">Even if we did have the resources to measure the entire population, by the time we finished measuring it, the population has already changed!<\/span><\/li>\r\n<\/ul>\r\n<h2>Example of Large and Dynamic Population<\/h2>\r\nOur previous Global Warming example would be relevant:\r\n<ul>\r\n \t<li>If we did speak to everyone at BCIT about their belief in global warming,<\/li>\r\n \t<li>By the time we finished collecting all the data,<\/li>\r\n \t<li>It is quite possible that opinions have changed.<\/li>\r\n<\/ul>\r\n<h2>When to Sample<\/h2>\r\nFor the reasons discussed above, sampling is a technique which is applied to study and measure characteristics of populations whereby the population\u00a0 itself is typical large and changing in nature.\r\n<h1>Census vs Sample<\/h1>\r\n\"While a census is an attempt to gather information about every member of the population, sampling gathers information only about a part, the sample, to represent the whole. Because a sample is only part of the population, we can study it more extensively than we can all of the members of the population.\"\r\n\r\n<a href=\"https:\/\/www.learner.org\/wp-content\/uploads\/2019\/03\/AgainstAllOdds_StudentGuide_Unit16-Census-and-Sampling.pdf\">https:\/\/www.learner.org\/wp-content\/uploads\/2019\/03\/AgainstAllOdds_StudentGuide_Unit16-Census-and-Sampling.pdf<\/a>\r\n<h1>How to Measure Population Characteristics Using Sampling<\/h1>\r\nSo how do we measure population characteristics using sampling?\r\n<ol>\r\n \t<li>First extract a portion of the data from the population (i.e., a representative truly random and unbiased sample). This is depicted by the arrow and smaller circle in Figure 43.1 that illustrates a representative sample from the population.<\/li>\r\n \t<li>Next, we calculate the sample characteristics (called \"statistics\") that we're interested in.<\/li>\r\n \t<li>Once we have the statistics we must close the loop (as shown in Figure 43.1) and use our statistics to \"estimate\" the population characteristics (i.e., parameters).<\/li>\r\n<\/ol>\r\nIn other words, the statistics are not what we really want - that's what we have - so we must close the loop by using the statistics to understand what the population parameters are.\r\n<h2>Estimating Population Parameters<\/h2>\r\nThe next several chapters of this textbook focus on what is called \"Estimation.\" We will be using sampling statistics to <strong><em>estimate <\/em><\/strong>population parameters for means and proportions.\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: An Introduction to Sampling<\/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=\"156\"]\r\n\r\n[h5p id=\"157\"]\r\n\r\n[h5p id=\"158\"]\r\n\r\nClick the sections below to reveal the solutions to the above exercises\r\n\r\n[h5p id=\"159\"]\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>In this section, you will learn:<\/p>\n<ul>\n<li>Why it matters to differentiate between samples and populations<\/li>\n<li>Some population and sample parameters of interest<\/li>\n<li>When sampling should be used<\/li>\n<li>The difference between a sample and a census<\/li>\n<li>How to measure population characteristics using sampling<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h2>Sample vs Population<\/h2>\n<p>Up to this point in time, it really has not mattered if we were dealing with a sample or a population. The only difference in this course so far has been:<\/p>\n<ul>\n<li>when calculating the standard deviation<\/li>\n<li>there were different formulas for calculating the standard deviation of a population<\/li>\n<li>compared to calculating the standard deviation of a sample.<\/li>\n<\/ul>\n<h1>Why it Matters to Differentiate Samples and Populations<\/h1>\n<p>It matters that we differentiate between sample and populations because of three important reasons:<\/p>\n<ol>\n<li>You will be working with problems that contain information about the sample <em>and<\/em> the population.<\/li>\n<li>If you don&#8217;t clearly distinguish between the two, you can&#8217;t possibly be successful solving the problems.<\/li>\n<li>If you don&#8217;t distinguish the two elements, you probably don&#8217;t really understand why we&#8217;re solving the problem in the first place!<\/li>\n<\/ol>\n<h1>Population and Sample Parameters<\/h1>\n<p>Figure 43.1 below illustrates conceptually what we&#8217;re trying to do with sampling:<\/p>\n<ul>\n<li>The large circle on the left depicts the population (these are things we really want to know).<\/li>\n<li>The symbols inside represent population characteristics or <strong><em>population <\/em><\/strong><strong><em>parameters <\/em><\/strong>(these represent characteristics of the population and are considered to be the truth of the population)<\/li>\n<li>For example, what do BCIT students really think about something? Do they really believe that Global Warming is a problem or not?<\/li>\n<\/ul>\n<figure id=\"attachment_2176\" aria-describedby=\"caption-attachment-2176\" style=\"width: 390px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/SamplevsPopulation.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2176\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/SamplevsPopulation.jpg\" alt=\"Figure comparing sample versus population characteristics, and metrics.\" width=\"390\" height=\"160\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/SamplevsPopulation.jpg 898w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/SamplevsPopulation-300x123.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/SamplevsPopulation-768x314.jpg 768w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/SamplevsPopulation-65x27.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/SamplevsPopulation-225x92.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2023\/12\/SamplevsPopulation-350x143.jpg 350w\" sizes=\"auto, (max-width: 390px) 100vw, 390px\" \/><\/a><figcaption id=\"caption-attachment-2176\" class=\"wp-caption-text\">Figure 43.1 Population versus Sample Characteristics<\/figcaption><\/figure>\n<h2>Population Characteristics of Interest<\/h2>\n<p>Some population characteristics that we are interested in include:<\/p>\n<ul>\n<li>the population mean: [latex]\\mu[\/latex] (stated as &#8220;mu&#8221;)<\/li>\n<li>the population standard deviation: [latex]\\sigma[\/latex] (stated as &#8220;sigma&#8221;)<\/li>\n<li>the population slope: [latex]\\beta[\/latex] (stated as &#8220;beta&#8221;)<\/li>\n<li>the population percentage: [latex]\\pi[\/latex] (stated as &#8220;pie&#8221;)<\/li>\n<\/ul>\n<p>In the situation where the population is rather small:<\/p>\n<ul>\n<li>It may be possible to measure such characteristics directly.<\/li>\n<li>But this is not where sampling is used.<\/li>\n<\/ul>\n<h2>Sample Characteristics Recorded<\/h2>\n<p>The most common sample metrics or (statistics) that we will record are:<\/p>\n<ul>\n<li>the sample mean: [latex]\\overline{x}[\/latex] (stated as &#8220;x-bar&#8221;)<\/li>\n<li>the sample standard deviation: [latex]s[\/latex]<\/li>\n<li>the sample slope: [latex]b_1[\/latex]<\/li>\n<li>the sample percentage: [latex]\\overline{p}[\/latex] (stated as &#8220;p-bar&#8221;)<\/li>\n<\/ul>\n<h1>When is Sampling Used?<\/h1>\n<h2>For large populations<\/h2>\n<ul>\n<li>We might not have the resources to collect data from the entire population.<\/li>\n<li>It is especially problematic if the population is also dynamic in nature.<\/li>\n<li><span style=\"text-align: initial\">Even if we did have the resources to measure the entire population, by the time we finished measuring it, the population has already changed!<\/span><\/li>\n<\/ul>\n<h2>Example of Large and Dynamic Population<\/h2>\n<p>Our previous Global Warming example would be relevant:<\/p>\n<ul>\n<li>If we did speak to everyone at BCIT about their belief in global warming,<\/li>\n<li>By the time we finished collecting all the data,<\/li>\n<li>It is quite possible that opinions have changed.<\/li>\n<\/ul>\n<h2>When to Sample<\/h2>\n<p>For the reasons discussed above, sampling is a technique which is applied to study and measure characteristics of populations whereby the population\u00a0 itself is typical large and changing in nature.<\/p>\n<h1>Census vs Sample<\/h1>\n<p>&#8220;While a census is an attempt to gather information about every member of the population, sampling gathers information only about a part, the sample, to represent the whole. Because a sample is only part of the population, we can study it more extensively than we can all of the members of the population.&#8221;<\/p>\n<p><a href=\"https:\/\/www.learner.org\/wp-content\/uploads\/2019\/03\/AgainstAllOdds_StudentGuide_Unit16-Census-and-Sampling.pdf\">https:\/\/www.learner.org\/wp-content\/uploads\/2019\/03\/AgainstAllOdds_StudentGuide_Unit16-Census-and-Sampling.pdf<\/a><\/p>\n<h1>How to Measure Population Characteristics Using Sampling<\/h1>\n<p>So how do we measure population characteristics using sampling?<\/p>\n<ol>\n<li>First extract a portion of the data from the population (i.e., a representative truly random and unbiased sample). This is depicted by the arrow and smaller circle in Figure 43.1 that illustrates a representative sample from the population.<\/li>\n<li>Next, we calculate the sample characteristics (called &#8220;statistics&#8221;) that we&#8217;re interested in.<\/li>\n<li>Once we have the statistics we must close the loop (as shown in Figure 43.1) and use our statistics to &#8220;estimate&#8221; the population characteristics (i.e., parameters).<\/li>\n<\/ol>\n<p>In other words, the statistics are not what we really want &#8211; that&#8217;s what we have &#8211; so we must close the loop by using the statistics to understand what the population parameters are.<\/p>\n<h2>Estimating Population Parameters<\/h2>\n<p>The next several chapters of this textbook focus on what is called &#8220;Estimation.&#8221; We will be using sampling statistics to <strong><em>estimate <\/em><\/strong>population parameters for means and proportions.<\/p>\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: An Introduction to Sampling<\/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-156\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-156\" class=\"h5p-iframe\" data-content-id=\"156\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Sample Versus Population Metrics Key Takeaways 1\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-157\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-157\" class=\"h5p-iframe\" data-content-id=\"157\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Sample Versus Population Metrics Key Takeaways 2\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-158\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-158\" class=\"h5p-iframe\" data-content-id=\"158\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Sample Versus Population Metrics Key Takeaways 3\"><\/iframe><\/div>\n<\/div>\n<p>Click the sections below to reveal the solutions to the above exercises<\/p>\n<div id=\"h5p-159\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-159\" class=\"h5p-iframe\" data-content-id=\"159\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Sample Versus Population Metrics 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":883,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-241","chapter","type-chapter","status-publish","hentry"],"part":320,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/241","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\/883"}],"version-history":[{"count":16,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/241\/revisions"}],"predecessor-version":[{"id":2188,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/241\/revisions\/2188"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/320"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/241\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=241"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=241"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=241"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=241"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}