{"id":1973,"date":"2021-09-16T19:24:03","date_gmt":"2021-09-16T19:24:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1973"},"modified":"2022-02-18T05:23:42","modified_gmt":"2022-02-18T05:23:42","slug":"distinguishing-between-population-parameters-and-sample-statistics","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/distinguishing-between-population-parameters-and-sample-statistics\/","title":{"raw":"Distinguishing between Population Parameters and Sample Statistics","rendered":"Distinguishing between Population Parameters and Sample Statistics"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Distinguish between a population and a sample<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Distinguish between a parameter and a statistic<\/li>\r\n<\/ul>\r\n<\/div>\r\nRecall that a <em>population<\/em> is the entire group of individuals or objects that we want to study. Usually, it is not possible to study the whole population, so we collect data from a part of the population, called a <em>sample<\/em>. We use the sample to draw conclusions about the population.\r\n\r\nFor example, suppose our research question is \u201cWhat is the average amount of money spent on textbooks per semester by full-time students at Seattle Central?\u201d We cannot interview every full-time student at Seattle Central because it would take too much time and cost too much money. We therefore carefully select a sample of full-time students at Seattle Central to represent the population of all full-time students at that college. Then we collect data from the sample to estimate the average amount spent on textbooks.\r\n\r\nThis example illustrates how the research question guides the investigation. A well-stated research question contains information about:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The population (full-time students at Seattle Central).<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The information we will collect from each individual in the sample. We call this the <strong>variable<\/strong>. The variable is what we plan to measure (amount of money spent on textbooks per semester), and is often represented by <em>X<\/em>.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">A numerical characteristic about the population related to this variable (the <em>average<\/em> amount of money spent on textbooks per semester).<\/li>\r\n<\/ul>\r\nA numerical characteristic about a population is called a <strong>parameter<\/strong>. In the example above we are interested in the average, or mean amount of money all students spent on textbooks per semester.\u00a0 A population mean is represented by [latex]\\mu[\/latex].\r\n\r\nWe use information from the sample to estimate the population mean. A numerical characteristic of a sample is called a <strong>sample statistic<\/strong>. It seems natural that our first estimate of the population mean, [latex]\\mu[\/latex] is the sample mean [latex]\\overline{x}[\/latex]. If a statement refers to a mean or average, it indicates that [latex]\\mu[\/latex] or [latex]\\overline{x}[\/latex] is being considered.\r\n\r\nAnother type of research question about a population we might be interested in is whether the majority of students qualify for federal student loans.\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The population (full-time students at Seattle Central).<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The information we will collect from each individual in the sample is the <strong>variable<\/strong>. The variable is whether or not each student in the sample qualifies for federal student loans.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The numerical characteristic about the population related to this variable is the proportion of all students at Seattle Central that qualify for student loans.<\/li>\r\n<\/ul>\r\nA population proportion is represented by [latex]\\varrho[\/latex]. We\u00a0estimate a population proportion from a sample proportion. A sample proportion is represented by [latex]\\hat{\\varrho}[\/latex].\u00a0If a statement quotes a percentage or a fraction, a proportion is being considered. If 3 out of 5 people in our sample agree with a statement, this corresponds to the sample proportion [latex]\\hat{p} = \\frac{3}{5} = 0.6 = 60%[\/latex].\r\n\r\nSome examples of research questions and corresponding to variables of interest are given in the following table.\r\n<div align=\"left\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Research Question<\/strong><\/td>\r\n<td><strong>Population Parameter and Notation<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>What is the average number of hours students work per week?<\/td>\r\n<td>Mean, [latex]\\mu[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>What percentage of students commute?<\/td>\r\n<td>Proportion, [latex]\\varrho[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Do the majority of students participate in athletics?<\/td>\r\n<td>Proportion, [latex]\\varrho[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Do athletes have a higher grade point average than non-athletes?<\/td>\r\n<td>Means, [latex]\\mu _\\mathrm{athletes} \\ \\mathrm{and} \\ \\mu _\\mathrm{nonathletes} [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn deciding whether information refers to a sample or a population, consider whether all members of the population have been included. If so, a numerical characteristic refers to a population parameter. The word \"every\" or \"each\" in a statement can be a clue that an entire population has been included.\r\n\r\nIf only a portion of the population has been included, a numerical characteristic refers to a sample statistic. If a statement refers to a relatively small number of what would be a large population, the information likely refers to a sample.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDetermine whether the number refers to a population parameter or a sample statistic.\r\n<ol>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Based on records of all employees at a small company, the average annual salary is $52,539.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">In a survey of 50 athletes at a large university, 78% said they were happy they chose to continue competing at the collegiate level.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">In a study of 63 law firms in the United States, the average hourly billing rate was $279.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">706 of the 2,223 passengers on the Titanic survived.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"729725\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"729725\"]\r\n<ol>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Since the statement referred to an average calculated on all employee salaries, $52,539 refers to the population mean, [latex]\\mu[\/latex].<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Since a large university likely has many more than 50 athletes, and the statement refers to a survey, 78% refers to a sample statistic, [latex]\\hat{p}[\/latex].<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">There are many law firms in the United States, so the 63 law firms represent a sample. Then the average hourly billing rate, $279, represents a sample mean [latex]\\overline{x}[\/latex].<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The statement refers to all passengers on the Titanic, so 706 out of 2,223, or 31.8%, refers to a population proportion, [latex]\\varrho[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]33101[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Distinguish between a population and a sample<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Distinguish between a parameter and a statistic<\/li>\n<\/ul>\n<\/div>\n<p>Recall that a <em>population<\/em> is the entire group of individuals or objects that we want to study. Usually, it is not possible to study the whole population, so we collect data from a part of the population, called a <em>sample<\/em>. We use the sample to draw conclusions about the population.<\/p>\n<p>For example, suppose our research question is \u201cWhat is the average amount of money spent on textbooks per semester by full-time students at Seattle Central?\u201d We cannot interview every full-time student at Seattle Central because it would take too much time and cost too much money. We therefore carefully select a sample of full-time students at Seattle Central to represent the population of all full-time students at that college. Then we collect data from the sample to estimate the average amount spent on textbooks.<\/p>\n<p>This example illustrates how the research question guides the investigation. A well-stated research question contains information about:<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The population (full-time students at Seattle Central).<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The information we will collect from each individual in the sample. We call this the <strong>variable<\/strong>. The variable is what we plan to measure (amount of money spent on textbooks per semester), and is often represented by <em>X<\/em>.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">A numerical characteristic about the population related to this variable (the <em>average<\/em> amount of money spent on textbooks per semester).<\/li>\n<\/ul>\n<p>A numerical characteristic about a population is called a <strong>parameter<\/strong>. In the example above we are interested in the average, or mean amount of money all students spent on textbooks per semester.\u00a0 A population mean is represented by [latex]\\mu[\/latex].<\/p>\n<p>We use information from the sample to estimate the population mean. A numerical characteristic of a sample is called a <strong>sample statistic<\/strong>. It seems natural that our first estimate of the population mean, [latex]\\mu[\/latex] is the sample mean [latex]\\overline{x}[\/latex]. If a statement refers to a mean or average, it indicates that [latex]\\mu[\/latex] or [latex]\\overline{x}[\/latex] is being considered.<\/p>\n<p>Another type of research question about a population we might be interested in is whether the majority of students qualify for federal student loans.<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The population (full-time students at Seattle Central).<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The information we will collect from each individual in the sample is the <strong>variable<\/strong>. The variable is whether or not each student in the sample qualifies for federal student loans.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The numerical characteristic about the population related to this variable is the proportion of all students at Seattle Central that qualify for student loans.<\/li>\n<\/ul>\n<p>A population proportion is represented by [latex]\\varrho[\/latex]. We\u00a0estimate a population proportion from a sample proportion. A sample proportion is represented by [latex]\\hat{\\varrho}[\/latex].\u00a0If a statement quotes a percentage or a fraction, a proportion is being considered. If 3 out of 5 people in our sample agree with a statement, this corresponds to the sample proportion [latex]\\hat{p} = \\frac{3}{5} = 0.6 = 60%[\/latex].<\/p>\n<p>Some examples of research questions and corresponding to variables of interest are given in the following table.<\/p>\n<div style=\"text-align: left;\">\n<table>\n<tbody>\n<tr>\n<td><strong>Research Question<\/strong><\/td>\n<td><strong>Population Parameter and Notation<\/strong><\/td>\n<\/tr>\n<tr>\n<td>What is the average number of hours students work per week?<\/td>\n<td>Mean, [latex]\\mu[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>What percentage of students commute?<\/td>\n<td>Proportion, [latex]\\varrho[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Do the majority of students participate in athletics?<\/td>\n<td>Proportion, [latex]\\varrho[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Do athletes have a higher grade point average than non-athletes?<\/td>\n<td>Means, [latex]\\mu _\\mathrm{athletes} \\ \\mathrm{and} \\ \\mu _\\mathrm{nonathletes}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In deciding whether information refers to a sample or a population, consider whether all members of the population have been included. If so, a numerical characteristic refers to a population parameter. The word &#8220;every&#8221; or &#8220;each&#8221; in a statement can be a clue that an entire population has been included.<\/p>\n<p>If only a portion of the population has been included, a numerical characteristic refers to a sample statistic. If a statement refers to a relatively small number of what would be a large population, the information likely refers to a sample.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Determine whether the number refers to a population parameter or a sample statistic.<\/p>\n<ol>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Based on records of all employees at a small company, the average annual salary is $52,539.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">In a survey of 50 athletes at a large university, 78% said they were happy they chose to continue competing at the collegiate level.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">In a study of 63 law firms in the United States, the average hourly billing rate was $279.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">706 of the 2,223 passengers on the Titanic survived.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q729725\">Show Answer<\/span><\/p>\n<div id=\"q729725\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Since the statement referred to an average calculated on all employee salaries, $52,539 refers to the population mean, [latex]\\mu[\/latex].<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Since a large university likely has many more than 50 athletes, and the statement refers to a survey, 78% refers to a sample statistic, [latex]\\hat{p}[\/latex].<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">There are many law firms in the United States, so the 63 law firms represent a sample. Then the average hourly billing rate, $279, represents a sample mean [latex]\\overline{x}[\/latex].<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The statement refers to all passengers on the Titanic, so 706 out of 2,223, or 31.8%, refers to a population proportion, [latex]\\varrho[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm33101\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=33101&theme=oea&iframe_resize_id=ohm33101&show_question_numbers\" width=\"100%\" height=\"150\"><\/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-1973\">\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>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><li>QID 33101. <strong>Authored by<\/strong>: Lippman, D. <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>","protected":false},"author":169134,"menu_order":4,"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\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"QID 33101\",\"author\":\"Lippman, D\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1973","chapter","type-chapter","status-publish","hentry"],"part":269,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1973","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":11,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1973\/revisions"}],"predecessor-version":[{"id":3749,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1973\/revisions\/3749"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/269"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1973\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1973"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1973"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1973"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1973"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}