{"id":1078,"date":"2021-06-08T21:42:45","date_gmt":"2021-06-09T01:42:45","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&#038;p=1078"},"modified":"2025-04-25T18:23:58","modified_gmt":"2025-04-25T22:23:58","slug":"10-1-population-proportion-and-the-sample-proportion","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/10-1-population-proportion-and-the-sample-proportion\/","title":{"raw":"10.1 Population Proportion and the Sample Proportion","rendered":"10.1 Population Proportion and the Sample Proportion"},"content":{"raw":"Recall that the population mean [latex]\\mu = \\frac{\\sum x_i}{N}[\/latex] is a population parameter used to describe the population, where N is the population size (number of individuals in the population). The population proportion\r\n<p align=\"center\">[latex]p = \\frac{\\text{\\# of individuals having a certain attribute}}{\\text{population size}} = \\frac{\\text{\\# of successes}}{N}[\/latex]<\/p>\r\nis another parameter used to describe the population. For example, the proportion of female students at MacEwan is defined as\r\n<p align=\"center\">[latex]p = \\frac{\\text{\\# of female students at MacEwan}}{\\text{total number of students at MacEwan}} = \\frac{\\text{\\# of successes}}{N}.[\/latex]<\/p>\r\nIn this instance, picking a female student is regarded as a success.\r\n\r\nJust as the sample mean [latex]\\bar{x} = \\frac{\\sum x_i}{n}[\/latex] is used to estimate the population mean [latex]\\mu[\/latex], the sample proportion [latex]\\hat{p}[\/latex] is used to estimate the population proportion <em style=\"margin: 0px; padding: 0px; border: 0px; font-weight: 400; font-style: italic; font-family: arial; font-size: 16px; color: #000000; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none;\">p,<\/em> where\r\n<p align=\"center\">[latex]\\hat{p} = \\frac{\\text{\\# of individuals having a certain attribute in the sample}}{\\text{sample size}} = \\frac{\\text{\\# of successes in the sample}}{n}.[\/latex]<\/p>\r\nHere are several examples:\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Examples<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul type=\"disc\">\r\n \t<li>A random sample of <em>n<\/em> = 100 students is obtained from MacEwan University. Of the 100 students in the sample, 65 are female. The sample proportion [latex]\\hat{p} = \\frac{x}{n} = \\frac{65}{100}[\/latex] provides a point estimate of [latex]p[\/latex], the proportion of female students at MacEwan.<\/li>\r\n \t<li>A random sample of <em>n<\/em> = 1000 judo matches is obtained, and it is determined that 510 of the matches are won by the athletes wearing a blue uniform. The sample proportion [latex]\\hat{p} = \\frac{x}{n} = \\frac{510}{1000}[\/latex] is a point estimate of [latex]p[\/latex], the proportion of winners in blue.<\/li>\r\n \t<li>A credit card company sends an advertisement to <em>n<\/em> = 500 randomly chosen customers and only 10 customers respond. The sample proportion [latex]\\hat{p} = \\frac{x}{n} = \\frac{10}{500}[\/latex] is a point estimate of [latex]p[\/latex], the proportion of respondents.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>","rendered":"<p>Recall that the population mean [latex]\\mu = \\frac{\\sum x_i}{N}[\/latex] is a population parameter used to describe the population, where N is the population size (number of individuals in the population). The population proportion<\/p>\n<p style=\"text-align: center;\">[latex]p = \\frac{\\text{\\# of individuals having a certain attribute}}{\\text{population size}} = \\frac{\\text{\\# of successes}}{N}[\/latex]<\/p>\n<p>is another parameter used to describe the population. For example, the proportion of female students at MacEwan is defined as<\/p>\n<p style=\"text-align: center;\">[latex]p = \\frac{\\text{\\# of female students at MacEwan}}{\\text{total number of students at MacEwan}} = \\frac{\\text{\\# of successes}}{N}.[\/latex]<\/p>\n<p>In this instance, picking a female student is regarded as a success.<\/p>\n<p>Just as the sample mean [latex]\\bar{x} = \\frac{\\sum x_i}{n}[\/latex] is used to estimate the population mean [latex]\\mu[\/latex], the sample proportion [latex]\\hat{p}[\/latex] is used to estimate the population proportion <em style=\"margin: 0px; padding: 0px; border: 0px; font-weight: 400; font-style: italic; font-family: arial; font-size: 16px; color: #000000; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none;\">p,<\/em> where<\/p>\n<p style=\"text-align: center;\">[latex]\\hat{p} = \\frac{\\text{\\# of individuals having a certain attribute in the sample}}{\\text{sample size}} = \\frac{\\text{\\# of successes in the sample}}{n}.[\/latex]<\/p>\n<p>Here are several examples:<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Examples<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul type=\"disc\">\n<li>A random sample of <em>n<\/em> = 100 students is obtained from MacEwan University. Of the 100 students in the sample, 65 are female. The sample proportion [latex]\\hat{p} = \\frac{x}{n} = \\frac{65}{100}[\/latex] provides a point estimate of [latex]p[\/latex], the proportion of female students at MacEwan.<\/li>\n<li>A random sample of <em>n<\/em> = 1000 judo matches is obtained, and it is determined that 510 of the matches are won by the athletes wearing a blue uniform. The sample proportion [latex]\\hat{p} = \\frac{x}{n} = \\frac{510}{1000}[\/latex] is a point estimate of [latex]p[\/latex], the proportion of winners in blue.<\/li>\n<li>A credit card company sends an advertisement to <em>n<\/em> = 500 randomly chosen customers and only 10 customers respond. The sample proportion [latex]\\hat{p} = \\frac{x}{n} = \\frac{10}{500}[\/latex] is a point estimate of [latex]p[\/latex], the proportion of respondents.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n","protected":false},"author":19,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1078","chapter","type-chapter","status-publish","hentry"],"part":1075,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1078","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":14,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1078\/revisions"}],"predecessor-version":[{"id":5434,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1078\/revisions\/5434"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/1075"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/1078\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=1078"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=1078"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=1078"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=1078"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}