{"id":1148,"date":"2023-06-22T01:56:48","date_gmt":"2023-06-22T01:56:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-distribution-of-a-sample-proportion-dig-deeper\/"},"modified":"2024-01-03T22:05:32","modified_gmt":"2024-01-03T22:05:32","slug":"sampling-distribution-of-a-sample-proportion-dig-deeper","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-distribution-of-a-sample-proportion-dig-deeper\/","title":{"raw":"Sampling Distribution of a Sample Proportion: Fresh Take","rendered":"Sampling Distribution of a Sample Proportion: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Use technology to create a sampling distribution of a sample proportion given [latex]n[\/latex] and [latex]p[\/latex].<\/li>\r\n\t<li>Calculate the mean and standard deviation for a sampling distribution of a sample proportion.<\/li>\r\n\t<li>Recognize the difference between the standard deviation and the standard error of a sample proportion.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>Every day, we often see articles in the newspaper reporting the result of a poll using\u00a0proportions or percentages.<\/p>\r\n<p>[footnote]https:\/\/www.cato.org\/blog\/new-poll-76-americans-oppose-student-debt-cancellation-it-drives-price-college-64-oppose-it[\/footnote]The Cato Institute 2022 Student Debt Cancellation National Survey was designed and conducted by the Cato Institute in collaboration with YouGov.\u00a0The title of the article is \"<strong>76% of Americans Oppose Student Debt Cancellation if It Drives up the Price of College, 64% Oppose if It Raises Taxes<\/strong>\".<\/p>\r\n<p>However, did you read the fine print? Or, in this case, the bottom of the article? The margin of error for the survey of [latex]2000[\/latex] Americans [latex]18[\/latex] years of age and older is [latex]+\/- 2.39[\/latex] percentage points at the [latex]95\\%[\/latex] level of confidence.<\/p>\r\n<p>We always need to be aware of the headline of an article and not take it as a factual information. Most of the information we obtained are from a sample and therefore we can only make inference about the population.<\/p>\r\n<p>Now, to calculate the margin of error, we need to first understand how to find the mean of the distribution and how to calculate interpret the standard error.<\/p>\r\n<section class=\"textbox recall\">\r\n<ul>\r\n\t<li>The <strong>estimated\u00a0<\/strong>mean of the distribution of sample proportions is [latex]\\hat{p}[\/latex].<\/li>\r\n\t<li>To distinguish it from the true standard deviation of sample proportions, we call the <strong>estimated<\/strong> standard deviation of sample proportions the <strong>standard error<\/strong> of [latex]\\hat{p}[\/latex]:<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center\"><span style=\"font-size: 1rem;text-align: center\">Standard Error: [latex]SE = \\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex]<\/span><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. Five hundred randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the [latex]500[\/latex] people surveyed, [latex]421[\/latex] responded yes \u2013 they own cell phones. Find the mean and standard deviation of the sampling distribution of this sample proportion.[reveal-answer q=\"235324\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"235324\"]<br \/>\r\nFirst of all, note that our value comes from a sample, therefore we will find the estimated mean and estimated standard deviation (standard error) for this sample proportion.\r\n\r\n<ul>\r\n\t<li>The <strong>estimated\u00a0<\/strong>mean = [latex]\\dfrac{421}{500}=0.842[\/latex]<\/li>\r\n\t<li>The\u00a0<strong>standard error<\/strong> = <span style=\"font-size: 1rem;text-align: center\">[latex]SE = \\sqrt{\\frac{0.842(1-0.842)}{500}}=\\sqrt{\\frac{0.842(0.158)}{500}} \\approx 0.0163[\/latex]<\/span><\/li>\r\n<\/ul>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use technology to create a sampling distribution of a sample proportion given [latex]n[\/latex] and [latex]p[\/latex].<\/li>\n<li>Calculate the mean and standard deviation for a sampling distribution of a sample proportion.<\/li>\n<li>Recognize the difference between the standard deviation and the standard error of a sample proportion.<\/li>\n<\/ul>\n<\/section>\n<p>Every day, we often see articles in the newspaper reporting the result of a poll using\u00a0proportions or percentages.<\/p>\n<p><a class=\"footnote\" title=\"https:\/\/www.cato.org\/blog\/new-poll-76-americans-oppose-student-debt-cancellation-it-drives-price-college-64-oppose-it\" id=\"return-footnote-1148-1\" href=\"#footnote-1148-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>The Cato Institute 2022 Student Debt Cancellation National Survey was designed and conducted by the Cato Institute in collaboration with YouGov.\u00a0The title of the article is &#8220;<strong>76% of Americans Oppose Student Debt Cancellation if It Drives up the Price of College, 64% Oppose if It Raises Taxes<\/strong>&#8220;.<\/p>\n<p>However, did you read the fine print? Or, in this case, the bottom of the article? The margin of error for the survey of [latex]2000[\/latex] Americans [latex]18[\/latex] years of age and older is [latex]+\/- 2.39[\/latex] percentage points at the [latex]95\\%[\/latex] level of confidence.<\/p>\n<p>We always need to be aware of the headline of an article and not take it as a factual information. Most of the information we obtained are from a sample and therefore we can only make inference about the population.<\/p>\n<p>Now, to calculate the margin of error, we need to first understand how to find the mean of the distribution and how to calculate interpret the standard error.<\/p>\n<section class=\"textbox recall\">\n<ul>\n<li>The <strong>estimated\u00a0<\/strong>mean of the distribution of sample proportions is [latex]\\hat{p}[\/latex].<\/li>\n<li>To distinguish it from the true standard deviation of sample proportions, we call the <strong>estimated<\/strong> standard deviation of sample proportions the <strong>standard error<\/strong> of [latex]\\hat{p}[\/latex]:<\/li>\n<\/ul>\n<p style=\"text-align: center\"><span style=\"font-size: 1rem;text-align: center\">Standard Error: [latex]SE = \\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex]<\/span><\/p>\n<\/section>\n<section class=\"textbox example\">Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. Five hundred randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the [latex]500[\/latex] people surveyed, [latex]421[\/latex] responded yes \u2013 they own cell phones. Find the mean and standard deviation of the sampling distribution of this sample proportion.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q235324\">Show answer<\/button><\/p>\n<div id=\"q235324\" class=\"hidden-answer\" style=\"display: none\">\nFirst of all, note that our value comes from a sample, therefore we will find the estimated mean and estimated standard deviation (standard error) for this sample proportion.<\/p>\n<ul>\n<li>The <strong>estimated\u00a0<\/strong>mean = [latex]\\dfrac{421}{500}=0.842[\/latex]<\/li>\n<li>The\u00a0<strong>standard error<\/strong> = <span style=\"font-size: 1rem;text-align: center\">[latex]SE = \\sqrt{\\frac{0.842(1-0.842)}{500}}=\\sqrt{\\frac{0.842(0.158)}{500}} \\approx 0.0163[\/latex]<\/span><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1148-1\">https:\/\/www.cato.org\/blog\/new-poll-76-americans-oppose-student-debt-cancellation-it-drives-price-college-64-oppose-it <a href=\"#return-footnote-1148-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":8,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1126,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1148"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1148\/revisions"}],"predecessor-version":[{"id":4748,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1148\/revisions\/4748"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1126"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1148\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1148"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1148"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1148"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1148"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}