{"id":1190,"date":"2023-06-22T01:59:53","date_gmt":"2023-06-22T01:59:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sample-size-for-proportions-learn-it-4\/"},"modified":"2025-05-16T03:16:27","modified_gmt":"2025-05-16T03:16:27","slug":"sample-size-for-proportions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sample-size-for-proportions-learn-it-4\/","title":{"raw":"Sample Size for Proportions: Learn It 4","rendered":"Sample Size for Proportions: Learn It 4"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li class=\"li1\">Find the required sample size for a desired margin of error and population's confidence interval.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Sample Size Needed for Proportions<\/h2>\r\n<p>Researchers can use the margin of error formula, [latex] ME = z^{*} \\cdot \\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex], to determine the minimum sample size needed to produce a given margin of error by solving for [latex] n [\/latex].<\/p>\r\n<p>The rearranged formula to find the sample size needed for proportion:<\/p>\r\n<p style=\"text-align: center;\">[latex] n = \\hat{p}(1-\\hat{p})(\\frac{z^{*}}{ME})^{2}[\/latex]<\/p>\r\n<p>Notice that this formula requires the researcher to know the value of [latex] \\hat{p} [\/latex], which is unknown. However, researchers often have preliminary data or prior research that can be used to estimate [latex] \\hat{p} [\/latex].<\/p>\r\n<section class=\"textbox proTip\">If there is no way to estimate [latex] \\hat{p} [\/latex], researchers will find the largest possible n by setting [latex] \\hat{p} [\/latex] to [latex]0.5[\/latex]. (Try it! The largest value you can get for [latex] \\hat{p}(1-\\hat{p})[\/latex]) is [latex]0.25[\/latex] when you set [latex] \\hat{p} [\/latex] to [latex]0.5[\/latex].)<\/section>\r\n<p>We can also use our statistical tool to help us calculate the sample size.<\/p>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/inference_prop\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\r\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/inference_prop\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1673[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1675[\/ohm2_question]<\/section>\r\n<section class=\"textbox proTip\">Using the conservative [latex] \\hat{p} = 0.5 [\/latex] approach always yields a larger than necessary sample size.<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li class=\"li1\">Find the required sample size for a desired margin of error and population&#8217;s confidence interval.<\/li>\n<\/ul>\n<\/section>\n<h2>Sample Size Needed for Proportions<\/h2>\n<p>Researchers can use the margin of error formula, [latex]ME = z^{*} \\cdot \\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}[\/latex], to determine the minimum sample size needed to produce a given margin of error by solving for [latex]n[\/latex].<\/p>\n<p>The rearranged formula to find the sample size needed for proportion:<\/p>\n<p style=\"text-align: center;\">[latex]n = \\hat{p}(1-\\hat{p})(\\frac{z^{*}}{ME})^{2}[\/latex]<\/p>\n<p>Notice that this formula requires the researcher to know the value of [latex]\\hat{p}[\/latex], which is unknown. However, researchers often have preliminary data or prior research that can be used to estimate [latex]\\hat{p}[\/latex].<\/p>\n<section class=\"textbox proTip\">If there is no way to estimate [latex]\\hat{p}[\/latex], researchers will find the largest possible n by setting [latex]\\hat{p}[\/latex] to [latex]0.5[\/latex]. (Try it! The largest value you can get for [latex]\\hat{p}(1-\\hat{p})[\/latex]) is [latex]0.25[\/latex] when you set [latex]\\hat{p}[\/latex] to [latex]0.5[\/latex].)<\/section>\n<p>We can also use our statistical tool to help us calculate the sample size.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/inference_prop\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/inference_prop\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1673\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1673&theme=lumen&iframe_resize_id=ohm1673&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1675\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1675&theme=lumen&iframe_resize_id=ohm1675&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox proTip\">Using the conservative [latex]\\hat{p} = 0.5[\/latex] approach always yields a larger than necessary sample size.<\/section>\n","protected":false},"author":8,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1163,"module-header":"learn_it","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\/1190"}],"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":8,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1190\/revisions"}],"predecessor-version":[{"id":6735,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1190\/revisions\/6735"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1163"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1190\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1190"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1190"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1190"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1190"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}