{"id":1160,"date":"2023-06-22T01:56:59","date_gmt":"2023-06-22T01:56:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-variability-dig-deeper\/"},"modified":"2025-05-16T02:58:54","modified_gmt":"2025-05-16T02:58:54","slug":"sampling-variability-dig-deeper","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/sampling-variability-dig-deeper\/","title":{"raw":"Sampling Variability: Fresh Take","rendered":"Sampling Variability: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Find the sample size needed for a sampling distribution to have a desired standard deviation.<\/li>\r\n\t<li>Check the conditions for normal approximation of a sampling distribution of a sample proportion.<\/li>\r\n\t<li>Use the normal distribution to calculate probabilities and percentiles from a sampling distribution.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1605[\/ohm2_question]<\/section>\r\n<h2><strong>Calculating the Sample Size\u00a0<\/strong><strong>[latex]n[\/latex]<\/strong><\/h2>\r\n<p>If researchers desire a specific standard error, then they can use the standard error formula to calculate the required sample size.<\/p>\r\n<p style=\"text-align: center;\">[latex]SE = \\sqrt{\\dfrac{p(1-p)}{n}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]SE^2 = \\dfrac{p(1-p)}{n}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]n \\times SE^2 = p(1-p)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]n = \\dfrac{p(1-p)}{SE^2}[\/latex]<\/p>\r\n<section class=\"textbox example\">Suppose a mobile phone company wants to determine the current percentage of customers aged 50+ who use text messaging on their cell phones.<br \/>\r\n<br \/>\r\nHow many customers aged 50+ should the company survey in order to be [latex]68\\%[\/latex] confident that the estimated (sample) proportion is within three percentage points of the true population proportion of customers aged 50+ who use text messaging on their cell phones? <strong>Answer: [latex]278[\/latex] customers.<\/strong><strong><br \/>\r\n<br \/>\r\nExplanation: <\/strong>Recall that [latex]68\\%[\/latex] of the data falls within [latex]1[\/latex] standard deviation from the mean. This means that [latex]3\\%[\/latex] is the standard deviation (or the standard error) desired.\r\n\r\n<p style=\"text-align: center;\">[latex]n = \\dfrac{p(1-p)}{SE^2}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">However, in order to find [latex]n[\/latex], we need to know the estimated (sample) proportion [latex]p[\/latex]. But, we do not know [latex]p[\/latex] yet.<\/p>\r\n<p>Since we multiply [latex]p[\/latex] and [latex](1-p)[\/latex] together, we make [latex]p[\/latex] equal to [latex]0.5[\/latex] because [latex]p(1-p)=(0.5)(0.5) = 0.25[\/latex] results in the largest possible product. (Try other product: [latex]p(1-p)=(0.6)(0.4) = 0.24[\/latex], [latex]p(1-p)=(0.7)(0.3) = 0.21[\/latex], and so on.)<\/p>\r\n<p style=\"text-align: center;\">[latex]n = \\dfrac{p(1-p)}{SE^2}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]n = \\dfrac{0.5(1-0.5)}{0.03^2}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]n \\approx 277.78[\/latex]<\/p>\r\n<p>Round the answer to the next higher value.<\/p>\r\n<p>The sample size should be <strong>[latex]278[\/latex]<\/strong> cell phone customers aged 50+ in order to be [latex]68\\%[\/latex] confident that the estimated (sample) proportion is within three percentage points of the true population proportion of all customers aged 50+ who use text messaging on their cell phones.<\/p>\r\n<\/section>\r\n<section class=\"textbox proTip\">In order to find [latex]n[\/latex], we need to know the estimated (sample) proportion [latex]p[\/latex]. If [latex]p[\/latex] is unknown, use [latex]p=0.5[\/latex].<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Find the sample size needed for a sampling distribution to have a desired standard deviation.<\/li>\n<li>Check the conditions for normal approximation of a sampling distribution of a sample proportion.<\/li>\n<li>Use the normal distribution to calculate probabilities and percentiles from a sampling distribution.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1605\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1605&theme=lumen&iframe_resize_id=ohm1605&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2><strong>Calculating the Sample Size\u00a0<\/strong><strong>[latex]n[\/latex]<\/strong><\/h2>\n<p>If researchers desire a specific standard error, then they can use the standard error formula to calculate the required sample size.<\/p>\n<p style=\"text-align: center;\">[latex]SE = \\sqrt{\\dfrac{p(1-p)}{n}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]SE^2 = \\dfrac{p(1-p)}{n}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]n \\times SE^2 = p(1-p)[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]n = \\dfrac{p(1-p)}{SE^2}[\/latex]<\/p>\n<section class=\"textbox example\">Suppose a mobile phone company wants to determine the current percentage of customers aged 50+ who use text messaging on their cell phones.<\/p>\n<p>How many customers aged 50+ should the company survey in order to be [latex]68\\%[\/latex] confident that the estimated (sample) proportion is within three percentage points of the true population proportion of customers aged 50+ who use text messaging on their cell phones? <strong>Answer: [latex]278[\/latex] customers.<\/strong><strong><\/p>\n<p>Explanation: <\/strong>Recall that [latex]68\\%[\/latex] of the data falls within [latex]1[\/latex] standard deviation from the mean. This means that [latex]3\\%[\/latex] is the standard deviation (or the standard error) desired.<\/p>\n<p style=\"text-align: center;\">[latex]n = \\dfrac{p(1-p)}{SE^2}[\/latex]<\/p>\n<p style=\"text-align: left;\">However, in order to find [latex]n[\/latex], we need to know the estimated (sample) proportion [latex]p[\/latex]. But, we do not know [latex]p[\/latex] yet.<\/p>\n<p>Since we multiply [latex]p[\/latex] and [latex](1-p)[\/latex] together, we make [latex]p[\/latex] equal to [latex]0.5[\/latex] because [latex]p(1-p)=(0.5)(0.5) = 0.25[\/latex] results in the largest possible product. (Try other product: [latex]p(1-p)=(0.6)(0.4) = 0.24[\/latex], [latex]p(1-p)=(0.7)(0.3) = 0.21[\/latex], and so on.)<\/p>\n<p style=\"text-align: center;\">[latex]n = \\dfrac{p(1-p)}{SE^2}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]n = \\dfrac{0.5(1-0.5)}{0.03^2}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]n \\approx 277.78[\/latex]<\/p>\n<p>Round the answer to the next higher value.<\/p>\n<p>The sample size should be <strong>[latex]278[\/latex]<\/strong> cell phone customers aged 50+ in order to be [latex]68\\%[\/latex] confident that the estimated (sample) proportion is within three percentage points of the true population proportion of all customers aged 50+ who use text messaging on their cell phones.<\/p>\n<\/section>\n<section class=\"textbox proTip\">In order to find [latex]n[\/latex], we need to know the estimated (sample) proportion [latex]p[\/latex]. If [latex]p[\/latex] is unknown, use [latex]p=0.5[\/latex].<\/section>\n","protected":false},"author":8,"menu_order":28,"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\/1160"}],"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\/1160\/revisions"}],"predecessor-version":[{"id":6715,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1160\/revisions\/6715"}],"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\/1160\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1160"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1160"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1160"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}