{"id":1946,"date":"2021-09-16T17:27:33","date_gmt":"2021-09-16T17:27:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1946"},"modified":"2023-12-05T09:27:22","modified_gmt":"2023-12-05T09:27:22","slug":"a-single-population-mean-using-the-normal-distribution-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/a-single-population-mean-using-the-normal-distribution-2\/","title":{"raw":"Changing the Confidence Level or Sample Size","rendered":"Changing the Confidence Level or Sample Size"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<section>\r\n<ul id=\"list12315\">\r\n \t<li>Explain how the margin of error changes when changing the confidence level or sample size<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Interval Width<\/h3>\r\nTo calculate the width of an interval, take the larger number and subtract the smaller number, this will tell you how many numbers are in the interval, which is how wide the interval is. For example, the width of the interval [latex](2,10)[\/latex] is [latex]10-2=8[\/latex].\r\n\r\n<\/div>\r\n<h2>Changing the Confidence Level or Sample Size<\/h2>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 4<\/h3>\r\nSuppose we change the original problem in Example 2 by using a 95% confidence level. Find a 95% confidence interval for the true (population) mean statistics exam score.\r\n[reveal-answer q=\"304750\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"304750\"]\r\n\r\nTo find the confidence interval, you need the sample mean, [latex]\\displaystyle\\overline{x}[\/latex], and the <em data-redactor-tag=\"em\">EBM<\/em>.\r\n<p class=\"p1\">[latex]\\displaystyle\\overline{x}[\/latex] = 68<\/p>\r\n<p class=\"p1\">EBM = ([latex]\\displaystyle{z}_{\\frac{{\\alpha}}{{2}}})(\\frac{{\\sigma}}{{\\sqrt{n}}}[\/latex])<\/p>\r\n<p class=\"p1\">[latex]\\displaystyle{\\sigma}={3}[\/latex]<\/p>\r\n<p class=\"p1\">n = 36<\/p>\r\n<em data-redactor-tag=\"em\">CL<\/em> = 0.95 so <em data-redactor-tag=\"em\">\u03b1<\/em> = 1 \u2013 <em data-redactor-tag=\"em\">CL<\/em> = 1 \u2013 0.95 = 0.05\r\n\r\n[latex]\\dfrac{a}{2}[\/latex] = 0.025\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]z_{\\frac{a}{2}}[\/latex] =\u00a0<em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.025<\/sub>\r\n\r\nThe area to the right of <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.025<\/sub> is 0.025 and the area to the left of <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.025<\/sub> is 1 \u2013 0.025 = 0.975.\r\n\r\n[latex]z_{\\frac{a}{2}}[\/latex] =\u00a0<em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.025<\/sub> = 1.96\r\n\r\nwhen using invnorm(0.975,0,1) on the TI-83, 83+, or 84+ calculators. (This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the standard normal distribution.)\r\n<p id=\"element-88\" class=\" \"><em data-effect=\"italics\">EBM<\/em>\u00a0= (1.96)[latex]\\left ( \\dfrac{3}{\\sqrt{36}} \\right )[\/latex]\u00a0= 0.98<\/p>\r\n<p id=\"element-956\" class=\" \"><span style=\"white-space: nowrap;\">[latex]\\overline{x}[\/latex]<\/span> \u2013\u00a0<em data-effect=\"italics\">EBM<\/em>\u00a0= 68 \u2013 0.98 = 67.02<\/p>\r\n<p id=\"element-64\" class=\" \"><span style=\"white-space: nowrap;\">[latex]\\overline{x}[\/latex]<\/span> +\u00a0<em data-effect=\"italics\">EBM<\/em>\u00a0= 68 + 0.98 = 68.98<\/p>\r\nNotice that the <em data-redactor-tag=\"em\">EBM<\/em> is larger for a 95% confidence level in the original problem.\r\n\r\n[\/hidden-answer]\r\n<h4>Interpretation<\/h4>\r\nWe estimate with 95% confidence that the true population mean for all statistics exam scores is between 67.02 and 68.98.\r\n<h4>Explanation of 95% Confidence Level<\/h4>\r\nNinety-five percent of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score.\r\n<h4>Comparing the Results<\/h4>\r\nThe 90% confidence interval is (67.18, 68.82). The 95% confidence interval is (67.02, 68.98). The 95% confidence interval is wider. If you look at the graphs, because the area 0.95 is larger than the area 0.90, it makes sense that the 95% confidence interval is wider. To be more confident that the confidence interval actually does contain the true value of the population mean for all statistics exam scores, the confidence interval necessarily needs to be wider.\r\n\r\n<img class=\"aligncenter wp-image-1941 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/16153623\/66bc6e14b0b45f5bc4ac4a9a53b64fd664eef649.jpeg\" alt=\"Graphs of changing the confidence interval from 09% to 95%\" width=\"731\" height=\"229\" \/>\r\n<h4>Summary: Effect of Changing the Confidence Level<\/h4>\r\n<ul>\r\n \t<li>Increasing the confidence level increases the error bound, making the confidence interval wider.<\/li>\r\n \t<li>Decreasing the confidence level decreases the error bound, making the confidence interval narrower.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it 4<\/h3>\r\nRefer to the pizza-delivery Try It 2 exercise. The population standard deviation is six minutes and the sample mean deliver time is 36 minutes. Use a sample size of 20. Find a 95% confidence interval estimate for the true mean pizza delivery time.\r\n[reveal-answer q=\"407318\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"407318\"]\r\n\r\n(33.37, 38.63)\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 5<\/h3>\r\nSuppose we change the original problem in Example 2 to see what happens to the error bound if the sample size is changed.\r\n\r\nLeave everything the same except the sample size. Use the original 90% confidence level. What happens to the error bound and the confidence interval if we increase the sample size and use <em data-redactor-tag=\"em\">n<\/em> = 100 instead of <em data-redactor-tag=\"em\">n<\/em> = 36? What happens if we decrease the sample size to <em data-redactor-tag=\"em\">n<\/em> = 25 instead of <em data-redactor-tag=\"em\">n<\/em> = 36?\r\n<ul>\r\n \t<li>[latex]\\overline{{x}}[\/latex] = 68<\/li>\r\n \t<li><em data-redactor-tag=\"em\">EBM<\/em> = [latex]{\\left ( z_\\frac{a}{2} \\right )}{\\left ( \\frac{\\sigma}{\\sqrt n} \\right )}[\/latex]<\/li>\r\n \t<li><em data-redactor-tag=\"em\">\u03c3<\/em> = 3<\/li>\r\n \t<li>The confidence level is 90% (<em data-redactor-tag=\"em\">CL<\/em>=0.90)<\/li>\r\n \t<li>[latex]{z_\\frac{a}{2}}[\/latex] = z<sub>0.05<\/sub> = 1.645.<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"44270\"]Show Answer and Summary[\/reveal-answer]\r\n[hidden-answer a=\"44270\"]\r\n\r\n<strong>Solution A<\/strong>\r\n\r\nIf we <strong data-redactor-tag=\"strong\">increase<\/strong> the sample size <em data-redactor-tag=\"em\">n<\/em> to 100, we <strong data-redactor-tag=\"strong\">decrease<\/strong> the error bound.\r\n\r\nWhen [latex]n=100: EBM = (Z_{\\frac{\\alpha}{2}})(\\frac{\\alpha}{\\sqrt{n}}) = (1.645)(\\frac{3}{\\sqrt{100}}) = 0.4935[\/latex].\r\n\r\n<strong>Solution B<\/strong>\r\n\r\nIf we <strong data-redactor-tag=\"strong\">decrease<\/strong> the sample size <em data-redactor-tag=\"em\">n<\/em> to 25, we <strong data-redactor-tag=\"strong\">increase<\/strong> the error bound.\r\n\r\nWhen [latex]n=25: EBM = (Z_{\\frac{\\alpha}{2}})(\\frac{\\alpha}{\\sqrt{n}}) = (1.645)(\\frac{3}{\\sqrt{25}}) = 0.987[\/latex].\r\n<h4>Summary: Effect of Changing the Sample Size<\/h4>\r\n<ul>\r\n \t<li>Increasing the sample size causes the error bound to decrease, making the confidence interval narrower.<\/li>\r\n \t<li>Decreasing the sample size causes the error bound to increase, making the confidence interval wider.<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It 5<\/h3>\r\nRefer to the pizza-delivery Try It 2 exercise. The mean delivery time is 36 minutes and the population standard deviation is six minutes. Assume the sample size is changed to 50 restaurants with the same sample mean. Find a 90% confidence interval estimate for the population mean delivery time.\r\n[reveal-answer q=\"130501\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"130501\"]\r\n\r\n(34.6041, 37.3958)\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<section>\n<ul id=\"list12315\">\n<li>Explain how the margin of error changes when changing the confidence level or sample size<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall: Interval Width<\/h3>\n<p>To calculate the width of an interval, take the larger number and subtract the smaller number, this will tell you how many numbers are in the interval, which is how wide the interval is. For example, the width of the interval [latex](2,10)[\/latex] is [latex]10-2=8[\/latex].<\/p>\n<\/div>\n<h2>Changing the Confidence Level or Sample Size<\/h2>\n<div class=\"textbox exercises\">\n<h3>Example 4<\/h3>\n<p>Suppose we change the original problem in Example 2 by using a 95% confidence level. Find a 95% confidence interval for the true (population) mean statistics exam score.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q304750\">Show Answer<\/span><\/p>\n<div id=\"q304750\" class=\"hidden-answer\" style=\"display: none\">\n<p>To find the confidence interval, you need the sample mean, [latex]\\displaystyle\\overline{x}[\/latex], and the <em data-redactor-tag=\"em\">EBM<\/em>.<\/p>\n<p class=\"p1\">[latex]\\displaystyle\\overline{x}[\/latex] = 68<\/p>\n<p class=\"p1\">EBM = ([latex]\\displaystyle{z}_{\\frac{{\\alpha}}{{2}}})(\\frac{{\\sigma}}{{\\sqrt{n}}}[\/latex])<\/p>\n<p class=\"p1\">[latex]\\displaystyle{\\sigma}={3}[\/latex]<\/p>\n<p class=\"p1\">n = 36<\/p>\n<p><em data-redactor-tag=\"em\">CL<\/em> = 0.95 so <em data-redactor-tag=\"em\">\u03b1<\/em> = 1 \u2013 <em data-redactor-tag=\"em\">CL<\/em> = 1 \u2013 0.95 = 0.05<\/p>\n<p>[latex]\\dfrac{a}{2}[\/latex] = 0.025\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]z_{\\frac{a}{2}}[\/latex] =\u00a0<em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.025<\/sub><\/p>\n<p>The area to the right of <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.025<\/sub> is 0.025 and the area to the left of <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.025<\/sub> is 1 \u2013 0.025 = 0.975.<\/p>\n<p>[latex]z_{\\frac{a}{2}}[\/latex] =\u00a0<em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.025<\/sub> = 1.96<\/p>\n<p>when using invnorm(0.975,0,1) on the TI-83, 83+, or 84+ calculators. (This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the standard normal distribution.)<\/p>\n<p id=\"element-88\" class=\"\"><em data-effect=\"italics\">EBM<\/em>\u00a0= (1.96)[latex]\\left ( \\dfrac{3}{\\sqrt{36}} \\right )[\/latex]\u00a0= 0.98<\/p>\n<p id=\"element-956\" class=\"\"><span style=\"white-space: nowrap;\">[latex]\\overline{x}[\/latex]<\/span> \u2013\u00a0<em data-effect=\"italics\">EBM<\/em>\u00a0= 68 \u2013 0.98 = 67.02<\/p>\n<p id=\"element-64\" class=\"\"><span style=\"white-space: nowrap;\">[latex]\\overline{x}[\/latex]<\/span> +\u00a0<em data-effect=\"italics\">EBM<\/em>\u00a0= 68 + 0.98 = 68.98<\/p>\n<p>Notice that the <em data-redactor-tag=\"em\">EBM<\/em> is larger for a 95% confidence level in the original problem.<\/p>\n<\/div>\n<\/div>\n<h4>Interpretation<\/h4>\n<p>We estimate with 95% confidence that the true population mean for all statistics exam scores is between 67.02 and 68.98.<\/p>\n<h4>Explanation of 95% Confidence Level<\/h4>\n<p>Ninety-five percent of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score.<\/p>\n<h4>Comparing the Results<\/h4>\n<p>The 90% confidence interval is (67.18, 68.82). The 95% confidence interval is (67.02, 68.98). The 95% confidence interval is wider. If you look at the graphs, because the area 0.95 is larger than the area 0.90, it makes sense that the 95% confidence interval is wider. To be more confident that the confidence interval actually does contain the true value of the population mean for all statistics exam scores, the confidence interval necessarily needs to be wider.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1941 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/16153623\/66bc6e14b0b45f5bc4ac4a9a53b64fd664eef649.jpeg\" alt=\"Graphs of changing the confidence interval from 09% to 95%\" width=\"731\" height=\"229\" \/><\/p>\n<h4>Summary: Effect of Changing the Confidence Level<\/h4>\n<ul>\n<li>Increasing the confidence level increases the error bound, making the confidence interval wider.<\/li>\n<li>Decreasing the confidence level decreases the error bound, making the confidence interval narrower.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it 4<\/h3>\n<p>Refer to the pizza-delivery Try It 2 exercise. The population standard deviation is six minutes and the sample mean deliver time is 36 minutes. Use a sample size of 20. Find a 95% confidence interval estimate for the true mean pizza delivery time.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q407318\">Show Answer<\/span><\/p>\n<div id=\"q407318\" class=\"hidden-answer\" style=\"display: none\">\n<p>(33.37, 38.63)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example 5<\/h3>\n<p>Suppose we change the original problem in Example 2 to see what happens to the error bound if the sample size is changed.<\/p>\n<p>Leave everything the same except the sample size. Use the original 90% confidence level. What happens to the error bound and the confidence interval if we increase the sample size and use <em data-redactor-tag=\"em\">n<\/em> = 100 instead of <em data-redactor-tag=\"em\">n<\/em> = 36? What happens if we decrease the sample size to <em data-redactor-tag=\"em\">n<\/em> = 25 instead of <em data-redactor-tag=\"em\">n<\/em> = 36?<\/p>\n<ul>\n<li>[latex]\\overline{{x}}[\/latex] = 68<\/li>\n<li><em data-redactor-tag=\"em\">EBM<\/em> = [latex]{\\left ( z_\\frac{a}{2} \\right )}{\\left ( \\frac{\\sigma}{\\sqrt n} \\right )}[\/latex]<\/li>\n<li><em data-redactor-tag=\"em\">\u03c3<\/em> = 3<\/li>\n<li>The confidence level is 90% (<em data-redactor-tag=\"em\">CL<\/em>=0.90)<\/li>\n<li>[latex]{z_\\frac{a}{2}}[\/latex] = z<sub>0.05<\/sub> = 1.645.<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44270\">Show Answer and Summary<\/span><\/p>\n<div id=\"q44270\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Solution A<\/strong><\/p>\n<p>If we <strong data-redactor-tag=\"strong\">increase<\/strong> the sample size <em data-redactor-tag=\"em\">n<\/em> to 100, we <strong data-redactor-tag=\"strong\">decrease<\/strong> the error bound.<\/p>\n<p>When [latex]n=100: EBM = (Z_{\\frac{\\alpha}{2}})(\\frac{\\alpha}{\\sqrt{n}}) = (1.645)(\\frac{3}{\\sqrt{100}}) = 0.4935[\/latex].<\/p>\n<p><strong>Solution B<\/strong><\/p>\n<p>If we <strong data-redactor-tag=\"strong\">decrease<\/strong> the sample size <em data-redactor-tag=\"em\">n<\/em> to 25, we <strong data-redactor-tag=\"strong\">increase<\/strong> the error bound.<\/p>\n<p>When [latex]n=25: EBM = (Z_{\\frac{\\alpha}{2}})(\\frac{\\alpha}{\\sqrt{n}}) = (1.645)(\\frac{3}{\\sqrt{25}}) = 0.987[\/latex].<\/p>\n<h4>Summary: Effect of Changing the Sample Size<\/h4>\n<ul>\n<li>Increasing the sample size causes the error bound to decrease, making the confidence interval narrower.<\/li>\n<li>Decreasing the sample size causes the error bound to increase, making the confidence interval wider.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It 5<\/h3>\n<p>Refer to the pizza-delivery Try It 2 exercise. The mean delivery time is 36 minutes and the population standard deviation is six minutes. Assume the sample size is changed to 50 restaurants with the same sample mean. Find a 90% confidence interval estimate for the population mean delivery time.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q130501\">Show Answer<\/span><\/p>\n<div id=\"q130501\" class=\"hidden-answer\" style=\"display: none\">\n<p>(34.6041, 37.3958)<\/p>\n<\/div>\n<\/div>\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-1946\">\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>A Single Population Mean using the Normal Distribution. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-1-a-single-population-mean-using-the-normal-distribution\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-1-a-single-population-mean-using-the-normal-distribution<\/a>. <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>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/a>. <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>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/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":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"A Single Population Mean using the Normal Distribution\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-1-a-single-population-mean-using-the-normal-distribution\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free 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