{"id":275,"date":"2021-07-14T15:59:04","date_gmt":"2021-07-14T15:59:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/answers-to-selected-exercises-7\/"},"modified":"2023-12-05T09:31:25","modified_gmt":"2023-12-05T09:31:25","slug":"answers-to-selected-exercises-7","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/answers-to-selected-exercises-7\/","title":{"raw":"Answers to Selected Exercises","rendered":"Answers to Selected Exercises"},"content":{"raw":"<h2>A Single Population Mean using the Normal Distribution \u2013 Practice<\/h2>\r\n1.\r\n<ul data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ul data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li>244<\/li>\r\n \t<li>15<\/li>\r\n \t<li>50<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n3.\u00a0[latex]N\\left(244,\\frac{15}{\\sqrt{50}}\\right)[\/latex]\r\n\r\n5. As the sample size increases, there will be less variability in the mean, so the interval size decreases.\r\n\r\n7. <em>X<\/em> is the time in minutes it takes to complete the U.S. Census short form. [latex]\\overline{X}[\/latex] is the mean time it took a sample of 200 people to complete the U.S. Census short form.\r\n\r\n9.\r\n\r\nCI: (7.9441, 8.4559)\r\n<figure id=\"eip-idm5552576\"><span id=\"eip-idm128260720\" data-type=\"media\" data-alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 8.2 on the horizontal axis. A central region is shaded between points 7.94 and 8.46.\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214626\/CNX_Stats_C08_M02_item002annoN.jpg\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 8.2 on the horizontal axis. A central region is shaded between points 7.94 and 8.46.\" width=\"380\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\r\n<p id=\"eip-idm5550160\"><em data-effect=\"italics\">EBM<\/em> = 0.26<\/p>\r\n11.\u00a0The level of confidence would decrease because decreasing <em data-effect=\"italics\">n<\/em> makes the confidence interval wider, so at the same error bound, the confidence level decreases.\r\n\r\n13. 1.\u00a0[latex]\\overline{x}[\/latex]= 2.2, 2.\u00a0<em data-effect=\"italics\">\u03c3<\/em> = 0.2, 3.\u00a0<em data-effect=\"italics\">n<\/em> = 20\r\n\r\n15. [latex]\\overline{X}[\/latex] is the mean weight of a sample of 20 heads of lettuce.\r\n\r\n17.\r\n<p id=\"fs-idm25106704\"><em data-effect=\"italics\">EBM<\/em> = 0.07<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\nCI: (2.1264, 2.2736)\r\n<figure id=\"fs-idp82050400\"><span id=\"eip-idp84209776\" data-type=\"media\" data-alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 2.2 on the horizontal axis. A central region is shaded between points 2.13 and 2.27.\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214628\/CNX_Stats_C08_M02_item003annoN.jpg\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 2.2 on the horizontal axis. A central region is shaded between points 2.13 and 2.27.\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\r\n19.\u00a0The interval is greater because the level of confidence increased. If the only change made in the analysis is a change in confidence level, then all we are doing is changing how much area is being calculated for the normal distribution. Therefore, a larger confidence level results in larger areas and larger intervals.\r\n\r\n21.\u00a0The confidence level would increase.\r\n\r\n23.\u00a030.4\r\n\r\n25.\u00a0<em data-effect=\"italics\">\u03c3<\/em>\r\n\r\n27.\u00a0<em data-effect=\"italics\">\u03bc<\/em>\r\n\r\n29.\u00a0normal\r\n\r\n31.\u00a00.025\r\n\r\n33.\u00a0(24.52,36.28)\r\n\r\n35.\u00a0We are 95% confident that the true mean age for Winger Foothill College students is between 24.52 and 36.28.\r\n\r\n37.\u00a0The error bound for the mean would decrease because as the CL decreases, you need less area under the normal curve (which translates into a smaller interval) to capture the true population mean.\r\n\r\n<section class=\" focusable\" tabindex=\"-1\">\r\n<h2 id=\"fs-idm23029072\" class=\"solution\" data-type=\"solution\">A Single Population Mean using the Student t Distribution \u2014 Practice<\/h2>\r\n39.\u00a0<em data-effect=\"italics\">X<\/em> is the number of hours a patient waits in the emergency room before being called back to be examined. [latex]\\overline{X}[\/latex]\u00a0is the mean wait time of 70 patients in the emergency room.\r\n\r\n41.\r\n<div class=\"exercise\" data-type=\"exercise\"><section>\r\n<div id=\"eip-501\" class=\"solution ui-solution-visible\" data-type=\"solution\"><section class=\"ui-body\">\r\n<p id=\"eip-334\">CI: (1.3808, 1.6192)<\/p>\r\n\r\n<figure id=\"fs-idm150372688\"><span id=\"eip-idp2023440\" data-type=\"media\" data-alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 1.5 on the horizontal axis. A central region is shaded between points 1.38 and 1.62.\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214637\/CNX_Stats_C08_M03_item001annoN.jpg\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 1.5 on the horizontal axis. A central region is shaded between points 1.38 and 1.62.\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\r\n<p id=\"eip-idm67620928\"><em data-effect=\"italics\">EBM<\/em> = 0.12<\/p>\r\n\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div class=\"exercise\" data-type=\"exercise\"><\/div>\r\n<div class=\"problem\" data-type=\"problem\">\r\n\r\n\u00a043.\r\n<ol id=\"fs-idm117900288\" data-number-style=\"lower-alpha\">\r\n \t<li>[latex]\\overline{x}[\/latex]= 151<\/li>\r\n \t<li>[latex]{x}_{s}[\/latex] = 32<\/li>\r\n \t<li><em data-effect=\"italics\">n<\/em> = 108<\/li>\r\n \t<li><em data-effect=\"italics\">n<\/em> \u2013 1 = 107<\/li>\r\n<\/ol>\r\n45. [latex]\\overline{X}[\/latex] the mean number of hours spent watching television per month from a sample of 108 Americans.\r\n\r\n47. CI: (142.92, 159.08)\r\n<div data-type=\"newline\"><\/div>\r\n<figure id=\"fs-idm53713472\"><span id=\"eip-idm24726752\" data-type=\"media\" data-alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 151 on the horizontal axis. A central region is shaded between points 142.92 and 159.08.\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214639\/CNX_Stats_C08_M03_item002annoN.jpg\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 151 on the horizontal axis. A central region is shaded between points 142.92 and 159.08.\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\r\n<\/div>\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\n<div class=\"problem\" data-type=\"problem\">\r\n\r\n<em data-effect=\"italics\">EBM<\/em> = 8.08\r\n\r\n49.\r\n<ol id=\"list-827634\" data-number-style=\"lower-alpha\">\r\n \t<li>3.26<\/li>\r\n \t<li>1.02<\/li>\r\n \t<li>39<\/li>\r\n<\/ol>\r\n51. [latex]\\mu[\/latex]\r\n\r\n53. [latex]{t}_{38}[\/latex]\r\n\r\n55. 0.25\r\n\r\n<\/div>\r\n57.\u00a0(2.93, 3.59)\r\n\r\n59.\u00a0We are 95% confident that the true mean number of colors for national flags is between 2.93 colors and 3.59 colors.\r\n\r\n60.\u00a0The error bound would become EBM = 0.245. This error bound decreases because as sample sizes increase, variability decreases and we need less interval length to capture the true mean.\r\n\r\n<\/section>\r\n<h2 data-type=\"solution\">A Population Proportion \u2014 Practice<\/h2>\r\n63. The sample size needed would increase. As the confidence level increases,\u00a0<span id=\"MathJax-Element-71-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;&amp;#x3B1;&lt;\/mi&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mi&gt;\u03b1&lt;\/mi&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><\/span>\r\n\r\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><semantics><annotation-xml encoding=\"MathML-Content\"><mi>\u03b1\u00a0<\/mi><\/annotation-xml><\/semantics><\/math>\r\n\r\ndecreases and [latex]z_{\\frac{\\alpha}{2}}[\/latex]\u00a0increases. To maintain the same error bound, the size of the sample needs to increase.\r\n\r\n65.\u00a0<em data-effect=\"italics\">X<\/em> is the number of \u201csuccesses\u201d where the woman makes the majority of the purchasing decisions for the household. <em data-effect=\"italics\">P<\/em>\u2032 is the percentage of households sampled where the woman makes the majority of the purchasing decisions for the household.\r\n\r\n67. CI: (0.5321, 0.6679)\r\n<figure id=\"fs-idp73910688\"><span id=\"eip-idp62068224\" data-type=\"media\" data-alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 0.6 on the horizontal axis. A central region is shaded between points 0.5321 and 0.6679.\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214647\/CNX_Stats_C08_M04_item002annoN.jpg\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 0.6 on the horizontal axis. A central region is shaded between points 0.5321 and 0.6679.\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\r\n<p id=\"eip-idp97751072\"><em data-effect=\"italics\">EBM<\/em>: 0.0679<\/p>\r\n69.\u00a0<em data-effect=\"italics\">X<\/em> is the number of \u201csuccesses\u201d where an executive prefers a truck. <em data-effect=\"italics\">P<\/em>\u2032 is the percentage of executives sampled who prefer a truck.\r\n\r\n71. CI: (0.19432, 0.33068)\r\n<figure id=\"fs-idm39446528\"><span id=\"eip-idp150796640\" data-type=\"media\" data-alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 0.26 on the horizontal axis. A central region is shaded between points 0.1943 and 0.3307.\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214650\/CNX_Stats_C08_M04_item002anno.jpg\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 0.26 on the horizontal axis. A central region is shaded between points 0.1943 and 0.3307.\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\r\n<p id=\"eip-idp173934176\"><em data-effect=\"italics\">EBM<\/em>: 0.0707<\/p>\r\n73.\u00a0<span style=\"font-size: 1rem; text-align: initial;\">The sampling error means that the true mean can be 2% above or below the sample mean.<\/span>\r\n\r\n75.\u00a0<em data-effect=\"italics\">P<\/em>\u2032 is the proportion of voters sampled who said the economy is the most important issue in the upcoming election.\r\n\r\n79.\u00a0The number of girls, ages 8 to 12, in the 5 p.m. Monday night beginning ice-skating class.\r\n\r\n81.\r\n<ul data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ul data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol id=\"fs-idp45823456\" data-number-style=\"lower-alpha\">\r\n \t<li><em data-effect=\"italics\">x<\/em> = 64<\/li>\r\n \t<li><em data-effect=\"italics\">n<\/em> = 80<\/li>\r\n \t<li><em data-effect=\"italics\">p<\/em>\u2032 = 0.8<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n83. p\r\n\r\n85. P'<span id=\"MathJax-Span-7219\" class=\"mo\">~<\/span><span id=\"MathJax-Span-7220\" class=\"mi\">N[latex]\\left(0.8,\\sqrt{\\frac{{(0.8)(0.2)}}{{80}}}\\right)[\/latex].\u00a0(0.72171, 0.87829).<\/span>\r\n\r\n87. 0.04\r\n\r\n89.\u00a0(0.72; 0.88)\r\n\r\n91.\u00a0With 92% confidence, we estimate the proportion of girls, ages 8 to 12, in a beginning ice-skating class at the Ice Chalet to be between 72% and 88%.\r\n\r\n93.\u00a0The error bound would increase. Assuming all other variables are kept constant, as the confidence level increases, the area under the curve corresponding to the confidence level becomes larger, which creates a wider interval and thus a larger error.\r\n\r\n<section id=\"fs-idm34727072\" class=\"free-response focusable\" tabindex=\"-1\" data-depth=\"1\">\r\n<div id=\"eip-77\" class=\"exercise\" data-type=\"exercise\"><section class=\" focusable\" tabindex=\"-1\">\r\n<div class=\"problem\" data-type=\"problem\">\r\n<h2>A Single Population Mean using the Normal Distribution \u2014 Homework<\/h2>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-idm104841536\" class=\"free-response focusable\" tabindex=\"-1\" data-depth=\"1\">\r\n<div id=\"exer1\" class=\"exercise\" data-type=\"exercise\"><section class=\" focusable\" tabindex=\"-1\">\r\n<div id=\"id4953529\" class=\"problem\" data-type=\"problem\">95.<\/div>\r\n<div data-type=\"problem\">a.<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div class=\"exercise\" data-type=\"exercise\"><section>\r\n<div class=\"solution ui-solution-visible\" data-type=\"solution\"><section class=\"ui-body\">\r\n<div id=\"list_s1\" data-type=\"list\" data-list-type=\"enumerated\" data-number-style=\"lower-alpha\">\r\n<div data-type=\"item\">\r\n<div id=\"list_s2\" data-type=\"list\" data-list-type=\"enumerated\" data-number-style=\"lower-roman\">\r\n<ol>\r\n \t<li data-type=\"item\">71<\/li>\r\n \t<li data-type=\"item\">3<\/li>\r\n \t<li data-type=\"item\">48<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"item\">b. X is the height of a Swiss male, and is the mean height from a sample of 48 Swiss males.<\/div>\r\n<div data-type=\"item\">c. Normal. We know the standard deviation for the population, and the sample size is greater than 30.<\/div>\r\n<div data-type=\"item\">\r\n<div id=\"list_s3\" data-type=\"list\" data-list-type=\"enumerated\" data-number-style=\"lower-roman\">\r\n<div data-type=\"item\">d. CI: (70.151, 71.49)<\/div>\r\n<div data-type=\"item\">\r\n<figure id=\"figure1234\"><span id=\"m1\" data-type=\"media\" data-alt=\"\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214630\/fig-ch08_09_01.jpg\" alt=\"\" width=\"380\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\r\n<\/div>\r\n<div data-type=\"item\"><em data-effect=\"italics\">EBM<\/em> = 0.849<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"item\">e. The confidence interval will decrease in size, because the sample size increased. Recall, when all factors remain unchanged, an increase in sample size decreases variability. Thus, we do not need as large an interval to capture the true population mean.<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div class=\"problem\" data-type=\"problem\"><\/div>\r\n<div class=\"problem\" data-type=\"problem\">97.<\/div>\r\n<div class=\"problem\" data-type=\"problem\">\r\n\r\na.\r\n<ol>\r\n \t<li>[latex]\\overline{x}[\/latex]\u00a0= 23.6<\/li>\r\n \t<li>\u03c3 = 7<\/li>\r\n \t<li>n = 100<\/li>\r\n<\/ol>\r\nb. X is the time needed to complete an individual tax form. [latex]\\overline{X}[\/latex]\u00a0is the mean time to complete tax forms from a sample of 100 customers.\r\n\r\nc. N(23.6,[latex]\\frac{{7}}{{\\sqrt{100}}}[\/latex]) because we know sigma.\r\n\r\nd.\r\n<ol>\r\n \t<li>(22.228, 24.972)<\/li>\r\n \t<li>\r\n<figure id=\"eip-idp4235472\"><span id=\"eip-id1706129a\" data-type=\"media\" data-alt=\"\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214633\/xx.jpg\" alt=\"\" width=\"400\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\r\n<\/li>\r\n \t<li><em data-effect=\"italics\">EBM<\/em> = 1.372<\/li>\r\n<\/ol>\r\ne. It will need to change the sample size. The firm needs to determine what the confidence level should be, then apply the error bound formula to determine the necessary sample size.\r\n\r\nf. The confidence level would increase as a result of a larger interval. Smaller sample sizes result in more variability. To capture the true population mean, we need to have a larger interval.\r\n\r\ng. According to the error bound formula, the firm needs to survey 206 people. Since we increase the confidence level, we need to increase either our error bound or the sample size.\r\n\r\n99.\r\n\r\na.\r\n<div data-type=\"item\">\r\n<div id=\"listaishdaskkdhghdkdlsslskd\" data-type=\"list\" data-list-type=\"enumerated\" data-number-style=\"lower-roman\">\r\n<ol>\r\n \t<li data-type=\"item\">7.9<\/li>\r\n \t<li data-type=\"item\">2.5<\/li>\r\n \t<li data-type=\"item\">20<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"item\">b. <em>X<\/em> is the number of letters a single camper will send home. [latex]\\overline{X}[\/latex]is the mean number of letters sent home from a sample of 20 campers.<\/div>\r\n<div data-type=\"item\">c. N 7.9([latex]\\frac{{2.5}}{{\\sqrt{20}}}[\/latex])<\/div>\r\n<div data-type=\"item\">\r\n<div id=\"eip-id1172668181207\" data-type=\"list\" data-list-type=\"enumerated\" data-number-style=\"lower-roman\">\r\n<div data-type=\"item\">d.<\/div>\r\n<ol>\r\n \t<li data-type=\"item\">CI: (6.98, 8.82)<span id=\"eip-id1169152053385\" data-type=\"media\" data-alt=\"\"><\/span><\/li>\r\n \t<li data-type=\"item\"><span id=\"eip-id1169152053385\" data-type=\"media\" data-alt=\"\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214635\/CNX_Stats_C08_M01_item001.jpg\" alt=\"\" width=\"400\" data-media-type=\"image\/png\" \/><\/span><\/li>\r\n \t<li data-type=\"item\"><em data-effect=\"italics\">EBM<\/em>: 0.92<\/li>\r\n<\/ol>\r\n<div data-type=\"item\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"item\">e. The error bound and confidence interval will decrease.<\/div>\r\n<\/div>\r\n101.\r\n\r\na. [latex]\\overline{x}[\/latex] = $568,873\r\n\r\nb.\u00a0[latex]CL=0.95 \\alpha = 1-0.95 = 0.05 z_{\\frac{\\alpha}{2}} = 1.96[\/latex]\r\n\r\n[latex]EBM = z_{0.025} \\frac{\\sigma}{\\sqrt{n}} = 1.96 \\frac{909200}{\\sqrt{40}} = \\$ 281,764[\/latex]\r\n\r\nc. [latex]\\overline{x} - EBM = 568,873 - 281,764 = 287,109[\/latex]\r\n\r\n[latex]\\overline{x} + 568,873 + 281,764 = 850,637[\/latex]\r\n\r\nd.\u00a0We estimate with 95% confidence that the mean amount of contributions received from all individuals by House candidates is between $287,109 and $850,637.\r\n\r\n&nbsp;\r\n\r\n103.\r\n\r\nUse the formula for <em data-effect=\"italics\">EBM<\/em>, solved for <em data-effect=\"italics\">n<\/em>:\r\n\r\nn = [latex]\\frac{{{z}^{2}{\\sigma}^{2}}}{{{EBM}^{2}}}[\/latex]\r\n<p id=\"eip-idm89942784\">From the statement of the problem, you know that <em data-effect=\"italics\">\u03c3<\/em> = 2.5, and you need<em data-effect=\"italics\">EBM<\/em> = 1.<\/p>\r\n<p id=\"eip-idp13947008\"><em data-effect=\"italics\">z<\/em> = <em data-effect=\"italics\">z<\/em><sub>0.035<\/sub> = 1.812<\/p>\r\n<p id=\"eip-idm56424256\">(This is the value of <em data-effect=\"italics\">z<\/em> for which the area under the density curve to the <strong><em data-effect=\"italics\"> right <\/em><\/strong>of <em data-effect=\"italics\"> z <\/em> is 0.035.)<\/p>\r\n<p id=\"eip-idp85126192\">n = [latex]\\frac{{{z}^{2}{\\sigma}^{2}}}{{{EBM}^{2}}}[\/latex] = [latex]\\frac{{{1.812}^{2}{2.5}^{2}}}{{{1}^{2}}}\\approx{20.52}[\/latex]<\/p>\r\n<p id=\"eip-idm82999392\">You need to measure at least 21 male students to achieve your goal.<\/p>\r\n\r\n<h2>A Single Population Mean using the Student t Distribution \u2014 Homework<\/h2>\r\n<div id=\"element-542\" class=\"exercise\" data-type=\"exercise\"><section class=\" focusable\" tabindex=\"-1\">\r\n<div id=\"eip-idm61120128\" class=\"solution\" data-type=\"solution\"><\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-idp293092128\" class=\"free-response focusable\" tabindex=\"-1\" data-depth=\"1\">\r\n<div id=\"exer2\" class=\"exercise\" data-type=\"exercise\"><section class=\" focusable\" tabindex=\"-1\">\r\n<div id=\"id6354520\" class=\"problem\" data-type=\"problem\"><\/div>\r\n<\/section><\/div>\r\n<\/section>105.\r\n\r\na.\r\n<div class=\"exercise\" data-type=\"exercise\"><section>\r\n<div class=\"solution ui-solution-visible\" data-type=\"solution\"><section class=\"ui-body\">\r\n<ol id=\"list_s4\" data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol id=\"list_s5\" data-number-style=\"lower-roman\">\r\n \t<li>8629<\/li>\r\n \t<li>6944<\/li>\r\n \t<li>35<\/li>\r\n \t<li>34<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<span id=\"MathJax-Element-432-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-5314\" class=\"math\"><span id=\"MathJax-Span-5315\" class=\"mrow\"><span id=\"MathJax-Span-5316\" class=\"semantics\"><span id=\"MathJax-Span-5317\" class=\"mrow\"><span id=\"MathJax-Span-5318\" class=\"mrow\"><span id=\"MathJax-Span-5319\" class=\"msub\"><span id=\"MathJax-Span-5320\" class=\"mi\">b. [latex]t_{<\/span><span id=\"MathJax-Span-5321\" class=\"mrow\"><span id=\"MathJax-Span-5322\" class=\"mn\">34}[\/latex]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\r\n<ol id=\"list_s7\" data-number-style=\"lower-roman\">\r\n \t<li>CI: (6244, 11,014)<\/li>\r\n \t<li>\r\n<figure id=\"fs-idp12588848\"><span id=\"mmm\" data-type=\"media\" data-alt=\"\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214642\/fig-ch08_10_01.jpg\" alt=\"\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\r\n<\/li>\r\n \t<li>EB = 2385<\/li>\r\n<\/ol>\r\nd. It will become smaller\r\n\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div class=\"problem\" data-type=\"problem\">107.<\/div>\r\n<div data-type=\"problem\">a.<\/div>\r\n<div data-type=\"problem\">\r\n<ol>\r\n \t<li><span id=\"MathJax-Element-90-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; overflow: initial; display: inline-table; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;mo&gt;&amp;#xAF;&lt;\/mo&gt;&lt;\/mover&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;mo&gt;\u00af&lt;\/mo&gt;&lt;\/mover&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-760\" class=\"math\"><span id=\"MathJax-Span-761\" class=\"mrow\"><span id=\"MathJax-Span-762\" class=\"semantics\"><span id=\"MathJax-Span-763\" class=\"mrow\"><span id=\"MathJax-Span-764\" class=\"mover\"><span id=\"MathJax-Span-765\" class=\"mi\">[latex]\\overline{x}<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0= 2.51[\/latex]<\/li>\r\n \t<li><span style=\"white-space: nowrap;\">[latex]s_{x} = 0.318[\/latex]<\/span><\/li>\r\n \t<li><em data-effect=\"italics\">n<\/em>\u00a0= 9<\/li>\r\n \t<li><em data-effect=\"italics\">n<\/em>\u00a0- 1 = 8<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\nb. the effective length of time for a tranquilizer\r\n\r\nc. the mean effective length of time of tranquilizers from a sample of nine patients\r\n\r\nd. We need to use a Student\u2019s t-distribution, because we do not know the population standard deviation.\r\n\r\ne.\r\n\r\n&nbsp;\r\n<ol>\r\n \t<li>CI: (2.27, 2.76)<\/li>\r\n \t<li>Check student's solution.<\/li>\r\n \t<li><em data-effect=\"italics\">EBM<\/em>: 0.25<\/li>\r\n<\/ol>\r\nf.\u00a0 If we were to sample many groups of nine patients, 95% of the samples would contain the true population mean length of time.\r\n\r\n<\/div>\r\n<div class=\"problem\" data-type=\"problem\">\r\n\r\n109. [latex]\\overline{x}[\/latex]= $,854.23, s =\u00a0$521,130.41\r\n<p id=\"fs-idm60939744\">Note that we are not given the population standard deviation, only the standard deviation of the sample.<\/p>\r\n<p id=\"eip-idm63664256\">There are 30 measures in the sample, so <em data-effect=\"italics\">n<\/em> = 30, and <em data-effect=\"italics\">df<\/em> = 30 - 1 = 29<\/p>\r\n<p id=\"fs-idm117819488\"><em data-effect=\"italics\">CL<\/em> = 0.96, so <em data-effect=\"italics\">\u03b1<\/em> = 1 - <em data-effect=\"italics\">CL<\/em> = 1 - 0.96 = 0.04<\/p>\r\n[latex]\\frac{{\\alpha}}{{2}}=0.02={t}_{0.02}-2.150[\/latex]\r\n\r\nEBM =\u00a0[latex]{t}_{\\frac{\\alpha}{2}}\\left(\\frac{s}{\\sqrt{n}}\\right)=2.150\\left(\\frac{521,130.41}{\\sqrt{30}}\\right)[\/latex]\u00a0~ $204,561.66\r\n\r\n[latex]\\overline{x}[\/latex]-EBM =\u00a0$251,854.23 - $204,561.66 = $47,292.57\r\n\r\n[latex]\\overline{x}[\/latex]+EBM =\u00a0$251,854.23+ $204,561.66 = $456,415.89\r\n<p id=\"eip-idm43369008\">We estimate with 96% confidence that the mean amount of money raised by all Leadership PACs during the 2011\u20132012 election cycle lies between $47,292.57 and $456,415.89.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<p id=\"eip-idm19923248\">Alternate Solution<\/p>\r\n\r\n<div id=\"fs-idm33267104\" class=\"note statistics calculator\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><section>\r\n<p id=\"fs-idm130936496\">Enter the data as a list.<\/p>\r\n<p id=\"fs-idm33521776\">Press <code>STAT<\/code> and arrow over to <code>TESTS<\/code>.<\/p>\r\n<p id=\"fs-idm74740208\">Arrow down to <code>8:TInterval<\/code>.<\/p>\r\n<p id=\"fs-idm137623792\">Press <code>ENTER<\/code>.<\/p>\r\n<p id=\"fs-idm117237136\">Arrow to Data and press <code>ENTER<\/code>.<\/p>\r\n<p id=\"fs-idm121069488\">Arrow down and enter the name of the list where the data is stored.<\/p>\r\n<p id=\"fs-idm76431648\">Enter <code>Freq<\/code>: 1<\/p>\r\n<p id=\"fs-idm157410432\">Enter <code>C-Level<\/code>: 0.96<\/p>\r\n<p id=\"fs-idm14318528\">Arrow down to <code>Calculate<\/code> and press <code>Enter<\/code>.<\/p>\r\n<p id=\"fs-idm131790576\">The 96% confidence interval is ($47,262, $456,447).<\/p>\r\n\r\n<\/section><\/div>\r\n<p id=\"eip-idp167827024\">The difference between solutions arises from rounding differences.<\/p>\r\n\r\n<\/div>\r\n111.\r\n\r\na.\r\n<ul data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ul data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol>\r\n \t<li style=\"list-style-type: none;\">\r\n<ol id=\"fs-idp16278992\" data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol id=\"fs-idm150491184\" data-number-style=\"lower-roman\">\r\n \t<li>[latex]\\overline{x}[\/latex]=<\/li>\r\n \t<li>s<sub>x<\/sub> =<\/li>\r\n \t<li><em data-effect=\"italics\">n<\/em> =<\/li>\r\n \t<li><em data-effect=\"italics\">n<\/em> - 1 =<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nb. <em>X<\/em> is the number of unoccupied seats on a single flight. [latex]\\overline{X}[\/latex] is the mean number of unoccupied seats from a sample of 225 flights.\r\n\r\nc. We will use a Student\u2019s t-distribution, because we do not know the population standard deviation.\r\n\r\nd.\r\n<ol id=\"fs-idm19293168\" data-number-style=\"lower-roman\">\r\n \t<li>CI: (11.12 , 12.08)<\/li>\r\n \t<li>Check student's solution.<\/li>\r\n \t<li><em data-effect=\"italics\">EBM<\/em>: 0.48<\/li>\r\n<\/ol>\r\n113.\r\n\r\na.\r\n<ul data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ul data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol id=\"fs-idm144387520\" data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol id=\"fs-idm104467472\" data-number-style=\"lower-roman\">\r\n \t<li>CI: (7.64 , 9.36)<\/li>\r\n \t<li>\r\n<figure id=\"fs-idm59520752\"><span id=\"mmm3\" data-type=\"media\" data-alt=\"\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214645\/fig-ch08_12_01.jpg\" alt=\"\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\r\n<\/li>\r\n \t<li>\r\n<figure><span data-type=\"media\" data-alt=\"\"><em data-effect=\"italics\">EBM<\/em>: 0.86<\/span><\/figure>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nb. The sample should have been increased.\r\n\r\nc. Answers will vary.\r\n\r\nd. Answers will vary.\r\n\r\ne. Answers will vary.\r\n\r\n115.\u00a0(12.32, 14.29)\r\n<h2>A Population Proportion<\/h2>\r\n117.\r\n<ul data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ul data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol id=\"fs-idm22599856\" data-number-style=\"lower-alpha\">\r\n \t<li>1,068<\/li>\r\n \t<li>The sample size would need to be increased since the critical value increases as the confidence level increases.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n119.\r\n<ul data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ul data-number-style=\"lower-alpha\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol id=\"list_solution\" data-number-style=\"lower-alpha\">\r\n \t<li>\r\n<p id=\"fs-idm2730480\"><em data-effect=\"italics\">X<\/em> = the number of people who feel that the president is doing an acceptable job;\u00a0<em data-effect=\"italics\">P<\/em>\u2032 = the proportion of people in a sample who feel that the president is doing an acceptable job.<\/p>\r\n<\/li>\r\n \t<li>N[latex]\\left(0.61,\\sqrt{\\frac{{(0.61)(0.39)}}{{1200}}}\\right)[\/latex]\r\n<ol id=\"list_solution1\" data-number-style=\"lower-roman\">\r\n \t<li>CI: (0.59, 0.63)<\/li>\r\n \t<li>Check student\u2019s solution<\/li>\r\n \t<li><em data-effect=\"italics\">EBM<\/em>: 0.02<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n121.\r\n\r\na.\r\n\r\n&nbsp;\r\n<ol>\r\n \t<li>(0.72, 0.82)<\/li>\r\n \t<li>(0.65, 0.76)<\/li>\r\n \t<li>(0.60, 0.72)<\/li>\r\n<\/ol>\r\nb. Yes, the intervals (0.72, 0.82) and (0.65, 0.76) overlap, and the intervals (0.65, 0.76) and (0.60, 0.72) overlap.\r\n\r\nc. We can say that there does not appear to be a significant difference between the proportion of Asian adults who say that their families would welcome a White person into their families and the proportion of Asian adults who say that their families would welcome a Latino person into their families.\r\n\r\nd. We can say that there is a significant difference between the proportion of Asian adults who say that their families would welcome a White person into their families and the proportion of Asian adults who say that their families would welcome a Black person into their families.\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n123.\r\n<div data-type=\"item\">a. <em data-effect=\"italics\">X<\/em> = the number of adult Americans who feel that crime is the main problem; <em data-effect=\"italics\">P\u2032<\/em> = the proportion of adult Americans who feel that crime is the main problem<\/div>\r\n<div data-type=\"item\">\r\n\r\nb. Since we are estimating a proportion, given <em data-effect=\"italics\">P\u2032<\/em> = 0.2 and <em data-effect=\"italics\">n<\/em> = 1000, the distribution we should use is [latex]N\\left(0.2,\\sqrt{\\frac{\\left(0.2\\right)}{\\left(0.8\\right)}}{1000}\\right)[\/latex].\r\n<div data-type=\"item\">\r\n<div id=\"fs-idm23582576\" data-type=\"list\" data-list-type=\"enumerated\" data-number-style=\"lower-roman\">\r\n<div data-type=\"item\">c.<\/div>\r\n<ol>\r\n \t<li data-type=\"item\">CI: (0.18, 0.22)<\/li>\r\n \t<li data-type=\"item\">Check student\u2019s solution.<\/li>\r\n \t<li data-type=\"item\"><em data-effect=\"italics\">EBM<\/em>: 0.02<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"item\">d. One way to lower the sampling error is to increase the sample size.<\/div>\r\n<div data-type=\"item\">e. The stated \u201c\u00b1 3%\u201d represents the maximum error bound. This means that those doing the study are reporting a maximum error of 3%. Thus, they estimate the percentage of adult Americans who feel that crime is the main problem to be between 18% and 22%.<\/div>\r\n<div data-type=\"item\"><\/div>\r\n<\/div>\r\n<div data-type=\"item\">125. 0.79<\/div>\r\n<div data-type=\"item\">127. 0.030<\/div>\r\n<div data-type=\"item\">129. 0.6614<\/div>\r\n131.\r\n\r\na. p' = [latex]\\frac{{(0.55+0.49)}}{{2}}=0.52[\/latex], \u00a0<em data-effect=\"italics\">EBP<\/em> = 0.55 - 0.52 = 0.03\r\n\r\nb. No, the confidence interval includes values less than or equal to 0.50. It is possible that less than half of the population believe this.\r\n\r\nc.\u00a0<em data-effect=\"italics\">CL<\/em>\u00a0= 0.75, so\u00a0<em data-effect=\"italics\">\u03b1<\/em>\u00a0= 1 \u2013 0.75 = 0.25 and [latex]\\frac{\\alpha}{2} = 0.125 \\ z_{\\frac{\\alpha}{2}} = 1.150[\/latex].\u00a0(The area to the right of this\u00a0<em data-effect=\"italics\">z<\/em>\u00a0is 0.125, so the area to the left is 1 \u2013 0.125 = 0.875.)\r\n<div data-type=\"newline\">EBP=(1.150)[latex]\\sqrt{\\frac{0.52(0.48)}{1000}}[\/latex]\u22480.018<\/div>\r\n<div data-type=\"newline\"><\/div>\r\n(<em data-effect=\"italics\">p<\/em>\u2032 - <em data-effect=\"italics\">EBP<\/em>, <em data-effect=\"italics\">p<\/em>\u2032 + <em data-effect=\"italics\">EBP<\/em>) = (0.52 \u2013 0.018, 0.52 + 0.018) = (0.502, 0.538)\r\n<p id=\"eip-idp88904352\">Alternate Solution<\/p>\r\n\r\n<div id=\"fs-idp94228912\" class=\"note statistics calculator\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><section>\r\n<p id=\"fs-idm22151456\">STAT TESTS A: 1-PropZinterval with <em data-effect=\"italics\">x<\/em> = (0.52)(1,000), <em data-effect=\"italics\">n<\/em> = 1,000, CL = 0.75.<\/p>\r\n<p id=\"fs-idm20874816\">Answer is (0.502, 0.538)<\/p>\r\nd. Yes \u2013 this interval does not fall less than 0.50 so we can conclude that at least half of all American adults believe that major sports programs corrupt education \u2013 but we do so with only 75% confidence.<em data-effect=\"italics\">CL<\/em> = 0.75, so <em data-effect=\"italics\">\u03b1<\/em> = 1 \u2013 0.75 = 0.25 and [latex\\frac{{\\alpha}}{{2}}<span class=\"mo\">=<\/span><span class=\"mn\"><span class=\"mn\">0.125{z}_{\\frac{{\\alpha}}{{2}}}[\/latex]<span id=\"MathJax-Element-542-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-7299\" class=\"math\"><span id=\"MathJax-Span-7300\" class=\"mrow\"><span id=\"MathJax-Span-7301\" class=\"semantics\"><span id=\"MathJax-Span-7302\" class=\"mrow\"><span id=\"MathJax-Span-7303\" class=\"mrow\"><span class=\"mo\">=<\/span><span class=\"mn\">1.150<\/span><\/span><\/span><\/span><\/span><\/span><\/span>. (The area to the right of this <em data-effect=\"italics\">z<\/em> is 0.125, so the area to the left is 1 \u2013 0.125 = 0.875.)<\/span><\/span>\r\n\r\n<\/section>133.<em data-effect=\"italics\">CL<\/em>= 0.95, so<em data-effect=\"italics\">\u03b1<\/em>= 1 \u2013 0.95 = 0.05 and [latex]\\frac{\\alpha}{2} = 0.025 \\ z_{\\frac{\\alpha}{2}} = 1.96[\/latex].\u00a0Use\u00a0<em data-effect=\"italics\">p<\/em>\u2032 =\u00a0<em data-effect=\"italics\">q<\/em>\u2032 = 0.5.\r\n\r\n[latex]n=\\frac{z_{\\frac{\\alpha}{2}} ^{2} p' q'}{EBP^2} = \\frac{1.96^2 (0.5)(0.5)}{0.05^2} = 384.16[\/latex].\u00a0You need to interview at least 385 students to estimate the proportion to within 5% at 95% confidence.\r\n\r\n<\/div>","rendered":"<h2>A Single Population Mean using the Normal Distribution \u2013 Practice<\/h2>\n<p>1.<\/p>\n<ul data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ul data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ul>\n<li>244<\/li>\n<li>15<\/li>\n<li>50<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>3.\u00a0[latex]N\\left(244,\\frac{15}{\\sqrt{50}}\\right)[\/latex]<\/p>\n<p>5. As the sample size increases, there will be less variability in the mean, so the interval size decreases.<\/p>\n<p>7. <em>X<\/em> is the time in minutes it takes to complete the U.S. Census short form. [latex]\\overline{X}[\/latex] is the mean time it took a sample of 200 people to complete the U.S. Census short form.<\/p>\n<p>9.<\/p>\n<p>CI: (7.9441, 8.4559)<\/p>\n<figure id=\"eip-idm5552576\"><span id=\"eip-idm128260720\" data-type=\"media\" data-alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 8.2 on the horizontal axis. A central region is shaded between points 7.94 and 8.46.\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214626\/CNX_Stats_C08_M02_item002annoN.jpg\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 8.2 on the horizontal axis. A central region is shaded between points 7.94 and 8.46.\" width=\"380\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\n<p id=\"eip-idm5550160\"><em data-effect=\"italics\">EBM<\/em> = 0.26<\/p>\n<p>11.\u00a0The level of confidence would decrease because decreasing <em data-effect=\"italics\">n<\/em> makes the confidence interval wider, so at the same error bound, the confidence level decreases.<\/p>\n<p>13. 1.\u00a0[latex]\\overline{x}[\/latex]= 2.2, 2.\u00a0<em data-effect=\"italics\">\u03c3<\/em> = 0.2, 3.\u00a0<em data-effect=\"italics\">n<\/em> = 20<\/p>\n<p>15. [latex]\\overline{X}[\/latex] is the mean weight of a sample of 20 heads of lettuce.<\/p>\n<p>17.<\/p>\n<p id=\"fs-idm25106704\"><em data-effect=\"italics\">EBM<\/em> = 0.07<\/p>\n<div data-type=\"newline\"><\/div>\n<p>CI: (2.1264, 2.2736)<\/p>\n<figure id=\"fs-idp82050400\"><span id=\"eip-idp84209776\" data-type=\"media\" data-alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 2.2 on the horizontal axis. A central region is shaded between points 2.13 and 2.27.\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214628\/CNX_Stats_C08_M02_item003annoN.jpg\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 2.2 on the horizontal axis. A central region is shaded between points 2.13 and 2.27.\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\n<p>19.\u00a0The interval is greater because the level of confidence increased. If the only change made in the analysis is a change in confidence level, then all we are doing is changing how much area is being calculated for the normal distribution. Therefore, a larger confidence level results in larger areas and larger intervals.<\/p>\n<p>21.\u00a0The confidence level would increase.<\/p>\n<p>23.\u00a030.4<\/p>\n<p>25.\u00a0<em data-effect=\"italics\">\u03c3<\/em><\/p>\n<p>27.\u00a0<em data-effect=\"italics\">\u03bc<\/em><\/p>\n<p>29.\u00a0normal<\/p>\n<p>31.\u00a00.025<\/p>\n<p>33.\u00a0(24.52,36.28)<\/p>\n<p>35.\u00a0We are 95% confident that the true mean age for Winger Foothill College students is between 24.52 and 36.28.<\/p>\n<p>37.\u00a0The error bound for the mean would decrease because as the CL decreases, you need less area under the normal curve (which translates into a smaller interval) to capture the true population mean.<\/p>\n<section class=\"focusable\" tabindex=\"-1\">\n<h2 id=\"fs-idm23029072\" class=\"solution\" data-type=\"solution\">A Single Population Mean using the Student t Distribution \u2014 Practice<\/h2>\n<p>39.\u00a0<em data-effect=\"italics\">X<\/em> is the number of hours a patient waits in the emergency room before being called back to be examined. [latex]\\overline{X}[\/latex]\u00a0is the mean wait time of 70 patients in the emergency room.<\/p>\n<p>41.<\/p>\n<div class=\"exercise\" data-type=\"exercise\">\n<section>\n<div id=\"eip-501\" class=\"solution ui-solution-visible\" data-type=\"solution\">\n<section class=\"ui-body\">\n<p id=\"eip-334\">CI: (1.3808, 1.6192)<\/p>\n<figure id=\"fs-idm150372688\"><span id=\"eip-idp2023440\" data-type=\"media\" data-alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 1.5 on the horizontal axis. A central region is shaded between points 1.38 and 1.62.\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214637\/CNX_Stats_C08_M03_item001annoN.jpg\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 1.5 on the horizontal axis. A central region is shaded between points 1.38 and 1.62.\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\n<p id=\"eip-idm67620928\"><em data-effect=\"italics\">EBM<\/em> = 0.12<\/p>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"exercise\" data-type=\"exercise\"><\/div>\n<div class=\"problem\" data-type=\"problem\">\n<p>\u00a043.<\/p>\n<ol id=\"fs-idm117900288\" data-number-style=\"lower-alpha\">\n<li>[latex]\\overline{x}[\/latex]= 151<\/li>\n<li>[latex]{x}_{s}[\/latex] = 32<\/li>\n<li><em data-effect=\"italics\">n<\/em> = 108<\/li>\n<li><em data-effect=\"italics\">n<\/em> \u2013 1 = 107<\/li>\n<\/ol>\n<p>45. [latex]\\overline{X}[\/latex] the mean number of hours spent watching television per month from a sample of 108 Americans.<\/p>\n<p>47. CI: (142.92, 159.08)<\/p>\n<div data-type=\"newline\"><\/div>\n<figure id=\"fs-idm53713472\"><span id=\"eip-idm24726752\" data-type=\"media\" data-alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 151 on the horizontal axis. A central region is shaded between points 142.92 and 159.08.\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214639\/CNX_Stats_C08_M03_item002annoN.jpg\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 151 on the horizontal axis. A central region is shaded between points 142.92 and 159.08.\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\n<\/div>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<div class=\"problem\" data-type=\"problem\">\n<p><em data-effect=\"italics\">EBM<\/em> = 8.08<\/p>\n<p>49.<\/p>\n<ol id=\"list-827634\" data-number-style=\"lower-alpha\">\n<li>3.26<\/li>\n<li>1.02<\/li>\n<li>39<\/li>\n<\/ol>\n<p>51. [latex]\\mu[\/latex]<\/p>\n<p>53. [latex]{t}_{38}[\/latex]<\/p>\n<p>55. 0.25<\/p>\n<\/div>\n<p>57.\u00a0(2.93, 3.59)<\/p>\n<p>59.\u00a0We are 95% confident that the true mean number of colors for national flags is between 2.93 colors and 3.59 colors.<\/p>\n<p>60.\u00a0The error bound would become EBM = 0.245. This error bound decreases because as sample sizes increase, variability decreases and we need less interval length to capture the true mean.<\/p>\n<\/section>\n<h2 data-type=\"solution\">A Population Proportion \u2014 Practice<\/h2>\n<p>63. The sample size needed would increase. As the confidence level increases,\u00a0<span id=\"MathJax-Element-71-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; overflow: initial; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;&amp;#x3B1;&lt;\/mi&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mi&gt;\u03b1&lt;\/mi&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><\/span><\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><semantics><annotation-xml encoding=\"MathML-Content\"><mi>\u03b1\u00a0<\/mi><\/annotation-xml><\/semantics><\/math><\/p>\n<p>decreases and [latex]z_{\\frac{\\alpha}{2}}[\/latex]\u00a0increases. To maintain the same error bound, the size of the sample needs to increase.<\/p>\n<p>65.\u00a0<em data-effect=\"italics\">X<\/em> is the number of \u201csuccesses\u201d where the woman makes the majority of the purchasing decisions for the household. <em data-effect=\"italics\">P<\/em>\u2032 is the percentage of households sampled where the woman makes the majority of the purchasing decisions for the household.<\/p>\n<p>67. CI: (0.5321, 0.6679)<\/p>\n<figure id=\"fs-idp73910688\"><span id=\"eip-idp62068224\" data-type=\"media\" data-alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 0.6 on the horizontal axis. A central region is shaded between points 0.5321 and 0.6679.\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214647\/CNX_Stats_C08_M04_item002annoN.jpg\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 0.6 on the horizontal axis. A central region is shaded between points 0.5321 and 0.6679.\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\n<p id=\"eip-idp97751072\"><em data-effect=\"italics\">EBM<\/em>: 0.0679<\/p>\n<p>69.\u00a0<em data-effect=\"italics\">X<\/em> is the number of \u201csuccesses\u201d where an executive prefers a truck. <em data-effect=\"italics\">P<\/em>\u2032 is the percentage of executives sampled who prefer a truck.<\/p>\n<p>71. CI: (0.19432, 0.33068)<\/p>\n<figure id=\"fs-idm39446528\"><span id=\"eip-idp150796640\" data-type=\"media\" data-alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 0.26 on the horizontal axis. A central region is shaded between points 0.1943 and 0.3307.\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214650\/CNX_Stats_C08_M04_item002anno.jpg\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 0.26 on the horizontal axis. A central region is shaded between points 0.1943 and 0.3307.\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\n<p id=\"eip-idp173934176\"><em data-effect=\"italics\">EBM<\/em>: 0.0707<\/p>\n<p>73.\u00a0<span style=\"font-size: 1rem; text-align: initial;\">The sampling error means that the true mean can be 2% above or below the sample mean.<\/span><\/p>\n<p>75.\u00a0<em data-effect=\"italics\">P<\/em>\u2032 is the proportion of voters sampled who said the economy is the most important issue in the upcoming election.<\/p>\n<p>79.\u00a0The number of girls, ages 8 to 12, in the 5 p.m. Monday night beginning ice-skating class.<\/p>\n<p>81.<\/p>\n<ul data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ul data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ol id=\"fs-idp45823456\" data-number-style=\"lower-alpha\">\n<li><em data-effect=\"italics\">x<\/em> = 64<\/li>\n<li><em data-effect=\"italics\">n<\/em> = 80<\/li>\n<li><em data-effect=\"italics\">p<\/em>\u2032 = 0.8<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>83. p<\/p>\n<p>85. P&#8217;<span id=\"MathJax-Span-7219\" class=\"mo\">~<\/span><span id=\"MathJax-Span-7220\" class=\"mi\">N[latex]\\left(0.8,\\sqrt{\\frac{{(0.8)(0.2)}}{{80}}}\\right)[\/latex].\u00a0(0.72171, 0.87829).<\/span><\/p>\n<p>87. 0.04<\/p>\n<p>89.\u00a0(0.72; 0.88)<\/p>\n<p>91.\u00a0With 92% confidence, we estimate the proportion of girls, ages 8 to 12, in a beginning ice-skating class at the Ice Chalet to be between 72% and 88%.<\/p>\n<p>93.\u00a0The error bound would increase. Assuming all other variables are kept constant, as the confidence level increases, the area under the curve corresponding to the confidence level becomes larger, which creates a wider interval and thus a larger error.<\/p>\n<section id=\"fs-idm34727072\" class=\"free-response focusable\" tabindex=\"-1\" data-depth=\"1\">\n<div id=\"eip-77\" class=\"exercise\" data-type=\"exercise\">\n<section class=\"focusable\" tabindex=\"-1\">\n<div class=\"problem\" data-type=\"problem\">\n<h2>A Single Population Mean using the Normal Distribution \u2014 Homework<\/h2>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-idm104841536\" class=\"free-response focusable\" tabindex=\"-1\" data-depth=\"1\">\n<div id=\"exer1\" class=\"exercise\" data-type=\"exercise\">\n<section class=\"focusable\" tabindex=\"-1\">\n<div id=\"id4953529\" class=\"problem\" data-type=\"problem\">95.<\/div>\n<div data-type=\"problem\">a.<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div class=\"exercise\" data-type=\"exercise\">\n<section>\n<div class=\"solution ui-solution-visible\" data-type=\"solution\">\n<section class=\"ui-body\">\n<div id=\"list_s1\" data-type=\"list\" data-list-type=\"enumerated\" data-number-style=\"lower-alpha\">\n<div data-type=\"item\">\n<div id=\"list_s2\" data-type=\"list\" data-list-type=\"enumerated\" data-number-style=\"lower-roman\">\n<ol>\n<li data-type=\"item\">71<\/li>\n<li data-type=\"item\">3<\/li>\n<li data-type=\"item\">48<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div data-type=\"item\">b. X is the height of a Swiss male, and is the mean height from a sample of 48 Swiss males.<\/div>\n<div data-type=\"item\">c. Normal. We know the standard deviation for the population, and the sample size is greater than 30.<\/div>\n<div data-type=\"item\">\n<div id=\"list_s3\" data-type=\"list\" data-list-type=\"enumerated\" data-number-style=\"lower-roman\">\n<div data-type=\"item\">d. CI: (70.151, 71.49)<\/div>\n<div data-type=\"item\">\n<figure id=\"figure1234\"><span id=\"m1\" data-type=\"media\" data-alt=\"\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214630\/fig-ch08_09_01.jpg\" alt=\"\" width=\"380\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\n<\/div>\n<div data-type=\"item\"><em data-effect=\"italics\">EBM<\/em> = 0.849<\/div>\n<\/div>\n<\/div>\n<div data-type=\"item\">e. The confidence interval will decrease in size, because the sample size increased. Recall, when all factors remain unchanged, an increase in sample size decreases variability. Thus, we do not need as large an interval to capture the true population mean.<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"problem\" data-type=\"problem\"><\/div>\n<div class=\"problem\" data-type=\"problem\">97.<\/div>\n<div class=\"problem\" data-type=\"problem\">\n<p>a.<\/p>\n<ol>\n<li>[latex]\\overline{x}[\/latex]\u00a0= 23.6<\/li>\n<li>\u03c3 = 7<\/li>\n<li>n = 100<\/li>\n<\/ol>\n<p>b. X is the time needed to complete an individual tax form. [latex]\\overline{X}[\/latex]\u00a0is the mean time to complete tax forms from a sample of 100 customers.<\/p>\n<p>c. N(23.6,[latex]\\frac{{7}}{{\\sqrt{100}}}[\/latex]) because we know sigma.<\/p>\n<p>d.<\/p>\n<ol>\n<li>(22.228, 24.972)<\/li>\n<li>\n<figure id=\"eip-idp4235472\"><span id=\"eip-id1706129a\" data-type=\"media\" data-alt=\"\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214633\/xx.jpg\" alt=\"\" width=\"400\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\n<\/li>\n<li><em data-effect=\"italics\">EBM<\/em> = 1.372<\/li>\n<\/ol>\n<p>e. It will need to change the sample size. The firm needs to determine what the confidence level should be, then apply the error bound formula to determine the necessary sample size.<\/p>\n<p>f. The confidence level would increase as a result of a larger interval. Smaller sample sizes result in more variability. To capture the true population mean, we need to have a larger interval.<\/p>\n<p>g. According to the error bound formula, the firm needs to survey 206 people. Since we increase the confidence level, we need to increase either our error bound or the sample size.<\/p>\n<p>99.<\/p>\n<p>a.<\/p>\n<div data-type=\"item\">\n<div id=\"listaishdaskkdhghdkdlsslskd\" data-type=\"list\" data-list-type=\"enumerated\" data-number-style=\"lower-roman\">\n<ol>\n<li data-type=\"item\">7.9<\/li>\n<li data-type=\"item\">2.5<\/li>\n<li data-type=\"item\">20<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div data-type=\"item\">b. <em>X<\/em> is the number of letters a single camper will send home. [latex]\\overline{X}[\/latex]is the mean number of letters sent home from a sample of 20 campers.<\/div>\n<div data-type=\"item\">c. N 7.9([latex]\\frac{{2.5}}{{\\sqrt{20}}}[\/latex])<\/div>\n<div data-type=\"item\">\n<div id=\"eip-id1172668181207\" data-type=\"list\" data-list-type=\"enumerated\" data-number-style=\"lower-roman\">\n<div data-type=\"item\">d.<\/div>\n<ol>\n<li data-type=\"item\">CI: (6.98, 8.82)<span id=\"eip-id1169152053385\" data-type=\"media\" data-alt=\"\"><\/span><\/li>\n<li data-type=\"item\"><span id=\"eip-id1169152053385\" data-type=\"media\" data-alt=\"\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214635\/CNX_Stats_C08_M01_item001.jpg\" alt=\"\" width=\"400\" data-media-type=\"image\/png\" \/><\/span><\/li>\n<li data-type=\"item\"><em data-effect=\"italics\">EBM<\/em>: 0.92<\/li>\n<\/ol>\n<div data-type=\"item\"><\/div>\n<\/div>\n<\/div>\n<div data-type=\"item\">e. The error bound and confidence interval will decrease.<\/div>\n<\/div>\n<p>101.<\/p>\n<p>a. [latex]\\overline{x}[\/latex] = $568,873<\/p>\n<p>b.\u00a0[latex]CL=0.95 \\alpha = 1-0.95 = 0.05 z_{\\frac{\\alpha}{2}} = 1.96[\/latex]<\/p>\n<p>[latex]EBM = z_{0.025} \\frac{\\sigma}{\\sqrt{n}} = 1.96 \\frac{909200}{\\sqrt{40}} = \\$ 281,764[\/latex]<\/p>\n<p>c. [latex]\\overline{x} - EBM = 568,873 - 281,764 = 287,109[\/latex]<\/p>\n<p>[latex]\\overline{x} + 568,873 + 281,764 = 850,637[\/latex]<\/p>\n<p>d.\u00a0We estimate with 95% confidence that the mean amount of contributions received from all individuals by House candidates is between $287,109 and $850,637.<\/p>\n<p>&nbsp;<\/p>\n<p>103.<\/p>\n<p>Use the formula for <em data-effect=\"italics\">EBM<\/em>, solved for <em data-effect=\"italics\">n<\/em>:<\/p>\n<p>n = [latex]\\frac{{{z}^{2}{\\sigma}^{2}}}{{{EBM}^{2}}}[\/latex]<\/p>\n<p id=\"eip-idm89942784\">From the statement of the problem, you know that <em data-effect=\"italics\">\u03c3<\/em> = 2.5, and you need<em data-effect=\"italics\">EBM<\/em> = 1.<\/p>\n<p id=\"eip-idp13947008\"><em data-effect=\"italics\">z<\/em> = <em data-effect=\"italics\">z<\/em><sub>0.035<\/sub> = 1.812<\/p>\n<p id=\"eip-idm56424256\">(This is the value of <em data-effect=\"italics\">z<\/em> for which the area under the density curve to the <strong><em data-effect=\"italics\"> right <\/em><\/strong>of <em data-effect=\"italics\"> z <\/em> is 0.035.)<\/p>\n<p id=\"eip-idp85126192\">n = [latex]\\frac{{{z}^{2}{\\sigma}^{2}}}{{{EBM}^{2}}}[\/latex] = [latex]\\frac{{{1.812}^{2}{2.5}^{2}}}{{{1}^{2}}}\\approx{20.52}[\/latex]<\/p>\n<p id=\"eip-idm82999392\">You need to measure at least 21 male students to achieve your goal.<\/p>\n<h2>A Single Population Mean using the Student t Distribution \u2014 Homework<\/h2>\n<div id=\"element-542\" class=\"exercise\" data-type=\"exercise\">\n<section class=\"focusable\" tabindex=\"-1\">\n<div id=\"eip-idm61120128\" class=\"solution\" data-type=\"solution\"><\/div>\n<\/section>\n<\/div>\n<section id=\"fs-idp293092128\" class=\"free-response focusable\" tabindex=\"-1\" data-depth=\"1\">\n<div id=\"exer2\" class=\"exercise\" data-type=\"exercise\">\n<section class=\"focusable\" tabindex=\"-1\">\n<div id=\"id6354520\" class=\"problem\" data-type=\"problem\"><\/div>\n<\/section>\n<\/div>\n<\/section>\n<p>105.<\/p>\n<p>a.<\/p>\n<div class=\"exercise\" data-type=\"exercise\">\n<section>\n<div class=\"solution ui-solution-visible\" data-type=\"solution\">\n<section class=\"ui-body\">\n<ol id=\"list_s4\" data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ol id=\"list_s5\" data-number-style=\"lower-roman\">\n<li>8629<\/li>\n<li>6944<\/li>\n<li>35<\/li>\n<li>34<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><span id=\"MathJax-Element-432-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-5314\" class=\"math\"><span id=\"MathJax-Span-5315\" class=\"mrow\"><span id=\"MathJax-Span-5316\" class=\"semantics\"><span id=\"MathJax-Span-5317\" class=\"mrow\"><span id=\"MathJax-Span-5318\" class=\"mrow\"><span id=\"MathJax-Span-5319\" class=\"msub\"><span id=\"MathJax-Span-5320\" class=\"mi\">b. [latex]t_{<\/span><span id=\"MathJax-Span-5321\" class=\"mrow\"><span id=\"MathJax-Span-5322\" class=\"mn\">34}[\/latex]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<ol id=\"list_s7\" data-number-style=\"lower-roman\">\n<li>CI: (6244, 11,014)<\/li>\n<li>\n<figure id=\"fs-idp12588848\"><span id=\"mmm\" data-type=\"media\" data-alt=\"\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214642\/fig-ch08_10_01.jpg\" alt=\"\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\n<\/li>\n<li>EB = 2385<\/li>\n<\/ol>\n<p>d. It will become smaller<\/p>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"problem\" data-type=\"problem\">107.<\/div>\n<div data-type=\"problem\">a.<\/div>\n<div data-type=\"problem\">\n<ol>\n<li><span id=\"MathJax-Element-90-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; overflow: initial; display: inline-table; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;mo&gt;&amp;#xAF;&lt;\/mo&gt;&lt;\/mover&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;mo&gt;\u00af&lt;\/mo&gt;&lt;\/mover&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-760\" class=\"math\"><span id=\"MathJax-Span-761\" class=\"mrow\"><span id=\"MathJax-Span-762\" class=\"semantics\"><span id=\"MathJax-Span-763\" class=\"mrow\"><span id=\"MathJax-Span-764\" class=\"mover\"><span id=\"MathJax-Span-765\" class=\"mi\">[latex]\\overline{x}<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0= 2.51[\/latex]<\/li>\n<li><span style=\"white-space: nowrap;\">[latex]s_{x} = 0.318[\/latex]<\/span><\/li>\n<li><em data-effect=\"italics\">n<\/em>\u00a0= 9<\/li>\n<li><em data-effect=\"italics\">n<\/em>\u00a0&#8211; 1 = 8<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p>b. the effective length of time for a tranquilizer<\/p>\n<p>c. the mean effective length of time of tranquilizers from a sample of nine patients<\/p>\n<p>d. We need to use a Student\u2019s t-distribution, because we do not know the population standard deviation.<\/p>\n<p>e.<\/p>\n<p>&nbsp;<\/p>\n<ol>\n<li>CI: (2.27, 2.76)<\/li>\n<li>Check student&#8217;s solution.<\/li>\n<li><em data-effect=\"italics\">EBM<\/em>: 0.25<\/li>\n<\/ol>\n<p>f.\u00a0 If we were to sample many groups of nine patients, 95% of the samples would contain the true population mean length of time.<\/p>\n<\/div>\n<div class=\"problem\" data-type=\"problem\">\n<p>109. [latex]\\overline{x}[\/latex]= $,854.23, s =\u00a0$521,130.41<\/p>\n<p id=\"fs-idm60939744\">Note that we are not given the population standard deviation, only the standard deviation of the sample.<\/p>\n<p id=\"eip-idm63664256\">There are 30 measures in the sample, so <em data-effect=\"italics\">n<\/em> = 30, and <em data-effect=\"italics\">df<\/em> = 30 &#8211; 1 = 29<\/p>\n<p id=\"fs-idm117819488\"><em data-effect=\"italics\">CL<\/em> = 0.96, so <em data-effect=\"italics\">\u03b1<\/em> = 1 &#8211; <em data-effect=\"italics\">CL<\/em> = 1 &#8211; 0.96 = 0.04<\/p>\n<p>[latex]\\frac{{\\alpha}}{{2}}=0.02={t}_{0.02}-2.150[\/latex]<\/p>\n<p>EBM =\u00a0[latex]{t}_{\\frac{\\alpha}{2}}\\left(\\frac{s}{\\sqrt{n}}\\right)=2.150\\left(\\frac{521,130.41}{\\sqrt{30}}\\right)[\/latex]\u00a0~ $204,561.66<\/p>\n<p>[latex]\\overline{x}[\/latex]-EBM =\u00a0$251,854.23 &#8211; $204,561.66 = $47,292.57<\/p>\n<p>[latex]\\overline{x}[\/latex]+EBM =\u00a0$251,854.23+ $204,561.66 = $456,415.89<\/p>\n<p id=\"eip-idm43369008\">We estimate with 96% confidence that the mean amount of money raised by all Leadership PACs during the 2011\u20132012 election cycle lies between $47,292.57 and $456,415.89.<\/p>\n<div data-type=\"newline\"><\/div>\n<p id=\"eip-idm19923248\">Alternate Solution<\/p>\n<div id=\"fs-idm33267104\" class=\"note statistics calculator\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<section>\n<p id=\"fs-idm130936496\">Enter the data as a list.<\/p>\n<p id=\"fs-idm33521776\">Press <code>STAT<\/code> and arrow over to <code>TESTS<\/code>.<\/p>\n<p id=\"fs-idm74740208\">Arrow down to <code>8:TInterval<\/code>.<\/p>\n<p id=\"fs-idm137623792\">Press <code>ENTER<\/code>.<\/p>\n<p id=\"fs-idm117237136\">Arrow to Data and press <code>ENTER<\/code>.<\/p>\n<p id=\"fs-idm121069488\">Arrow down and enter the name of the list where the data is stored.<\/p>\n<p id=\"fs-idm76431648\">Enter <code>Freq<\/code>: 1<\/p>\n<p id=\"fs-idm157410432\">Enter <code>C-Level<\/code>: 0.96<\/p>\n<p id=\"fs-idm14318528\">Arrow down to <code>Calculate<\/code> and press <code>Enter<\/code>.<\/p>\n<p id=\"fs-idm131790576\">The 96% confidence interval is ($47,262, $456,447).<\/p>\n<\/section>\n<\/div>\n<p id=\"eip-idp167827024\">The difference between solutions arises from rounding differences.<\/p>\n<\/div>\n<p>111.<\/p>\n<p>a.<\/p>\n<ul data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ul data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ol>\n<li style=\"list-style-type: none;\">\n<ol id=\"fs-idp16278992\" data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ol id=\"fs-idm150491184\" data-number-style=\"lower-roman\">\n<li>[latex]\\overline{x}[\/latex]=<\/li>\n<li>s<sub>x<\/sub> =<\/li>\n<li><em data-effect=\"italics\">n<\/em> =<\/li>\n<li><em data-effect=\"italics\">n<\/em> &#8211; 1 =<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>b. <em>X<\/em> is the number of unoccupied seats on a single flight. [latex]\\overline{X}[\/latex] is the mean number of unoccupied seats from a sample of 225 flights.<\/p>\n<p>c. We will use a Student\u2019s t-distribution, because we do not know the population standard deviation.<\/p>\n<p>d.<\/p>\n<ol id=\"fs-idm19293168\" data-number-style=\"lower-roman\">\n<li>CI: (11.12 , 12.08)<\/li>\n<li>Check student&#8217;s solution.<\/li>\n<li><em data-effect=\"italics\">EBM<\/em>: 0.48<\/li>\n<\/ol>\n<p>113.<\/p>\n<p>a.<\/p>\n<ul data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ul data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ol id=\"fs-idm144387520\" data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ol id=\"fs-idm104467472\" data-number-style=\"lower-roman\">\n<li>CI: (7.64 , 9.36)<\/li>\n<li>\n<figure id=\"fs-idm59520752\"><span id=\"mmm3\" data-type=\"media\" data-alt=\"\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214645\/fig-ch08_12_01.jpg\" alt=\"\" width=\"380\" data-media-type=\"image\/jpeg\" \/><\/span><\/figure>\n<\/li>\n<li>\n<figure><span data-type=\"media\" data-alt=\"\"><em data-effect=\"italics\">EBM<\/em>: 0.86<\/span><\/figure>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>b. The sample should have been increased.<\/p>\n<p>c. Answers will vary.<\/p>\n<p>d. Answers will vary.<\/p>\n<p>e. Answers will vary.<\/p>\n<p>115.\u00a0(12.32, 14.29)<\/p>\n<h2>A Population Proportion<\/h2>\n<p>117.<\/p>\n<ul data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ul data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ol id=\"fs-idm22599856\" data-number-style=\"lower-alpha\">\n<li>1,068<\/li>\n<li>The sample size would need to be increased since the critical value increases as the confidence level increases.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>119.<\/p>\n<ul data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ul data-number-style=\"lower-alpha\">\n<li style=\"list-style-type: none;\">\n<ol id=\"list_solution\" data-number-style=\"lower-alpha\">\n<li>\n<p id=\"fs-idm2730480\"><em data-effect=\"italics\">X<\/em> = the number of people who feel that the president is doing an acceptable job;\u00a0<em data-effect=\"italics\">P<\/em>\u2032 = the proportion of people in a sample who feel that the president is doing an acceptable job.<\/p>\n<\/li>\n<li>N[latex]\\left(0.61,\\sqrt{\\frac{{(0.61)(0.39)}}{{1200}}}\\right)[\/latex]\n<ol id=\"list_solution1\" data-number-style=\"lower-roman\">\n<li>CI: (0.59, 0.63)<\/li>\n<li>Check student\u2019s solution<\/li>\n<li><em data-effect=\"italics\">EBM<\/em>: 0.02<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>121.<\/p>\n<p>a.<\/p>\n<p>&nbsp;<\/p>\n<ol>\n<li>(0.72, 0.82)<\/li>\n<li>(0.65, 0.76)<\/li>\n<li>(0.60, 0.72)<\/li>\n<\/ol>\n<p>b. Yes, the intervals (0.72, 0.82) and (0.65, 0.76) overlap, and the intervals (0.65, 0.76) and (0.60, 0.72) overlap.<\/p>\n<p>c. We can say that there does not appear to be a significant difference between the proportion of Asian adults who say that their families would welcome a White person into their families and the proportion of Asian adults who say that their families would welcome a Latino person into their families.<\/p>\n<p>d. We can say that there is a significant difference between the proportion of Asian adults who say that their families would welcome a White person into their families and the proportion of Asian adults who say that their families would welcome a Black person into their families.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>123.<\/p>\n<div data-type=\"item\">a. <em data-effect=\"italics\">X<\/em> = the number of adult Americans who feel that crime is the main problem; <em data-effect=\"italics\">P\u2032<\/em> = the proportion of adult Americans who feel that crime is the main problem<\/div>\n<div data-type=\"item\">\n<p>b. Since we are estimating a proportion, given <em data-effect=\"italics\">P\u2032<\/em> = 0.2 and <em data-effect=\"italics\">n<\/em> = 1000, the distribution we should use is [latex]N\\left(0.2,\\sqrt{\\frac{\\left(0.2\\right)}{\\left(0.8\\right)}}{1000}\\right)[\/latex].<\/p>\n<div data-type=\"item\">\n<div id=\"fs-idm23582576\" data-type=\"list\" data-list-type=\"enumerated\" data-number-style=\"lower-roman\">\n<div data-type=\"item\">c.<\/div>\n<ol>\n<li data-type=\"item\">CI: (0.18, 0.22)<\/li>\n<li data-type=\"item\">Check student\u2019s solution.<\/li>\n<li data-type=\"item\"><em data-effect=\"italics\">EBM<\/em>: 0.02<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div data-type=\"item\">d. One way to lower the sampling error is to increase the sample size.<\/div>\n<div data-type=\"item\">e. The stated \u201c\u00b1 3%\u201d represents the maximum error bound. This means that those doing the study are reporting a maximum error of 3%. Thus, they estimate the percentage of adult Americans who feel that crime is the main problem to be between 18% and 22%.<\/div>\n<div data-type=\"item\"><\/div>\n<\/div>\n<div data-type=\"item\">125. 0.79<\/div>\n<div data-type=\"item\">127. 0.030<\/div>\n<div data-type=\"item\">129. 0.6614<\/div>\n<p>131.<\/p>\n<p>a. p&#8217; = [latex]\\frac{{(0.55+0.49)}}{{2}}=0.52[\/latex], \u00a0<em data-effect=\"italics\">EBP<\/em> = 0.55 &#8211; 0.52 = 0.03<\/p>\n<p>b. No, the confidence interval includes values less than or equal to 0.50. It is possible that less than half of the population believe this.<\/p>\n<p>c.\u00a0<em data-effect=\"italics\">CL<\/em>\u00a0= 0.75, so\u00a0<em data-effect=\"italics\">\u03b1<\/em>\u00a0= 1 \u2013 0.75 = 0.25 and [latex]\\frac{\\alpha}{2} = 0.125 \\ z_{\\frac{\\alpha}{2}} = 1.150[\/latex].\u00a0(The area to the right of this\u00a0<em data-effect=\"italics\">z<\/em>\u00a0is 0.125, so the area to the left is 1 \u2013 0.125 = 0.875.)<\/p>\n<div data-type=\"newline\">EBP=(1.150)[latex]\\sqrt{\\frac{0.52(0.48)}{1000}}[\/latex]\u22480.018<\/div>\n<div data-type=\"newline\"><\/div>\n<p>(<em data-effect=\"italics\">p<\/em>\u2032 &#8211; <em data-effect=\"italics\">EBP<\/em>, <em data-effect=\"italics\">p<\/em>\u2032 + <em data-effect=\"italics\">EBP<\/em>) = (0.52 \u2013 0.018, 0.52 + 0.018) = (0.502, 0.538)<\/p>\n<p id=\"eip-idp88904352\">Alternate Solution<\/p>\n<div id=\"fs-idp94228912\" class=\"note statistics calculator\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<section>\n<p id=\"fs-idm22151456\">STAT TESTS A: 1-PropZinterval with <em data-effect=\"italics\">x<\/em> = (0.52)(1,000), <em data-effect=\"italics\">n<\/em> = 1,000, CL = 0.75.<\/p>\n<p id=\"fs-idm20874816\">Answer is (0.502, 0.538)<\/p>\n<p>d. Yes \u2013 this interval does not fall less than 0.50 so we can conclude that at least half of all American adults believe that major sports programs corrupt education \u2013 but we do so with only 75% confidence.<em data-effect=\"italics\">CL<\/em> = 0.75, so <em data-effect=\"italics\">\u03b1<\/em> = 1 \u2013 0.75 = 0.25 and [latex]<span id=\"MathJax-Element-542-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-7299\" class=\"math\"><span id=\"MathJax-Span-7300\" class=\"mrow\"><span id=\"MathJax-Span-7301\" class=\"semantics\"><span id=\"MathJax-Span-7302\" class=\"mrow\"><span id=\"MathJax-Span-7303\" class=\"mrow\"><span class=\"mo\">=<\/span><span class=\"mn\">1.150<\/span><\/span><\/span><\/span><\/span><\/span><\/span>. (The area to the right of this <em data-effect=\"italics\">z<\/em> is 0.125, so the area to the left is 1 \u2013 0.125 = 0.875.)<\/span><\/span>    <\/section>\n<p>133.<em data-effect=\"italics\">CL<\/em>= 0.95, so<em data-effect=\"italics\">\u03b1<\/em>= 1 \u2013 0.95 = 0.05 and [latex]\\frac{\\alpha}{2} = 0.025 \\ z_{\\frac{\\alpha}{2}} = 1.96[\/latex].\u00a0Use\u00a0<em data-effect=\"italics\">p<\/em>\u2032 =\u00a0<em data-effect=\"italics\">q<\/em>\u2032 = 0.5.<\/p>\n<p>[latex]n=\\frac{z_{\\frac{\\alpha}{2}} ^{2} p' q'}{EBP^2} = \\frac{1.96^2 (0.5)(0.5)}{0.05^2} = 384.16[\/latex].\u00a0You need to interview at least 385 students to estimate the proportion to within 5% at 95% confidence.<\/p>\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-275\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><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":23,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-275","chapter","type-chapter","status-publish","hentry"],"part":269,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/275","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/275\/revisions"}],"predecessor-version":[{"id":4063,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/275\/revisions\/4063"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/269"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/275\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=275"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=275"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=275"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=275"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}