{"id":292,"date":"2016-04-21T22:43:41","date_gmt":"2016-04-21T22:43:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstats1xmaster\/?post_type=chapter&#038;p=292"},"modified":"2017-10-18T21:38:23","modified_gmt":"2017-10-18T21:38:23","slug":"a-single-population-mean-using-the-normal-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-fmcc-introstats1\/chapter\/a-single-population-mean-using-the-normal-distribution\/","title":{"raw":"A Single Population Mean using the Normal Distribution","rendered":"A Single Population Mean using the Normal Distribution"},"content":{"raw":"A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of\u00a0[latex]\\displaystyle\\overline{{x}}={10}[\/latex] and we have constructed the 90% confidence interval (5, 15) where <em>EBM<\/em> = 5.\r\n<h2>Calculating the Confidence Interval<\/h2>\r\nTo construct a confidence interval for a single unknown population mean\u00a0<em>\u03bc<\/em>, <strong>where the population standard deviation is known<\/strong>,\u00a0we need [latex]\\overline{{x}}[\/latex] is the\u00a0<strong>point estimate<\/strong> of the unknown population mean <em>\u03bc<\/em>.\r\n\r\n<strong>The confidence interval estimate will have the form:<\/strong>\r\n\r\n(point estimate \u2013 error bound, point estimate + error bound) or, in symbols,\r\n[latex]\\displaystyle{(\\overline{{x}}-{E}{B}{M},\\overline{{x}}+{E}{B}{M})}[\/latex]\r\n\r\nThe margin of error (<em>EBM<\/em>) depends on the <strong>confidence level<\/strong> (abbreviated <em><strong data-redactor-tag=\"strong\">CL<\/strong><\/em>). The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. However, it is more accurate to state that the confidence level is the percent of confidence intervals that contain the true population parameter when repeated samples are taken. Most often, it is the choice of the person constructing the confidence interval to choose a confidence level of 90% or higher because that person wants to be reasonably certain of his or her conclusions.\r\n\r\nThere is another probability called alpha (<em>\u03b1<\/em>). <em>\u03b1<\/em> is related to the confidence level, <em>CL<\/em>. <em>\u03b1<\/em> is the probability that the interval does not contain the unknown population parameter.\r\n\r\nMathematically,\u00a0<em>\u03b1<\/em> + <em>CL<\/em> = 1.\r\n\r\nhttps:\/\/youtu.be\/KG921rfbTDw\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose we have collected data from a sample. We know the sample mean but we do not know the mean for the entire population.\r\n\r\nThe sample mean is seven, and the error bound for the mean is 2.5.\r\n\r\n[latex]\\displaystyle\\overline{{x}}={7}{\\quad\\text{and}\\quad}{E}{B}{M}={2.5}[\/latex]\r\n\r\nThe confidence interval is (7 \u2013 2.5, 7 + 2.5), and calculating the values gives (4.5, 9.5).\r\n\r\nIf the confidence level (<em>CL<\/em>) is 95%, then we say that, \"We estimate with 95% confidence that the true value of the population mean is between 4.5 and 9.5.\"\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nSuppose we have data from a sample. The sample mean is 15, and the error bound for the mean is 3.2.\r\n\r\nWhat is the confidence interval estimate for the population mean?\r\n\r\n(11.8, 18.2)\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nA confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of\u00a0[latex]\\displaystyle\\overline{{x}}={10}[\/latex], and we have constructed the 90% confidence interval (5, 15) where <em>EBM<\/em> = 5.\r\n\r\nTo get a 90% confidence interval, we must include the central 90% of the probability of the normal distribution. If we include the central 90%, we leave out a total of\u00a0<em>\u03b1<\/em> = 10% in both tails, or 5% in each tail, of the normal distribution.\r\n\r\n<img src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/mrwg-kzcwi37i#fixme#fixme#fixme\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 10 on the horizontal axis. The points 5 and 15 are labeled on the axis. Vertical lines are drawn from these points to the curve, and the region between the lines is shaded. The shaded region has area equal to 0.90.\" \/>\r\n\r\nTo capture the central 90%, we must go out 1.645 \"standard deviations\" on either side of the calculated sample mean. The value 1.645 is the\u00a0<em>z<\/em>-score from a standard normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail.\r\n\r\nIt is important that the \"standard deviation\" used must be appropriate for the parameter we are estimating, so in this section we need to use the standard deviation that applies to sample means, which is[latex]\\displaystyle\\frac{{\\sigma}}{{\\sqrt{n}}}[\/latex]. The fraction\u00a0[latex]\\displaystyle\\frac{{\\sigma}}{{\\sqrt{n}}}[\/latex],\u00a0is commonly called the \"standard error of the mean\" in order to distinguish clearly the standard deviation for a mean from the population standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em>.\r\n\r\n<strong data-redactor-tag=\"strong\">In summary, as a result of the central limit theorem:<\/strong>\r\n<ul>\r\n \t<li>[latex]\\displaystyle\\overline{X}[\/latex], that is,\u00a0[latex]\\displaystyle\\overline{X}{\\sim}{N}\\left({\\mu}_{x}, \\frac{{\\sigma}}{{\\sqrt{n}}}\\right)[\/latex]<\/li>\r\n \t<li><strong data-redactor-tag=\"strong\">When the population standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em> is known, we use a normal distribution to calculate the error bound.<\/strong><\/li>\r\n<\/ul>\r\n<h2>Calculating the Confidence Interval<\/h2>\r\nTo construct a confidence interval estimate for an unknown population mean, we need data from a random sample. The steps to construct and interpret the confidence interval are:\r\n<ul>\r\n \t<li>Calculate the sample mean [latex]\\displaystyle\\overline{{x}}[\/latex] from the sample data. Remember, in this section we already know the population standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em>.<\/li>\r\n \t<li>Find the <em data-redactor-tag=\"em\">z<\/em>-score that corresponds to the confidence level.<\/li>\r\n \t<li>Calculate the error bound <em data-redactor-tag=\"em\">EBM<\/em>.<\/li>\r\n \t<li>Construct the confidence interval.<\/li>\r\n \t<li>Write a sentence that interprets the estimate in the context of the situation in the problem. (Explain what the confidence interval means, in the words of the problem.)<\/li>\r\n<\/ul>\r\nWe will first examine each step in more detail, and then illustrate the process with some examples.\r\n<h2>Finding the <em data-redactor-tag=\"em\">z<\/em>-score for the Stated Confidence Level<\/h2>\r\nWhen we know the population standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em>, we use a standard normal distribution to calculate the error bound EBM and construct the confidence interval. We need to find the value of <em data-redactor-tag=\"em\">z<\/em> that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution <em data-redactor-tag=\"em\">Z<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(0, 1).\r\n\r\nThe confidence level, <em data-redactor-tag=\"em\">CL<\/em>, is the area in the middle of the standard normal distribution. <em data-redactor-tag=\"em\">CL<\/em> = 1 \u2013 <em data-redactor-tag=\"em\">\u03b1<\/em>, so <em data-redactor-tag=\"em\">\u03b1<\/em> is the area that is split equally between the two tails. Each of the tails contains an area equal to \u03b12.\r\n\r\nThe z-score that has an area to the right of \u03b12 is denoted by z\u03b12.\r\n\r\nFor example, when <em data-redactor-tag=\"em\">CL<\/em> = 0.95, <em data-redactor-tag=\"em\">\u03b1<\/em> = 0.05 and \u03b12 = 0.025; we write z\u03b12=z0.025.\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\nz\u03b12=z0.025=1.96, using a calculator, computer or a standard normal probability table.\r\n\r\n<code data-redactor-tag=\"code\">invNorm<\/code>(0.975, 0, 1) = 1.96\r\n<h3>\u00a0Note<\/h3>\r\nRemember to use the area to the LEFT of ; in this chapter the last two inputs in the invNorm command are 0, 1, because you are using a standard normal distribution <em data-redactor-tag=\"em\">Z<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(0, 1).\r\n\r\n<hr \/>\r\n\r\n<h2>Calculating the Error Bound (<em data-redactor-tag=\"em\">EBM<\/em>)<\/h2>\r\nThe error bound formula for an unknown population mean <em data-redactor-tag=\"em\">\u03bc<\/em> when the population standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em> is known is\r\n<ul>\r\n \t<li>EBM = ([latex]\\displaystyle{z}_{\\frac{{\\alpha}}{{2}}})(\\frac{{\\sigma}}{{\\sqrt{n}}})[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Constructing the Confidence Interval<\/h2>\r\n<ul>\r\n \t<li>The confidence interval estimate has the format (\r\n<p class=\"p1\">[latex]\\displaystyle\\overline{x}[\/latex]\u2013EBM,[latex]\\displaystyle\\overline{x}[\/latex]+EBM).<\/p>\r\n<\/li>\r\n<\/ul>\r\nThe graph gives a picture of the entire situation.\r\n\r\nCL + [latex]\\displaystyle\\frac{{\\alpha}}{{2}}+\\frac{{\\alpha}}{{2}}={\\text{CL}}+{\\alpha}=1[\/latex]\r\n\r\n<a href=\"https:\/\/courses.candelalearning.com\/introstats1xmaster\/wp-content\/uploads\/sites\/635\/2015\/06\/Screen-Shot-2015-06-08-at-2.02.04-PM.png\"><img class=\"aligncenter wp-image-536 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214612\/Screen-Shot-2015-06-08-at-2.02.04-PM.png\" alt=\"Graph of how to construct a confidence interval for CL = 1-alpha\" width=\"470\" height=\"285\" \/><\/a>\r\n<h2>Writing the Interpretation<\/h2>\r\nThe interpretation should clearly state the confidence level ( <em data-redactor-tag=\"em\">CL<\/em>), explain what population parameter is being estimated (here, a <strong data-redactor-tag=\"strong\">population mean<\/strong>), and state the confidence interval (both endpoints). \"We estimate with ___% confidence that the true population mean (include the context of the problem) is between ___ and ___ (include appropriate units).\"\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of three points. A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68. Find a confidence interval estimate for the population mean exam score (the mean score on all exams).\r\n\r\nFind a 90% confidence interval for the true (population) mean of statistics exam scores.\r\n<ul>\r\n \t<li>You can use technology to calculate the confidence interval directly.<\/li>\r\n \t<li>The first solution is shown step-by-step (Solution A).<\/li>\r\n \t<li>The second solution uses the TI-83, 83+, and 84+ calculators (Solution B).<\/li>\r\n<\/ul>\r\nSolution A:\r\n\r\nTo find the confidence interval, you need the sample mean, and the <em data-redactor-tag=\"em\">EBM<\/em>.\r\n<p class=\"p1\">[latex]\\overline{x}={68}{EBM}=({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 = 6<\/p>\r\nThe confidence level is 90% ( <em data-redactor-tag=\"em\">CL<\/em> = 0.90)\r\n\r\n<em data-redactor-tag=\"em\">CL<\/em> = 0.90 so <em data-redactor-tag=\"em\">\u03b1<\/em> = 1 \u2013 <em data-redactor-tag=\"em\">CL<\/em> = 1 \u2013 0.90 = 0.10\r\n\r\n[latex]\\displaystyle\\frac{{\\alpha}}{{2}}=0.05[\/latex], [latex]\\displaystyle{z}_{\\frac{{\\alpha}}{{2}}}={z}_{0.05}[\/latex]\r\n\r\nThe area to the right of <em data-redactor-tag=\"em\">z<\/em>0.05 is 0.05 and the area to the left of <em data-redactor-tag=\"em\">z<\/em>0.05is 1 \u2013 0.\u00a0[latex]\\displaystyle\\frac{{{z}_{\\alpha}}}{{2}}={z}_{0.05}=1.645[\/latex]\r\n\r\nUsing invNorm(0.95, 0, 1) on the TI-83,83+, and 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\r\nEBM = (1.645)([latex]\\displaystyle\\frac{{3}}{{\\sqrt{36}}}[\/latex])= 0.8225\r\n\r\n[latex]\\displaystyle\\overline{x}[\/latex]- EBM = 68 - 0.8225 = 67.1775\r\n\r\n[latex]\\displaystyle\\overline{x}[\/latex]+EBM = 68 + 0.8225 = 68.8225\r\n\r\nThe 90% confidence interval is (67.1775, 68.8225).\r\n\r\n&nbsp;\r\n\r\nSolution B:\r\n\r\nPress <code data-redactor-tag=\"code\">STAT<\/code> and arrow over to<code data-redactor-tag=\"code\">TESTS<\/code>.\r\n\r\nArrow down to <code data-redactor-tag=\"code\">7:ZInterval<\/code>.\r\n\r\nPress <code data-redactor-tag=\"code\">ENTER<\/code>.\r\n\r\nArrow to <code data-redactor-tag=\"code\">Stats<\/code> and press <code data-redactor-tag=\"code\">ENTER<\/code>.\r\n\r\nArrow down and enter three for <em data-redactor-tag=\"em\">\u03c3<\/em>, 68 for[latex]\\displaystyle\\overline{X}[\/latex], 36 for <em data-redactor-tag=\"em\">n<\/em>, and .90 for <code data-redactor-tag=\"code\">C-level<\/code>.\r\n\r\nArrow down to <code data-redactor-tag=\"code\">Calculate<\/code> and press <code data-redactor-tag=\"code\">ENTER<\/code>.\r\n\r\nThe confidence interval is (to three decimal places)(67.178, 68.822).\r\n\r\n<\/div>\r\n<h4>Interpretation<\/h4>\r\nWe estimate with 90% confidence that the true population mean exam score for all statistics students is between 67.18 and 68.82.\r\n<h4>Explanation of 90% Confidence Level<\/h4>\r\nNinety percent of all confidence intervals constructed in this way contain the true mean statistics exam score. For example, if we constructed 100 of these confidence intervals, we would expect 90 of them to contain the true population mean exam score.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nSuppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of six minutes. A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes.\r\n\r\nFind a 90% confidence interval estimate for the population mean delivery time.\r\n\r\n(34.1347, 37.8653)\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe Specific Absorption Rate (SAR) for a cell phone measures the amount of radio frequency (RF) energy absorbed by the user's body when using the handset. Every cell phone emits RF energy. Different phone models have different SAR measures. To receive certification from the Federal Communications Commission (FCC) for sale in the United States, the SAR level for a cell phone must be no more than 1.6 watts per kilogram. This table shows the highest SAR level for a random selection of cell phone models as measured by the FCC.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Phone Model<\/th>\r\n<th>SAR<\/th>\r\n<th>Phone Model<\/th>\r\n<th>SAR<\/th>\r\n<th>Phone Model<\/th>\r\n<th>SAR<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Apple iPhone 4S<\/td>\r\n<td>1.11<\/td>\r\n<td>LG Ally<\/td>\r\n<td>1.36<\/td>\r\n<td>Pantech Laser<\/td>\r\n<td>0.74<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>BlackBerry Pearl 8120<\/td>\r\n<td>1.48<\/td>\r\n<td>LG AX275<\/td>\r\n<td>1.34<\/td>\r\n<td>Samsung Character<\/td>\r\n<td>0.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>BlackBerry Tour 9630<\/td>\r\n<td>1.43<\/td>\r\n<td>LG Cosmos<\/td>\r\n<td>1.18<\/td>\r\n<td>Samsung Epic 4G Touch<\/td>\r\n<td>0.4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cricket TXTM8<\/td>\r\n<td>1.3<\/td>\r\n<td>LG CU515<\/td>\r\n<td>1.3<\/td>\r\n<td>Samsung M240<\/td>\r\n<td>0.867<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>HP\/Palm Centro<\/td>\r\n<td>1.09<\/td>\r\n<td>LG Trax CU575<\/td>\r\n<td>1.26<\/td>\r\n<td>Samsung Messager III SCH-R750<\/td>\r\n<td>0.68<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>HTC One V<\/td>\r\n<td>0.455<\/td>\r\n<td>Motorola Q9h<\/td>\r\n<td>1.29<\/td>\r\n<td>Samsung Nexus S<\/td>\r\n<td>0.51<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>HTC Touch Pro 2<\/td>\r\n<td>1.41<\/td>\r\n<td>Motorola Razr2 V8<\/td>\r\n<td>0.36<\/td>\r\n<td>Samsung SGH-A227<\/td>\r\n<td>1.13<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Huawei M835 Ideos<\/td>\r\n<td>0.82<\/td>\r\n<td>Motorola Razr2 V9<\/td>\r\n<td>0.52<\/td>\r\n<td>SGH-a107 GoPhone<\/td>\r\n<td>0.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Kyocera DuraPlus<\/td>\r\n<td>0.78<\/td>\r\n<td>Motorola V195s<\/td>\r\n<td>1.6<\/td>\r\n<td>Sony W350a<\/td>\r\n<td>1.48<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Kyocera K127 Marbl<\/td>\r\n<td>1.25<\/td>\r\n<td>Nokia 1680<\/td>\r\n<td>1.39<\/td>\r\n<td>T-Mobile Concord<\/td>\r\n<td>1.38<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFind a 98% confidence interval for the true (population) mean of the Specific Absorption Rates (SARs) for cell phones. Assume that the population standard deviation is <em data-redactor-tag=\"em\">\u03c3<\/em> = 0.337.\r\n\r\nSolution A:\r\n\r\nTo find the confidence interval, start by finding the point estimate: the sample mean.\r\n<p class=\"p1\">[latex]\\displaystyle\\overline{x}[\/latex] = 1.024<\/p>\r\nNext, find the <em data-redactor-tag=\"em\">EBM<\/em>. Because you are creating a 98% confidence interval, <em data-redactor-tag=\"em\">CL<\/em> = 0.98.\r\n<p class=\"p1\"><a href=\"https:\/\/courses.candelalearning.com\/introstats1xmaster\/wp-content\/uploads\/sites\/635\/2015\/06\/Screen-Shot-2015-06-08-at-2.23.07-PM.png\"><img class=\"aligncenter wp-image-537\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214614\/Screen-Shot-2015-06-08-at-2.23.07-PM.png\" alt=\"Graph of area under the curve to the right of z 0.01 is 0.01.\" width=\"550\" height=\"251\" \/><\/a><\/p>\r\n<p class=\"p1\">You need to find <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.01<\/sub> having the property that the area under the normal density curve to the right of <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.01<\/sub> is 0.01 and the area to the left is 0.99. Use your calculator, a computer, or a probability table for the standard normal distribution to find <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.01 <\/sub>= 2.326.<\/p>\r\n<p class=\"p1\">EBM = ([latex]\\displaystyle{z}_{0.01}\\frac{{\\sigma}}{{\\sqrt{n}}}=(2.236)\\frac{{0.337}}{{\\sqrt{30}}}=0.1431[\/latex]<\/p>\r\n<p class=\"p1\">To find the 98% confidence interval, find[latex]\\displaystyle\\overline{x}\\pm{EBM}[\/latex]<\/p>\r\n[latex]\\displaystyle\\overline{x}[\/latex]- EBM = 1.024 - 0.1431\u00a0=\u00a00.8809\r\n\r\n[latex]\\displaystyle\\overline{x}[\/latex]+EBM = 1.024 +0.1431\u00a0=\u00a01.1671\r\n\r\nWe estimate with 98% confidence that the true SAR mean for the population of cell phones in the United States is between 0.8809 and 1.1671 watts per kilogram.\r\n\r\nSolution B:\r\n<ul>\r\n \t<li>Press STAT and arrow over to TESTS.<\/li>\r\n \t<li>Arrow down to 7: ZInterval.<\/li>\r\n \t<li>Press ENTER.<\/li>\r\n \t<li>Arrow to Stats and press ENTER.<\/li>\r\n \t<li>Arrow down and enter the following values:\r\n<ul>\r\n \t<li><em data-redactor-tag=\"em\">\u03c3<\/em>: 0.337<\/li>\r\n \t<li>[latex]\\displaystyle\\overline{x}[\/latex]:\u00a01.024<\/li>\r\n \t<li><em data-redactor-tag=\"em\">n<\/em>: 30<\/li>\r\n \t<li><em data-redactor-tag=\"em\">C<\/em>-level: 0.98<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Arrow down to Calculate and press ENTER.<\/li>\r\n \t<li>The confidence interval is (to three decimal places) (0.881, 1.167).<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nThis table shows a different random sampling of 20 cell phone models. Use this data to calculate a 93% confidence interval for the true mean SAR for cell phones certified for use in the United States. As previously, assume that the population standard deviation is <em data-redactor-tag=\"em\">\u03c3<\/em> = 0.337.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Phone Model<\/th>\r\n<th>SAR<\/th>\r\n<th>Phone Model<\/th>\r\n<th>SAR<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Blackberry Pearl 8120<\/td>\r\n<td>1.48<\/td>\r\n<td>Nokia E71x<\/td>\r\n<td>1.53<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>HTC Evo Design 4G<\/td>\r\n<td>0.8<\/td>\r\n<td>Nokia N75<\/td>\r\n<td>0.68<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>HTC Freestyle<\/td>\r\n<td>1.15<\/td>\r\n<td>Nokia N79<\/td>\r\n<td>1.4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>LG Ally<\/td>\r\n<td>1.36<\/td>\r\n<td>Sagem Puma<\/td>\r\n<td>1.24<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>LG Fathom<\/td>\r\n<td>0.77<\/td>\r\n<td>Samsung Fascinate<\/td>\r\n<td>0.57<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>LG Optimus Vu<\/td>\r\n<td>0.462<\/td>\r\n<td>Samsung Infuse 4G<\/td>\r\n<td>0.2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Motorola Cliq XT<\/td>\r\n<td>1.36<\/td>\r\n<td>Samsung Nexus S<\/td>\r\n<td>0.51<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Motorola Droid Pro<\/td>\r\n<td>1.39<\/td>\r\n<td>Samsung Replenish<\/td>\r\n<td>0.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Motorola Droid Razr M<\/td>\r\n<td>1.3<\/td>\r\n<td>Sony W518a Walkman<\/td>\r\n<td>0.73<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Nokia 7705 Twist<\/td>\r\n<td>0.7<\/td>\r\n<td>ZTE C79<\/td>\r\n<td>0.869<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p class=\"p1\">[latex]\\displaystyle\\overline{x}[\/latex] = 0.940<\/p>\r\n<p class=\"p1\">[latex]\\displaystyle\\frac{{\\alpha}}{{2}}=\\frac{{1-CL}}{{2}}=\\frac{{1-0.93}}{{2}}[\/latex]=0.035<\/p>\r\n<p class=\"p1\">[latex]\\displaystyle{z}_{0.05}[\/latex]=1.812<\/p>\r\n<p class=\"p1\">EBM=[latex]\\displaystyle({z}_{0.05})(\\frac{{\\sigma}}{{\\sqrt{n}}})=(1.182)(\\frac{{0.337}}{{\\sqrt{20}}}[\/latex]=0.1365<\/p>\r\n[latex]\\displaystyle\\overline{x}[\/latex]- EBM = 0.940 - 0.1365 =\u00a00.8035\r\n\r\n[latex]\\displaystyle\\overline{x}[\/latex]+EBM =\u00a00.940 + 0.1365 =\u00a01.0765\r\n\r\nWe estimate with 93% confidence that the true SAR mean for the population of cell phones in the United States is between 0.8035 and 1.0765 watts per kilogram.\r\n\r\n<\/div>\r\nNotice the difference in the confidence intervals calculated in Example 3 and the Try It just completed. These intervals are different for several reasons: they were calculated from different samples, the samples were different sizes, and the intervals were calculated for different levels of confidence. Even though the intervals are different, they do not yield conflicting information. The effects of these kinds of changes are the subject of the next section in this chapter.\r\n<h2>Changing the Confidence Level or Sample Size<\/h2>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/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\r\nSolution:\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\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\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\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<\/div>\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<p class=\"p1\"><a href=\"https:\/\/courses.candelalearning.com\/introstats1xmaster\/wp-content\/uploads\/sites\/635\/2015\/06\/Screen-Shot-2015-06-08-at-3.06.23-PM.png\"><img class=\"aligncenter wp-image-540\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214616\/Screen-Shot-2015-06-08-at-3.06.23-PM.png\" alt=\"Graphs of changing the confidence interval from 09% to 95%\" width=\"734\" height=\"236\" \/><\/a><\/p>\r\n\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 class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nRefer back to the pizza-delivery Try It 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\r\n(33.37, 38.63)\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/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>= 68<\/li>\r\n \t<li><em data-redactor-tag=\"em\">EBM<\/em> =<\/li>\r\n \t<li><em data-redactor-tag=\"em\">\u03c3<\/em> = 3; The confidence level is 90% (<em data-redactor-tag=\"em\">CL<\/em>=0.90); .<\/li>\r\n<\/ul>\r\nSolution A:\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\nSolution B:\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<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<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nRefer back to the pizza-delivery Try It 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\r\n(34.6041, 37.3958)\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<hr \/>\r\n\r\n<h1>Working Backwards to Find the Error Bound or Sample Mean<\/h1>\r\nWhen we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. However, sometimes when we read statistical studies, the study may state the confidence interval only. If we know the confidence interval, we can work backwards to find both the error bound and the sample mean.\r\n<h3>Finding the Error Bound<\/h3>\r\n<ul>\r\n \t<li>From the upper value for the interval, subtract the sample mean,<\/li>\r\n \t<li>OR, from the upper value for the interval, subtract the lower value. Then divide the difference by two.<\/li>\r\n<\/ul>\r\n<h3>Finding the Sample Mean<\/h3>\r\n<ul>\r\n \t<li>Subtract the error bound from the upper value of the confidence interval,<\/li>\r\n \t<li>OR, average the upper and lower endpoints of the confidence interval.<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nNotice that there are two methods to perform each calculation. You can choose the method that is easier to use with the information you know.\r\n\r\nSuppose we know that a confidence interval is (67.18, 68.82) and we want to find the error bound. We may know that the sample mean is 68, or perhaps our source only gave the confidence interval and did not tell us the value of the sample mean.\r\n\r\nCalculate the Error Bound:\r\n<ul>\r\n \t<li>If we know that the sample mean is 68: <em data-redactor-tag=\"em\">EBM<\/em> = 68.82 \u2013 68 = 0.82.<\/li>\r\n \t<li>If we don't know the sample mean: .<\/li>\r\n<\/ul>\r\nCalculate the Sample Mean:\r\n<ul>\r\n \t<li>If we know the error bound: = 68.82 \u2013 0.82 = 68<\/li>\r\n \t<li>If we don't know the error bound: .<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\nSuppose we know that a confidence interval is (42.12, 47.88). Find the error bound and the sample mean.\r\n\r\nSample mean is 45, error bound is 2.88\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<hr \/>\r\n\r\n<h1>Calculating the Sample Size <em data-redactor-tag=\"em\">n<\/em><\/h1>\r\nIf researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.\r\n\r\nThe error bound formula for a population mean when the population standard deviation is known is\r\n\r\nThe formula for sample size is , found by solving the error bound formula for <em data-redactor-tag=\"em\">n<\/em>.\r\n\r\nIn this formula, <em data-redactor-tag=\"em\">z<\/em> is , corresponding to the desired confidence level. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe population standard deviation for the age of Foothill College students is 15 years. If we want to be 95% confident that the sample mean age is within two years of the true population mean age of Foothill College students, how many randomly selected Foothill College students must be surveyed?\r\n<ul>\r\n \t<li>From the problem, we know that <em data-redactor-tag=\"em\">\u03c3<\/em> = 15 and <em data-redactor-tag=\"em\">EBM<\/em> = 2.<\/li>\r\n \t<li><em data-redactor-tag=\"em\">z<\/em> = <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.025<\/sub> = 1.96, because the confidence level is 95%.<\/li>\r\n \t<li>using the sample size equation.<\/li>\r\n \t<li>Use <em data-redactor-tag=\"em\">n<\/em> = 217: Always round the answer UP to the next higher integer to ensure that the sample size is large enough.<\/li>\r\n<\/ul>\r\nTherefore, 217 Foothill College students should be surveyed in order to be 95% confident that we are within two years of the true population mean age of Foothill College students.\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nThe population standard deviation for the height of high school basketball players is three inches. If we want to be 95% confident that the sample mean height is within one inch of the true population mean height, how many randomly selected students must be surveyed?\r\n\r\n35 students\r\n\r\n<\/div>\r\n\r\n<hr \/>\r\n\r\n<h2>References<\/h2>\r\n\"American Fact Finder.\" U.S. Census Bureau. Available online at http:\/\/factfinder2.census.gov\/faces\/nav\/jsf\/pages\/searchresults.xhtml?refresh=t (accessed July 2, 2013).\r\n\r\n\"Disclosure Data Catalog: Candidate Summary Report 2012.\" U.S. Federal Election Commission. Available online at http:\/\/www.fec.gov\/data\/index.jsp (accessed July 2, 2013).\r\n\r\n\"Headcount Enrollment Trends by Student Demographics Ten-Year Fall Trends to Most Recently Completed Fall.\" Foothill De Anza Community College District. Available online at http:\/\/research.fhda.edu\/factbook\/FH_Demo_Trends\/FoothillDemographicTrends.htm (accessed September 30,2013).\r\n\r\nKuczmarski, Robert J., Cynthia L. Ogden, Shumei S. Guo, Laurence M. Grummer-Strawn, Katherine M. Flegal, Zuguo Mei, Rong Wei, Lester R. Curtin, Alex F. Roche, Clifford L. Johnson. \"2000 CDC Growth Charts for the United States: Methods and Development.\" Centers for Disease Control and Prevention. Available online at http:\/\/www.cdc.gov\/growthcharts\/2000growthchart-us.pdf (accessed July 2, 2013).\r\n\r\nLa, Lynn, Kent German. \"Cell Phone Radiation Levels.\" c|net part of CBX Interactive Inc. Available online at http:\/\/reviews.cnet.com\/cell-phone-radiation-levels\/ (accessed July 2, 2013).\r\n\r\n\"Mean Income in the Past 12 Months (in 2011 Inflaction-Adjusted Dollars): 2011 American Community Survey 1-Year Estimates.\" American Fact Finder, U.S. Census Bureau. Available online at http:\/\/factfinder2.census.gov\/faces\/tableservices\/jsf\/pages\/productview.xhtml?pid=ACS_11_1YR_S1902&amp;prodType=table (accessed July 2, 2013).\r\n\r\n\"Metadata Description of Candidate Summary File.\" U.S. Federal Election Commission. Available online at http:\/\/www.fec.gov\/finance\/disclosure\/metadata\/metadataforcandidatesummary.shtml (accessed July 2, 2013).\r\n\r\n\"National Health and Nutrition Examination Survey.\" Centers for Disease Control and Prevention. Available online at http:\/\/www.cdc.gov\/nchs\/nhanes.htm (accessed July 2, 2013).\r\n<h1>Concept Review<\/h1>\r\nIn this module, we learned how to calculate the confidence interval for a single population mean where the population standard deviation is known. When estimating a population mean, the margin of error is called the error bound for a population mean ( <em data-redactor-tag=\"em\">EBM<\/em>). A confidence interval has the general form:\r\n\r\n(lower bound, upper bound) = (point estimate \u2013 <em data-redactor-tag=\"em\">EBM<\/em>, point estimate + <em data-redactor-tag=\"em\">EBM<\/em>)\r\n\r\nThe calculation of <em data-redactor-tag=\"em\">EBM<\/em> depends on the size of the sample and the level of confidence desired. The confidence level is the percent of all possible samples that can be expected to include the true population parameter. As the confidence level increases, the corresponding <em data-redactor-tag=\"em\">EBM<\/em> increases as well. As the sample size increases, the <em data-redactor-tag=\"em\">EBM<\/em> decreases. By the central limit theorem,\r\n\r\nGiven a confidence interval, you can work backwards to find the error bound ( <em data-redactor-tag=\"em\">EBM<\/em>) or the sample mean. To find the error bound, find the difference of the upper bound of the interval and the mean. If you do not know the sample mean, you can find the error bound by calculating half the difference of the upper and lower bounds. To find the sample mean given a confidence interval, find the difference of the upper bound and the error bound. If the error bound is unknown, then average the upper and lower bounds of the confidence interval to find the sample mean.\r\n\r\nSometimes researchers know in advance that they want to estimate a population mean within a specific margin of error for a given level of confidence. In that case, solve the <em data-redactor-tag=\"em\">EBM<\/em> formula for <em data-redactor-tag=\"em\">n<\/em> to discover the size of the sample that is needed to achieve this goal:\r\n<h1>Formula Review<\/h1>\r\n<p class=\"p1\">[latex]\\displaystyle\\overline{X}{\\sim}{N}\\left({\\mu}_{x}, \\frac{{\\sigma}}{{\\sqrt{n}}}\\right)[\/latex]. The distribution of sample means is normally distributed with mean equal to the population mean and standard deviation given by the population standard deviation divided by the square root of the sample size.<\/p>\r\nThe general form for a confidence interval for a single population mean, known standard deviation, normal distribution is given by\r\n\r\n(lower bound, upper bound) = (point estimate \u2013 <em data-redactor-tag=\"em\">EBM<\/em>, point estimate + <em data-redactor-tag=\"em\">EBM<\/em>)\r\n\r\n=([latex]\\displaystyle\\overline{x}[\/latex] - EBM, [latex]\\displaystyle\\overline{x}[\/latex]+EBM)\r\n\r\n=([latex]\\displaystyle\\overline{x}-{z}_{\\frac{{\\alpha}}{{\\sqrt{n}}}}, \\overline{x}+{z}_{\\frac{{\\alpha}}{{\\sqrt{n}}}}[\/latex])\r\n\r\nEBM =\u00a0[latex]\\displaystyle{z}_{\\frac{{\\alpha}}{{\\sqrt{n}}}}[\/latex]= the error bound for the mean, or the margin of error for a single population mean; this formula is used when the population standard deviation is known.\r\n<p id=\"eip-277\"><em data-effect=\"italics\">CL<\/em> = confidence level, or the proportion of confidence intervals created that are expected to contain the true population parameter<\/p>\r\n<p id=\"eip-401\"><em data-effect=\"italics\">\u03b1<\/em> = 1 \u2013 <em data-effect=\"italics\">CL<\/em> = the proportion of confidence intervals that will not contain the population<\/p>\r\n[latex]\\displaystyle{z}_{\\frac{{\\alpha}}{{\\sqrt{n}}}}[\/latex]= the <em data-effect=\"italics\">z<\/em>-score with the property that the area to the right of the z-score is \u00a0\u221d2 this is the <em data-effect=\"italics\">z<\/em>-score used in the calculation of <em data-effect=\"italics\">\"EBM\u00a0<\/em>where \u03b1 = 1 \u2013 <em data-effect=\"italics\">CL.<\/em>\r\n\r\nn = [latex]\\displaystyle\\frac{{{z}^{2}{\\sigma}^{2}}}{{{EBM}^{2}}}[\/latex] = the formula used to determine the sample size (<em data-effect=\"italics\">n<\/em>) needed to achieve a desired margin of error at a given level of confidence\r\n<p id=\"eip-73\">General form of a confidence interval<\/p>\r\n<p id=\"fs-idp49175936\">(lower value, upper value) = (point\u00a0estimate\u2212error bound, point estimate + error bound)<\/p>\r\nTo find the error bound when you know the confidence interval\r\n\r\nerror bound = upper value\u2212point\u00a0estimate OR error bound =[latex]\\displaystyle\\frac{{\\text{upper}{value}-{lower}{value}}}{{2}}[\/latex]\r\n<p id=\"eip-531\">Single Population Mean, Known Standard Deviation, Normal Distribution<\/p>\r\nUse the Normal Distribution for Means, Population Standard Deviation is Known <em data-effect=\"italics\">EBM=[latex]\\displaystyle{z}_{\\frac{{\\alpha}}{{2}}\\cdot\\frac{{\\sigma}}{{\\sqrt{n}}}}[\/latex]<\/em>\r\n\r\nThe confidence interval has the format\u00a0EBM =\u00a0([latex]\\displaystyle\\overline{x}[\/latex] - EBM, [latex]\\displaystyle\\overline{x}[\/latex]+EBM)","rendered":"<p>A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of\u00a0[latex]\\displaystyle\\overline{{x}}={10}[\/latex] and we have constructed the 90% confidence interval (5, 15) where <em>EBM<\/em> = 5.<\/p>\n<h2>Calculating the Confidence Interval<\/h2>\n<p>To construct a confidence interval for a single unknown population mean\u00a0<em>\u03bc<\/em>, <strong>where the population standard deviation is known<\/strong>,\u00a0we need [latex]\\overline{{x}}[\/latex] is the\u00a0<strong>point estimate<\/strong> of the unknown population mean <em>\u03bc<\/em>.<\/p>\n<p><strong>The confidence interval estimate will have the form:<\/strong><\/p>\n<p>(point estimate \u2013 error bound, point estimate + error bound) or, in symbols,<br \/>\n[latex]\\displaystyle{(\\overline{{x}}-{E}{B}{M},\\overline{{x}}+{E}{B}{M})}[\/latex]<\/p>\n<p>The margin of error (<em>EBM<\/em>) depends on the <strong>confidence level<\/strong> (abbreviated <em><strong data-redactor-tag=\"strong\">CL<\/strong><\/em>). The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. However, it is more accurate to state that the confidence level is the percent of confidence intervals that contain the true population parameter when repeated samples are taken. Most often, it is the choice of the person constructing the confidence interval to choose a confidence level of 90% or higher because that person wants to be reasonably certain of his or her conclusions.<\/p>\n<p>There is another probability called alpha (<em>\u03b1<\/em>). <em>\u03b1<\/em> is related to the confidence level, <em>CL<\/em>. <em>\u03b1<\/em> is the probability that the interval does not contain the unknown population parameter.<\/p>\n<p>Mathematically,\u00a0<em>\u03b1<\/em> + <em>CL<\/em> = 1.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Intro to Confidence Intervals for One Mean (Sigma Known)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KG921rfbTDw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose we have collected data from a sample. We know the sample mean but we do not know the mean for the entire population.<\/p>\n<p>The sample mean is seven, and the error bound for the mean is 2.5.<\/p>\n<p>[latex]\\displaystyle\\overline{{x}}={7}{\\quad\\text{and}\\quad}{E}{B}{M}={2.5}[\/latex]<\/p>\n<p>The confidence interval is (7 \u2013 2.5, 7 + 2.5), and calculating the values gives (4.5, 9.5).<\/p>\n<p>If the confidence level (<em>CL<\/em>) is 95%, then we say that, &#8220;We estimate with 95% confidence that the true value of the population mean is between 4.5 and 9.5.&#8221;<\/p>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Suppose we have data from a sample. The sample mean is 15, and the error bound for the mean is 3.2.<\/p>\n<p>What is the confidence interval estimate for the population mean?<\/p>\n<p>(11.8, 18.2)<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of\u00a0[latex]\\displaystyle\\overline{{x}}={10}[\/latex], and we have constructed the 90% confidence interval (5, 15) where <em>EBM<\/em> = 5.<\/p>\n<p>To get a 90% confidence interval, we must include the central 90% of the probability of the normal distribution. If we include the central 90%, we leave out a total of\u00a0<em>\u03b1<\/em> = 10% in both tails, or 5% in each tail, of the normal distribution.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/mrwg-kzcwi37i#fixme#fixme#fixme\" alt=\"This is a normal distribution curve. The peak of the curve coincides with the point 10 on the horizontal axis. The points 5 and 15 are labeled on the axis. Vertical lines are drawn from these points to the curve, and the region between the lines is shaded. The shaded region has area equal to 0.90.\" \/><\/p>\n<p>To capture the central 90%, we must go out 1.645 &#8220;standard deviations&#8221; on either side of the calculated sample mean. The value 1.645 is the\u00a0<em>z<\/em>-score from a standard normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail.<\/p>\n<p>It is important that the &#8220;standard deviation&#8221; used must be appropriate for the parameter we are estimating, so in this section we need to use the standard deviation that applies to sample means, which is[latex]\\displaystyle\\frac{{\\sigma}}{{\\sqrt{n}}}[\/latex]. The fraction\u00a0[latex]\\displaystyle\\frac{{\\sigma}}{{\\sqrt{n}}}[\/latex],\u00a0is commonly called the &#8220;standard error of the mean&#8221; in order to distinguish clearly the standard deviation for a mean from the population standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em>.<\/p>\n<p><strong data-redactor-tag=\"strong\">In summary, as a result of the central limit theorem:<\/strong><\/p>\n<ul>\n<li>[latex]\\displaystyle\\overline{X}[\/latex], that is,\u00a0[latex]\\displaystyle\\overline{X}{\\sim}{N}\\left({\\mu}_{x}, \\frac{{\\sigma}}{{\\sqrt{n}}}\\right)[\/latex]<\/li>\n<li><strong data-redactor-tag=\"strong\">When the population standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em> is known, we use a normal distribution to calculate the error bound.<\/strong><\/li>\n<\/ul>\n<h2>Calculating the Confidence Interval<\/h2>\n<p>To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. The steps to construct and interpret the confidence interval are:<\/p>\n<ul>\n<li>Calculate the sample mean [latex]\\displaystyle\\overline{{x}}[\/latex] from the sample data. Remember, in this section we already know the population standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em>.<\/li>\n<li>Find the <em data-redactor-tag=\"em\">z<\/em>-score that corresponds to the confidence level.<\/li>\n<li>Calculate the error bound <em data-redactor-tag=\"em\">EBM<\/em>.<\/li>\n<li>Construct the confidence interval.<\/li>\n<li>Write a sentence that interprets the estimate in the context of the situation in the problem. (Explain what the confidence interval means, in the words of the problem.)<\/li>\n<\/ul>\n<p>We will first examine each step in more detail, and then illustrate the process with some examples.<\/p>\n<h2>Finding the <em data-redactor-tag=\"em\">z<\/em>-score for the Stated Confidence Level<\/h2>\n<p>When we know the population standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em>, we use a standard normal distribution to calculate the error bound EBM and construct the confidence interval. We need to find the value of <em data-redactor-tag=\"em\">z<\/em> that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution <em data-redactor-tag=\"em\">Z<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(0, 1).<\/p>\n<p>The confidence level, <em data-redactor-tag=\"em\">CL<\/em>, is the area in the middle of the standard normal distribution. <em data-redactor-tag=\"em\">CL<\/em> = 1 \u2013 <em data-redactor-tag=\"em\">\u03b1<\/em>, so <em data-redactor-tag=\"em\">\u03b1<\/em> is the area that is split equally between the two tails. Each of the tails contains an area equal to \u03b12.<\/p>\n<p>The z-score that has an area to the right of \u03b12 is denoted by z\u03b12.<\/p>\n<p>For example, when <em data-redactor-tag=\"em\">CL<\/em> = 0.95, <em data-redactor-tag=\"em\">\u03b1<\/em> = 0.05 and \u03b12 = 0.025; we write z\u03b12=z0.025.<\/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>z\u03b12=z0.025=1.96, using a calculator, computer or a standard normal probability table.<\/p>\n<p><code data-redactor-tag=\"code\">invNorm<\/code>(0.975, 0, 1) = 1.96<\/p>\n<h3>\u00a0Note<\/h3>\n<p>Remember to use the area to the LEFT of ; in this chapter the last two inputs in the invNorm command are 0, 1, because you are using a standard normal distribution <em data-redactor-tag=\"em\">Z<\/em> ~ <em data-redactor-tag=\"em\">N<\/em>(0, 1).<\/p>\n<hr \/>\n<h2>Calculating the Error Bound (<em data-redactor-tag=\"em\">EBM<\/em>)<\/h2>\n<p>The error bound formula for an unknown population mean <em data-redactor-tag=\"em\">\u03bc<\/em> when the population standard deviation <em data-redactor-tag=\"em\">\u03c3<\/em> is known is<\/p>\n<ul>\n<li>EBM = ([latex]\\displaystyle{z}_{\\frac{{\\alpha}}{{2}}})(\\frac{{\\sigma}}{{\\sqrt{n}}})[\/latex]<\/li>\n<\/ul>\n<h2>Constructing the Confidence Interval<\/h2>\n<ul>\n<li>The confidence interval estimate has the format (\n<p class=\"p1\">[latex]\\displaystyle\\overline{x}[\/latex]\u2013EBM,[latex]\\displaystyle\\overline{x}[\/latex]+EBM).<\/p>\n<\/li>\n<\/ul>\n<p>The graph gives a picture of the entire situation.<\/p>\n<p>CL + [latex]\\displaystyle\\frac{{\\alpha}}{{2}}+\\frac{{\\alpha}}{{2}}={\\text{CL}}+{\\alpha}=1[\/latex]<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/introstats1xmaster\/wp-content\/uploads\/sites\/635\/2015\/06\/Screen-Shot-2015-06-08-at-2.02.04-PM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-536 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214612\/Screen-Shot-2015-06-08-at-2.02.04-PM.png\" alt=\"Graph of how to construct a confidence interval for CL = 1-alpha\" width=\"470\" height=\"285\" \/><\/a><\/p>\n<h2>Writing the Interpretation<\/h2>\n<p>The interpretation should clearly state the confidence level ( <em data-redactor-tag=\"em\">CL<\/em>), explain what population parameter is being estimated (here, a <strong data-redactor-tag=\"strong\">population mean<\/strong>), and state the confidence interval (both endpoints). &#8220;We estimate with ___% confidence that the true population mean (include the context of the problem) is between ___ and ___ (include appropriate units).&#8221;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of three points. A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68. Find a confidence interval estimate for the population mean exam score (the mean score on all exams).<\/p>\n<p>Find a 90% confidence interval for the true (population) mean of statistics exam scores.<\/p>\n<ul>\n<li>You can use technology to calculate the confidence interval directly.<\/li>\n<li>The first solution is shown step-by-step (Solution A).<\/li>\n<li>The second solution uses the TI-83, 83+, and 84+ calculators (Solution B).<\/li>\n<\/ul>\n<p>Solution A:<\/p>\n<p>To find the confidence interval, you need the sample mean, and the <em data-redactor-tag=\"em\">EBM<\/em>.<\/p>\n<p class=\"p1\">[latex]\\overline{x}={68}{EBM}=({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 = 6<\/p>\n<p>The confidence level is 90% ( <em data-redactor-tag=\"em\">CL<\/em> = 0.90)<\/p>\n<p><em data-redactor-tag=\"em\">CL<\/em> = 0.90 so <em data-redactor-tag=\"em\">\u03b1<\/em> = 1 \u2013 <em data-redactor-tag=\"em\">CL<\/em> = 1 \u2013 0.90 = 0.10<\/p>\n<p>[latex]\\displaystyle\\frac{{\\alpha}}{{2}}=0.05[\/latex], [latex]\\displaystyle{z}_{\\frac{{\\alpha}}{{2}}}={z}_{0.05}[\/latex]<\/p>\n<p>The area to the right of <em data-redactor-tag=\"em\">z<\/em>0.05 is 0.05 and the area to the left of <em data-redactor-tag=\"em\">z<\/em>0.05is 1 \u2013 0.\u00a0[latex]\\displaystyle\\frac{{{z}_{\\alpha}}}{{2}}={z}_{0.05}=1.645[\/latex]<\/p>\n<p>Using invNorm(0.95, 0, 1) on the TI-83,83+, and 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>EBM = (1.645)([latex]\\displaystyle\\frac{{3}}{{\\sqrt{36}}}[\/latex])= 0.8225<\/p>\n<p>[latex]\\displaystyle\\overline{x}[\/latex]&#8211; EBM = 68 &#8211; 0.8225 = 67.1775<\/p>\n<p>[latex]\\displaystyle\\overline{x}[\/latex]+EBM = 68 + 0.8225 = 68.8225<\/p>\n<p>The 90% confidence interval is (67.1775, 68.8225).<\/p>\n<p>&nbsp;<\/p>\n<p>Solution B:<\/p>\n<p>Press <code data-redactor-tag=\"code\">STAT<\/code> and arrow over to<code data-redactor-tag=\"code\">TESTS<\/code>.<\/p>\n<p>Arrow down to <code data-redactor-tag=\"code\">7:ZInterval<\/code>.<\/p>\n<p>Press <code data-redactor-tag=\"code\">ENTER<\/code>.<\/p>\n<p>Arrow to <code data-redactor-tag=\"code\">Stats<\/code> and press <code data-redactor-tag=\"code\">ENTER<\/code>.<\/p>\n<p>Arrow down and enter three for <em data-redactor-tag=\"em\">\u03c3<\/em>, 68 for[latex]\\displaystyle\\overline{X}[\/latex], 36 for <em data-redactor-tag=\"em\">n<\/em>, and .90 for <code data-redactor-tag=\"code\">C-level<\/code>.<\/p>\n<p>Arrow down to <code data-redactor-tag=\"code\">Calculate<\/code> and press <code data-redactor-tag=\"code\">ENTER<\/code>.<\/p>\n<p>The confidence interval is (to three decimal places)(67.178, 68.822).<\/p>\n<\/div>\n<h4>Interpretation<\/h4>\n<p>We estimate with 90% confidence that the true population mean exam score for all statistics students is between 67.18 and 68.82.<\/p>\n<h4>Explanation of 90% Confidence Level<\/h4>\n<p>Ninety percent of all confidence intervals constructed in this way contain the true mean statistics exam score. For example, if we constructed 100 of these confidence intervals, we would expect 90 of them to contain the true population mean exam score.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Suppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of six minutes. A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes.<\/p>\n<p>Find a 90% confidence interval estimate for the population mean delivery time.<\/p>\n<p>(34.1347, 37.8653)<\/p>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The Specific Absorption Rate (SAR) for a cell phone measures the amount of radio frequency (RF) energy absorbed by the user&#8217;s body when using the handset. Every cell phone emits RF energy. Different phone models have different SAR measures. To receive certification from the Federal Communications Commission (FCC) for sale in the United States, the SAR level for a cell phone must be no more than 1.6 watts per kilogram. This table shows the highest SAR level for a random selection of cell phone models as measured by the FCC.<\/p>\n<table>\n<thead>\n<tr>\n<th>Phone Model<\/th>\n<th>SAR<\/th>\n<th>Phone Model<\/th>\n<th>SAR<\/th>\n<th>Phone Model<\/th>\n<th>SAR<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Apple iPhone 4S<\/td>\n<td>1.11<\/td>\n<td>LG Ally<\/td>\n<td>1.36<\/td>\n<td>Pantech Laser<\/td>\n<td>0.74<\/td>\n<\/tr>\n<tr>\n<td>BlackBerry Pearl 8120<\/td>\n<td>1.48<\/td>\n<td>LG AX275<\/td>\n<td>1.34<\/td>\n<td>Samsung Character<\/td>\n<td>0.5<\/td>\n<\/tr>\n<tr>\n<td>BlackBerry Tour 9630<\/td>\n<td>1.43<\/td>\n<td>LG Cosmos<\/td>\n<td>1.18<\/td>\n<td>Samsung Epic 4G Touch<\/td>\n<td>0.4<\/td>\n<\/tr>\n<tr>\n<td>Cricket TXTM8<\/td>\n<td>1.3<\/td>\n<td>LG CU515<\/td>\n<td>1.3<\/td>\n<td>Samsung M240<\/td>\n<td>0.867<\/td>\n<\/tr>\n<tr>\n<td>HP\/Palm Centro<\/td>\n<td>1.09<\/td>\n<td>LG Trax CU575<\/td>\n<td>1.26<\/td>\n<td>Samsung Messager III SCH-R750<\/td>\n<td>0.68<\/td>\n<\/tr>\n<tr>\n<td>HTC One V<\/td>\n<td>0.455<\/td>\n<td>Motorola Q9h<\/td>\n<td>1.29<\/td>\n<td>Samsung Nexus S<\/td>\n<td>0.51<\/td>\n<\/tr>\n<tr>\n<td>HTC Touch Pro 2<\/td>\n<td>1.41<\/td>\n<td>Motorola Razr2 V8<\/td>\n<td>0.36<\/td>\n<td>Samsung SGH-A227<\/td>\n<td>1.13<\/td>\n<\/tr>\n<tr>\n<td>Huawei M835 Ideos<\/td>\n<td>0.82<\/td>\n<td>Motorola Razr2 V9<\/td>\n<td>0.52<\/td>\n<td>SGH-a107 GoPhone<\/td>\n<td>0.3<\/td>\n<\/tr>\n<tr>\n<td>Kyocera DuraPlus<\/td>\n<td>0.78<\/td>\n<td>Motorola V195s<\/td>\n<td>1.6<\/td>\n<td>Sony W350a<\/td>\n<td>1.48<\/td>\n<\/tr>\n<tr>\n<td>Kyocera K127 Marbl<\/td>\n<td>1.25<\/td>\n<td>Nokia 1680<\/td>\n<td>1.39<\/td>\n<td>T-Mobile Concord<\/td>\n<td>1.38<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Find a 98% confidence interval for the true (population) mean of the Specific Absorption Rates (SARs) for cell phones. Assume that the population standard deviation is <em data-redactor-tag=\"em\">\u03c3<\/em> = 0.337.<\/p>\n<p>Solution A:<\/p>\n<p>To find the confidence interval, start by finding the point estimate: the sample mean.<\/p>\n<p class=\"p1\">[latex]\\displaystyle\\overline{x}[\/latex] = 1.024<\/p>\n<p>Next, find the <em data-redactor-tag=\"em\">EBM<\/em>. Because you are creating a 98% confidence interval, <em data-redactor-tag=\"em\">CL<\/em> = 0.98.<\/p>\n<p class=\"p1\"><a href=\"https:\/\/courses.candelalearning.com\/introstats1xmaster\/wp-content\/uploads\/sites\/635\/2015\/06\/Screen-Shot-2015-06-08-at-2.23.07-PM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-537\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214614\/Screen-Shot-2015-06-08-at-2.23.07-PM.png\" alt=\"Graph of area under the curve to the right of z 0.01 is 0.01.\" width=\"550\" height=\"251\" \/><\/a><\/p>\n<p class=\"p1\">You need to find <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.01<\/sub> having the property that the area under the normal density curve to the right of <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.01<\/sub> is 0.01 and the area to the left is 0.99. Use your calculator, a computer, or a probability table for the standard normal distribution to find <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.01 <\/sub>= 2.326.<\/p>\n<p class=\"p1\">EBM = ([latex]\\displaystyle{z}_{0.01}\\frac{{\\sigma}}{{\\sqrt{n}}}=(2.236)\\frac{{0.337}}{{\\sqrt{30}}}=0.1431[\/latex]<\/p>\n<p class=\"p1\">To find the 98% confidence interval, find[latex]\\displaystyle\\overline{x}\\pm{EBM}[\/latex]<\/p>\n<p>[latex]\\displaystyle\\overline{x}[\/latex]&#8211; EBM = 1.024 &#8211; 0.1431\u00a0=\u00a00.8809<\/p>\n<p>[latex]\\displaystyle\\overline{x}[\/latex]+EBM = 1.024 +0.1431\u00a0=\u00a01.1671<\/p>\n<p>We estimate with 98% confidence that the true SAR mean for the population of cell phones in the United States is between 0.8809 and 1.1671 watts per kilogram.<\/p>\n<p>Solution B:<\/p>\n<ul>\n<li>Press STAT and arrow over to TESTS.<\/li>\n<li>Arrow down to 7: ZInterval.<\/li>\n<li>Press ENTER.<\/li>\n<li>Arrow to Stats and press ENTER.<\/li>\n<li>Arrow down and enter the following values:\n<ul>\n<li><em data-redactor-tag=\"em\">\u03c3<\/em>: 0.337<\/li>\n<li>[latex]\\displaystyle\\overline{x}[\/latex]:\u00a01.024<\/li>\n<li><em data-redactor-tag=\"em\">n<\/em>: 30<\/li>\n<li><em data-redactor-tag=\"em\">C<\/em>-level: 0.98<\/li>\n<\/ul>\n<\/li>\n<li>Arrow down to Calculate and press ENTER.<\/li>\n<li>The confidence interval is (to three decimal places) (0.881, 1.167).<\/li>\n<\/ul>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>This table shows a different random sampling of 20 cell phone models. Use this data to calculate a 93% confidence interval for the true mean SAR for cell phones certified for use in the United States. As previously, assume that the population standard deviation is <em data-redactor-tag=\"em\">\u03c3<\/em> = 0.337.<\/p>\n<table>\n<thead>\n<tr>\n<th>Phone Model<\/th>\n<th>SAR<\/th>\n<th>Phone Model<\/th>\n<th>SAR<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Blackberry Pearl 8120<\/td>\n<td>1.48<\/td>\n<td>Nokia E71x<\/td>\n<td>1.53<\/td>\n<\/tr>\n<tr>\n<td>HTC Evo Design 4G<\/td>\n<td>0.8<\/td>\n<td>Nokia N75<\/td>\n<td>0.68<\/td>\n<\/tr>\n<tr>\n<td>HTC Freestyle<\/td>\n<td>1.15<\/td>\n<td>Nokia N79<\/td>\n<td>1.4<\/td>\n<\/tr>\n<tr>\n<td>LG Ally<\/td>\n<td>1.36<\/td>\n<td>Sagem Puma<\/td>\n<td>1.24<\/td>\n<\/tr>\n<tr>\n<td>LG Fathom<\/td>\n<td>0.77<\/td>\n<td>Samsung Fascinate<\/td>\n<td>0.57<\/td>\n<\/tr>\n<tr>\n<td>LG Optimus Vu<\/td>\n<td>0.462<\/td>\n<td>Samsung Infuse 4G<\/td>\n<td>0.2<\/td>\n<\/tr>\n<tr>\n<td>Motorola Cliq XT<\/td>\n<td>1.36<\/td>\n<td>Samsung Nexus S<\/td>\n<td>0.51<\/td>\n<\/tr>\n<tr>\n<td>Motorola Droid Pro<\/td>\n<td>1.39<\/td>\n<td>Samsung Replenish<\/td>\n<td>0.3<\/td>\n<\/tr>\n<tr>\n<td>Motorola Droid Razr M<\/td>\n<td>1.3<\/td>\n<td>Sony W518a Walkman<\/td>\n<td>0.73<\/td>\n<\/tr>\n<tr>\n<td>Nokia 7705 Twist<\/td>\n<td>0.7<\/td>\n<td>ZTE C79<\/td>\n<td>0.869<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"p1\">[latex]\\displaystyle\\overline{x}[\/latex] = 0.940<\/p>\n<p class=\"p1\">[latex]\\displaystyle\\frac{{\\alpha}}{{2}}=\\frac{{1-CL}}{{2}}=\\frac{{1-0.93}}{{2}}[\/latex]=0.035<\/p>\n<p class=\"p1\">[latex]\\displaystyle{z}_{0.05}[\/latex]=1.812<\/p>\n<p class=\"p1\">EBM=[latex]\\displaystyle({z}_{0.05})(\\frac{{\\sigma}}{{\\sqrt{n}}})=(1.182)(\\frac{{0.337}}{{\\sqrt{20}}}[\/latex]=0.1365<\/p>\n<p>[latex]\\displaystyle\\overline{x}[\/latex]&#8211; EBM = 0.940 &#8211; 0.1365 =\u00a00.8035<\/p>\n<p>[latex]\\displaystyle\\overline{x}[\/latex]+EBM =\u00a00.940 + 0.1365 =\u00a01.0765<\/p>\n<p>We estimate with 93% confidence that the true SAR mean for the population of cell phones in the United States is between 0.8035 and 1.0765 watts per kilogram.<\/p>\n<\/div>\n<p>Notice the difference in the confidence intervals calculated in Example 3 and the Try It just completed. These intervals are different for several reasons: they were calculated from different samples, the samples were different sizes, and the intervals were calculated for different levels of confidence. Even though the intervals are different, they do not yield conflicting information. The effects of these kinds of changes are the subject of the next section in this chapter.<\/p>\n<h2>Changing the Confidence Level or Sample Size<\/h2>\n<div class=\"textbox exercises\">\n<h3>Example<\/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<p>Solution:<\/p>\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>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>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>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<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 class=\"p1\"><a href=\"https:\/\/courses.candelalearning.com\/introstats1xmaster\/wp-content\/uploads\/sites\/635\/2015\/06\/Screen-Shot-2015-06-08-at-3.06.23-PM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-540\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214616\/Screen-Shot-2015-06-08-at-3.06.23-PM.png\" alt=\"Graphs of changing the confidence interval from 09% to 95%\" width=\"734\" height=\"236\" \/><\/a><\/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 class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Refer back to the pizza-delivery Try It 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<p>(33.37, 38.63)<\/p>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/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>= 68<\/li>\n<li><em data-redactor-tag=\"em\">EBM<\/em> =<\/li>\n<li><em data-redactor-tag=\"em\">\u03c3<\/em> = 3; The confidence level is 90% (<em data-redactor-tag=\"em\">CL<\/em>=0.90); .<\/li>\n<\/ul>\n<p>Solution A:<\/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>Solution B:<\/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<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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Refer back to the pizza-delivery Try It 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<p>(34.6041, 37.3958)<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<hr \/>\n<h1>Working Backwards to Find the Error Bound or Sample Mean<\/h1>\n<p>When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. However, sometimes when we read statistical studies, the study may state the confidence interval only. If we know the confidence interval, we can work backwards to find both the error bound and the sample mean.<\/p>\n<h3>Finding the Error Bound<\/h3>\n<ul>\n<li>From the upper value for the interval, subtract the sample mean,<\/li>\n<li>OR, from the upper value for the interval, subtract the lower value. Then divide the difference by two.<\/li>\n<\/ul>\n<h3>Finding the Sample Mean<\/h3>\n<ul>\n<li>Subtract the error bound from the upper value of the confidence interval,<\/li>\n<li>OR, average the upper and lower endpoints of the confidence interval.<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Notice that there are two methods to perform each calculation. You can choose the method that is easier to use with the information you know.<\/p>\n<p>Suppose we know that a confidence interval is (67.18, 68.82) and we want to find the error bound. We may know that the sample mean is 68, or perhaps our source only gave the confidence interval and did not tell us the value of the sample mean.<\/p>\n<p>Calculate the Error Bound:<\/p>\n<ul>\n<li>If we know that the sample mean is 68: <em data-redactor-tag=\"em\">EBM<\/em> = 68.82 \u2013 68 = 0.82.<\/li>\n<li>If we don&#8217;t know the sample mean: .<\/li>\n<\/ul>\n<p>Calculate the Sample Mean:<\/p>\n<ul>\n<li>If we know the error bound: = 68.82 \u2013 0.82 = 68<\/li>\n<li>If we don&#8217;t know the error bound: .<\/li>\n<\/ul>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>Suppose we know that a confidence interval is (42.12, 47.88). Find the error bound and the sample mean.<\/p>\n<p>Sample mean is 45, error bound is 2.88<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<hr \/>\n<h1>Calculating the Sample Size <em data-redactor-tag=\"em\">n<\/em><\/h1>\n<p>If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.<\/p>\n<p>The error bound formula for a population mean when the population standard deviation is known is<\/p>\n<p>The formula for sample size is , found by solving the error bound formula for <em data-redactor-tag=\"em\">n<\/em>.<\/p>\n<p>In this formula, <em data-redactor-tag=\"em\">z<\/em> is , corresponding to the desired confidence level. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The population standard deviation for the age of Foothill College students is 15 years. If we want to be 95% confident that the sample mean age is within two years of the true population mean age of Foothill College students, how many randomly selected Foothill College students must be surveyed?<\/p>\n<ul>\n<li>From the problem, we know that <em data-redactor-tag=\"em\">\u03c3<\/em> = 15 and <em data-redactor-tag=\"em\">EBM<\/em> = 2.<\/li>\n<li><em data-redactor-tag=\"em\">z<\/em> = <em data-redactor-tag=\"em\">z<\/em><sub data-redactor-tag=\"sub\">0.025<\/sub> = 1.96, because the confidence level is 95%.<\/li>\n<li>using the sample size equation.<\/li>\n<li>Use <em data-redactor-tag=\"em\">n<\/em> = 217: Always round the answer UP to the next higher integer to ensure that the sample size is large enough.<\/li>\n<\/ul>\n<p>Therefore, 217 Foothill College students should be surveyed in order to be 95% confident that we are within two years of the true population mean age of Foothill College students.<\/p>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>The population standard deviation for the height of high school basketball players is three inches. If we want to be 95% confident that the sample mean height is within one inch of the true population mean height, how many randomly selected students must be surveyed?<\/p>\n<p>35 students<\/p>\n<\/div>\n<hr \/>\n<h2>References<\/h2>\n<p>&#8220;American Fact Finder.&#8221; U.S. Census Bureau. Available online at http:\/\/factfinder2.census.gov\/faces\/nav\/jsf\/pages\/searchresults.xhtml?refresh=t (accessed July 2, 2013).<\/p>\n<p>&#8220;Disclosure Data Catalog: Candidate Summary Report 2012.&#8221; U.S. Federal Election Commission. Available online at http:\/\/www.fec.gov\/data\/index.jsp (accessed July 2, 2013).<\/p>\n<p>&#8220;Headcount Enrollment Trends by Student Demographics Ten-Year Fall Trends to Most Recently Completed Fall.&#8221; Foothill De Anza Community College District. Available online at http:\/\/research.fhda.edu\/factbook\/FH_Demo_Trends\/FoothillDemographicTrends.htm (accessed September 30,2013).<\/p>\n<p>Kuczmarski, Robert J., Cynthia L. Ogden, Shumei S. Guo, Laurence M. Grummer-Strawn, Katherine M. Flegal, Zuguo Mei, Rong Wei, Lester R. Curtin, Alex F. Roche, Clifford L. Johnson. &#8220;2000 CDC Growth Charts for the United States: Methods and Development.&#8221; Centers for Disease Control and Prevention. Available online at http:\/\/www.cdc.gov\/growthcharts\/2000growthchart-us.pdf (accessed July 2, 2013).<\/p>\n<p>La, Lynn, Kent German. &#8220;Cell Phone Radiation Levels.&#8221; c|net part of CBX Interactive Inc. Available online at http:\/\/reviews.cnet.com\/cell-phone-radiation-levels\/ (accessed July 2, 2013).<\/p>\n<p>&#8220;Mean Income in the Past 12 Months (in 2011 Inflaction-Adjusted Dollars): 2011 American Community Survey 1-Year Estimates.&#8221; American Fact Finder, U.S. Census Bureau. Available online at http:\/\/factfinder2.census.gov\/faces\/tableservices\/jsf\/pages\/productview.xhtml?pid=ACS_11_1YR_S1902&amp;prodType=table (accessed July 2, 2013).<\/p>\n<p>&#8220;Metadata Description of Candidate Summary File.&#8221; U.S. Federal Election Commission. Available online at http:\/\/www.fec.gov\/finance\/disclosure\/metadata\/metadataforcandidatesummary.shtml (accessed July 2, 2013).<\/p>\n<p>&#8220;National Health and Nutrition Examination Survey.&#8221; Centers for Disease Control and Prevention. Available online at http:\/\/www.cdc.gov\/nchs\/nhanes.htm (accessed July 2, 2013).<\/p>\n<h1>Concept Review<\/h1>\n<p>In this module, we learned how to calculate the confidence interval for a single population mean where the population standard deviation is known. When estimating a population mean, the margin of error is called the error bound for a population mean ( <em data-redactor-tag=\"em\">EBM<\/em>). A confidence interval has the general form:<\/p>\n<p>(lower bound, upper bound) = (point estimate \u2013 <em data-redactor-tag=\"em\">EBM<\/em>, point estimate + <em data-redactor-tag=\"em\">EBM<\/em>)<\/p>\n<p>The calculation of <em data-redactor-tag=\"em\">EBM<\/em> depends on the size of the sample and the level of confidence desired. The confidence level is the percent of all possible samples that can be expected to include the true population parameter. As the confidence level increases, the corresponding <em data-redactor-tag=\"em\">EBM<\/em> increases as well. As the sample size increases, the <em data-redactor-tag=\"em\">EBM<\/em> decreases. By the central limit theorem,<\/p>\n<p>Given a confidence interval, you can work backwards to find the error bound ( <em data-redactor-tag=\"em\">EBM<\/em>) or the sample mean. To find the error bound, find the difference of the upper bound of the interval and the mean. If you do not know the sample mean, you can find the error bound by calculating half the difference of the upper and lower bounds. To find the sample mean given a confidence interval, find the difference of the upper bound and the error bound. If the error bound is unknown, then average the upper and lower bounds of the confidence interval to find the sample mean.<\/p>\n<p>Sometimes researchers know in advance that they want to estimate a population mean within a specific margin of error for a given level of confidence. In that case, solve the <em data-redactor-tag=\"em\">EBM<\/em> formula for <em data-redactor-tag=\"em\">n<\/em> to discover the size of the sample that is needed to achieve this goal:<\/p>\n<h1>Formula Review<\/h1>\n<p class=\"p1\">[latex]\\displaystyle\\overline{X}{\\sim}{N}\\left({\\mu}_{x}, \\frac{{\\sigma}}{{\\sqrt{n}}}\\right)[\/latex]. The distribution of sample means is normally distributed with mean equal to the population mean and standard deviation given by the population standard deviation divided by the square root of the sample size.<\/p>\n<p>The general form for a confidence interval for a single population mean, known standard deviation, normal distribution is given by<\/p>\n<p>(lower bound, upper bound) = (point estimate \u2013 <em data-redactor-tag=\"em\">EBM<\/em>, point estimate + <em data-redactor-tag=\"em\">EBM<\/em>)<\/p>\n<p>=([latex]\\displaystyle\\overline{x}[\/latex] &#8211; EBM, [latex]\\displaystyle\\overline{x}[\/latex]+EBM)<\/p>\n<p>=([latex]\\displaystyle\\overline{x}-{z}_{\\frac{{\\alpha}}{{\\sqrt{n}}}}, \\overline{x}+{z}_{\\frac{{\\alpha}}{{\\sqrt{n}}}}[\/latex])<\/p>\n<p>EBM =\u00a0[latex]\\displaystyle{z}_{\\frac{{\\alpha}}{{\\sqrt{n}}}}[\/latex]= the error bound for the mean, or the margin of error for a single population mean; this formula is used when the population standard deviation is known.<\/p>\n<p id=\"eip-277\"><em data-effect=\"italics\">CL<\/em> = confidence level, or the proportion of confidence intervals created that are expected to contain the true population parameter<\/p>\n<p id=\"eip-401\"><em data-effect=\"italics\">\u03b1<\/em> = 1 \u2013 <em data-effect=\"italics\">CL<\/em> = the proportion of confidence intervals that will not contain the population<\/p>\n<p>[latex]\\displaystyle{z}_{\\frac{{\\alpha}}{{\\sqrt{n}}}}[\/latex]= the <em data-effect=\"italics\">z<\/em>-score with the property that the area to the right of the z-score is \u00a0\u221d2 this is the <em data-effect=\"italics\">z<\/em>-score used in the calculation of <em data-effect=\"italics\">&#8220;EBM\u00a0<\/em>where \u03b1 = 1 \u2013 <em data-effect=\"italics\">CL.<\/em><\/p>\n<p>n = [latex]\\displaystyle\\frac{{{z}^{2}{\\sigma}^{2}}}{{{EBM}^{2}}}[\/latex] = the formula used to determine the sample size (<em data-effect=\"italics\">n<\/em>) needed to achieve a desired margin of error at a given level of confidence<\/p>\n<p id=\"eip-73\">General form of a confidence interval<\/p>\n<p id=\"fs-idp49175936\">(lower value, upper value) = (point\u00a0estimate\u2212error bound, point estimate + error bound)<\/p>\n<p>To find the error bound when you know the confidence interval<\/p>\n<p>error bound = upper value\u2212point\u00a0estimate OR error bound =[latex]\\displaystyle\\frac{{\\text{upper}{value}-{lower}{value}}}{{2}}[\/latex]<\/p>\n<p id=\"eip-531\">Single Population Mean, Known Standard Deviation, Normal Distribution<\/p>\n<p>Use the Normal Distribution for Means, Population Standard Deviation is Known <em data-effect=\"italics\">EBM=[latex]\\displaystyle{z}_{\\frac{{\\alpha}}{{2}}\\cdot\\frac{{\\sigma}}{{\\sqrt{n}}}}[\/latex]<\/em><\/p>\n<p>The confidence interval has the format\u00a0EBM =\u00a0([latex]\\displaystyle\\overline{x}[\/latex] &#8211; EBM, [latex]\\displaystyle\\overline{x}[\/latex]+EBM)<\/p>\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-292\">\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>A Single Population Mean using the Normal Distribution. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44:51\/Introductory_Statistics\">http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44:51\/Introductory_Statistics<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Introductory Statistics . <strong>Authored by<\/strong>: Barbara Illowski, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\">http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/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>: Download for free at http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Intro to Confidence Intervals for One Mean (Sigma Known). <strong>Authored by<\/strong>: jbstatistics. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KG921rfbTDw\">https:\/\/youtu.be\/KG921rfbTDw<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube LIcense<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t 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