{"id":271,"date":"2021-07-14T15:59:03","date_gmt":"2021-07-14T15:59:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/a-single-population-mean-using-the-normal-distribution\/"},"modified":"2023-12-05T09:27:04","modified_gmt":"2023-12-05T09:27:04","slug":"a-single-population-mean-using-the-normal-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/a-single-population-mean-using-the-normal-distribution\/","title":{"raw":"Estimating a Population Mean","rendered":"Estimating a Population Mean"},"content":{"raw":"
\r\n

Learning Outcomes<\/h3>\r\n
\r\n
    \r\n \t
  • Given a sample mean and error bound, create a confidence interval<\/li>\r\n \t
  • Using the formula for creating a confidence interval or technology, construct a confidence interval for a population mean based on a normal distribution<\/li>\r\n \t
  • Interpret a confidence interval for a population mean in context<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n
    \r\n

    Recall: INTERVAL NOTATION & DEFINITION<\/h3>\r\n\r\n\r\n\r\n\r\n\r\n
    Inequality<\/th>\r\nWords<\/th>\r\nInterval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    [latex]{a}\\lt{x}\\lt{ b}[\/latex]<\/td>\r\nall real numbers between\u00a0a<\/em> and b<\/em>, not including a and b<\/td>\r\n[latex]\\left(a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe main concept to remember is that parentheses represent solutions greater than or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. In statistics we use interval notation when we are writing a confidence interval. A confidence interval is an estimate of a population parameter. Since we are estimating the lower and upper limits we use parentheses to enclose the interval. A confidence interval can also be called an interval estimator.\r\n\r\n<\/div>\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 EBM<\/em> = 5.\r\n
    \r\n

    Recall: Convert a PERCENT TO A DECIMAL<\/h3>\r\n
      \r\n \t
    1. Write the percent as a ratio with the denominator 100.<\/li>\r\n \t
    2. Convert the fraction to a decimal by dividing the numerator by the denominator.<\/li>\r\n<\/ol>\r\n<\/div>\r\n

      Calculating the Confidence Interval<\/h2>\r\nTo construct a confidence interval for a single unknown population mean\u00a0\u03bc<\/em>,\u00a0where the population standard deviation is known<\/strong>, we need [latex]\\overline{{x}}[\/latex]\u00a0as an estimate for\u00a0\u03bc<\/em>\u00a0and we need the margin of error. Here, the margin of error (EBM<\/em>) is called the\u00a0error bound for a population mean<\/span>\u00a0(abbreviated\u00a0EBM<\/em><\/strong>). The sample mean [latex]\\overline{{x}}[\/latex]\u00a0is the\u00a0point estimate<\/strong>\u00a0of the unknown population mean\u00a0\u03bc<\/em>.\r\n\r\nThe confidence interval estimate will have the form:<\/strong>\r\n\r\n(point estimate \u2013 error bound, point estimate + error bound) or, in symbols, [latex]\\displaystyle{(\\overline{{x}}-{E}{B}{M},\\overline{{x}}+{E}{B}{M})}[\/latex]\r\n

      The margin of error (EBM<\/em>) depends on the\u00a0confidence level<\/span>\u00a0(abbreviated\u00a0CL<\/em><\/strong>). 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 their conclusions.<\/p>\r\nThere is another probability called alpha (\u03b1<\/em>). \u03b1<\/em> is related to the confidence level, CL<\/em>. \u03b1<\/em> is the probability that the interval does not contain the unknown population parameter.\r\n\r\nMathematically,\u00a0\u03b1<\/em> + CL<\/em> = 1.\r\n\r\nhttps:\/\/youtu.be\/KG921rfbTDw\r\n

      \r\n

      Example 1<\/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. The sample mean is seven, and the error bound for the mean is 2.5.\r\n\r\n[latex]\\overline{{x}} = {7}[\/latex] and [latex]{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 (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>\r\n
      \r\n

      try it 1<\/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[reveal-answer q=\"609234\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"609234\"]\r\n\r\n(11.8, 18.2)\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\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 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\u03b1<\/em> = 10% in both tails, or 5% in each tail, of the normal distribution.\r\n\r\n \r\n\r\n\"This\r\n

      To 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\u00a0z<\/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>\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 \u03c3<\/em>.\r\n\r\nIn summary, as a result of the central limit theorem:<\/strong>\r\n

        \r\n \t
      • [latex]\\displaystyle\\overline{X}[\/latex] is normally distributed, that is,\u00a0[latex]\\displaystyle\\overline{X}{\\sim}{N}\\left({\\mu}_{x}, \\frac{{\\sigma}}{{\\sqrt{n}}}\\right)[\/latex]<\/li>\r\n \t
      • When the population standard deviation \u03c3<\/em> is known, we use a normal distribution to calculate the error bound.<\/strong><\/li>\r\n<\/ul>\r\n

        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
          \r\n \t
        • Calculate the sample mean [latex]\\displaystyle\\overline{{x}}[\/latex] from the sample data. Remember, we already know the population standard deviation \u03c3\u00a0<\/em>in this section.<\/li>\r\n \t
        • Find the z<\/em>-score that corresponds to the confidence level.<\/li>\r\n \t
        • Calculate the error bound EBM<\/em>.<\/li>\r\n \t
        • Construct the confidence interval.<\/li>\r\n \t
        • 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

          Finding the z<\/em>-score for the Stated Confidence Level<\/h2>\r\nWhen we know the population standard deviation \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 z<\/em> that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution Z<\/em> ~ N<\/em>(0, 1).\r\n\r\nThe confidence level, CL<\/em>, is the area in the middle of the standard normal distribution. CL<\/em> = 1 \u2013 \u03b1<\/em>, so \u03b1<\/em> is the area that is split equally between the two tails. Each of the tails contains an area equal to [latex]\\dfrac{\u03b1}{2}[\/latex].\r\n\r\nThe z<\/em>-score that has an area to the right of [latex]\\dfrac{\u03b1}{2}[\/latex] is denoted by [latex]z_\\dfrac{\u03b1}{2}[\/latex].\r\n\r\nFor example, when CL<\/em> = 0.95, \u03b1<\/em> = 0.05 and [latex]\\dfrac{\u03b1}{2}[\/latex] = 0.025; we write [latex]z_\\dfrac{\u03b1}{2}[\/latex] =\u00a0z<\/em>0.025<\/sub>.\r\n\r\nThe area to the right of z<\/em>0.025<\/sub> is 0.025 and the area to the left of z<\/em>0.025<\/sub> is 1 \u2013 0.025 = 0.975.\r\n\r\n[latex]z_\\dfrac{\u03b1}{2}[\/latex] =\u00a0z<\/em>0.025\u00a0<\/sub>= 1.96, using a calculator, computer or a standard normal probability table.\r\n\r\n
          \r\n

          USING THE TI-83, 83+, 84, 84+ CALCULATOR<\/span><\/h3>\r\n<\/header>invNorm<\/code>(0.975, 0, 1) = 1.96\r\n

          Note<\/h3>\r\nRemember to use the area to the LEFT of [latex]z_\\dfrac{\u03b1}{2}[\/latex]; in this chapter the last two inputs in the invNorm command are 0, 1, because you are using a standard normal distribution Z<\/em> ~ N<\/em>(0, 1).\r\n

          Calculating the Error Bound (EBM<\/em>)<\/h2>\r\nThe error bound formula for an unknown population mean \u03bc<\/em> when the population standard deviation \u03c3<\/em> is known is\r\n
            \r\n \t
          • EBM = ([latex]\\displaystyle{z}_{\\frac{{\\alpha}}{{2}}})(\\frac{{\\sigma}}{{\\sqrt{n}}})[\/latex]<\/li>\r\n<\/ul>\r\n

            Constructing the Confidence Interval<\/h2>\r\n
              \r\n \t
            • The confidence interval estimate has the format ([latex]\\displaystyle\\overline{x}[\/latex] \u2013 EBM,[latex]\\displaystyle\\overline{x}[\/latex] + EBM).<\/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 \r\n\r\n\"Graph\r\n

              Writing the Interpretation<\/h2>\r\nThe interpretation should clearly state the confidence level (CL<\/em>), explain what population parameter is being estimated (here, a 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
              \r\n

              Example 2<\/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
                \r\n \t
              • You can use technology to calculate the confidence interval directly.<\/li>\r\n<\/ul>\r\n \r\n
                  \r\n \t
                • The first solution is shown step-by-step (Solution A).\r\n[reveal-answer q=\"986729\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"986729\"]Solution A:\r\nTo find the confidence interval, you need the sample mean,\u00a0[latex]\\overline{{x}}[\/latex], and the EBM.\r\n[latex]\\overline{x}[\/latex] = 68\r\n

                  [latex]{EBM}=({z}_{\\frac{{\\alpha}}{{2}}})(\\frac{{\\sigma}}{{\\sqrt{n}}})[\/latex]<\/p>\r\n[latex]\\displaystyle\\sigma=3[\/latex]\r\n\r\nn = 6\r\n\r\nThe confidence level is 90% ( CL = 0.90)\r\n\r\nCL = 0.90 so \u03b1 = 1 \u2013 CL = 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 [latex]z_{0.05}[\/latex] is 0.05 and the area to the left of [latex]z_{0.05}[\/latex] is 1 \u2013 0.05 = 0.95.\r\n\r\n[latex]z_\\frac{\\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[\/hidden-answer]<\/li>\r\n<\/ul>\r\n \r\n

                    \r\n \t
                  • The second solution uses the TI-83, 83+, and 84+ calculators (Solution B).<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"105256\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"105256\"]\r\n\r\n
                    \r\n

                    USING THE TI-83, 83+, 84, 84+ CALCULATOR<\/span><\/h4>\r\n<\/header>Press STAT<\/code> and arrow over to TESTS<\/code>.\r\nArrow down to 7:ZInterval<\/code>.\r\nPress ENTER<\/code>.\r\nArrow to Stats<\/code> and press ENTER<\/code>.\r\nArrow down and enter three for \u03c3<\/em>, 68 for[latex]\\displaystyle\\overline{X}[\/latex], 36 for n<\/em>, and .90 for C-level<\/code>.\r\nArrow down to Calculate<\/code> and press ENTER<\/code>.\r\nThe confidence interval is (to three decimal places)(67.178, 68.822).\r\n

                    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

                    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\r\n[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n
                    \r\n

                    try it 2<\/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[reveal-answer q=\"330453\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"330453\"]\r\n\r\n(34.1347, 37.8653)\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

                    <\/h3>\r\n
                    \r\n

                    Example 3<\/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\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                    Phone Model<\/th>\r\nSAR<\/th>\r\nPhone Model<\/th>\r\nSAR<\/th>\r\nPhone Model<\/th>\r\nSAR<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
                    Apple iPhone 4S<\/td>\r\n1.11<\/td>\r\nLG Ally<\/td>\r\n1.36<\/td>\r\nPantech Laser<\/td>\r\n0.74<\/td>\r\n<\/tr>\r\n
                    BlackBerry Pearl 8120<\/td>\r\n1.48<\/td>\r\nLG AX275<\/td>\r\n1.34<\/td>\r\nSamsung Character<\/td>\r\n0.5<\/td>\r\n<\/tr>\r\n
                    BlackBerry Tour 9630<\/td>\r\n1.43<\/td>\r\nLG Cosmos<\/td>\r\n1.18<\/td>\r\nSamsung Epic 4G Touch<\/td>\r\n0.4<\/td>\r\n<\/tr>\r\n
                    Cricket TXTM8<\/td>\r\n1.3<\/td>\r\nLG CU515<\/td>\r\n1.3<\/td>\r\nSamsung M240<\/td>\r\n0.867<\/td>\r\n<\/tr>\r\n
                    HP\/Palm Centro<\/td>\r\n1.09<\/td>\r\nLG Trax CU575<\/td>\r\n1.26<\/td>\r\nSamsung Messager III SCH-R750<\/td>\r\n0.68<\/td>\r\n<\/tr>\r\n
                    HTC One V<\/td>\r\n0.455<\/td>\r\nMotorola Q9h<\/td>\r\n1.29<\/td>\r\nSamsung Nexus S<\/td>\r\n0.51<\/td>\r\n<\/tr>\r\n
                    HTC Touch Pro 2<\/td>\r\n1.41<\/td>\r\nMotorola Razr2 V8<\/td>\r\n0.36<\/td>\r\nSamsung SGH-A227<\/td>\r\n1.13<\/td>\r\n<\/tr>\r\n
                    Huawei M835 Ideos<\/td>\r\n0.82<\/td>\r\nMotorola Razr2 V9<\/td>\r\n0.52<\/td>\r\nSGH-a107 GoPhone<\/td>\r\n0.3<\/td>\r\n<\/tr>\r\n
                    Kyocera DuraPlus<\/td>\r\n0.78<\/td>\r\nMotorola V195s<\/td>\r\n1.6<\/td>\r\nSony W350a<\/td>\r\n1.48<\/td>\r\n<\/tr>\r\n
                    Kyocera K127 Marbl<\/td>\r\n1.25<\/td>\r\nNokia 1680<\/td>\r\n1.39<\/td>\r\nT-Mobile Concord<\/td>\r\n1.38<\/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 \u03c3<\/em> = 0.337.[reveal-answer q=\"207583\"]Show Solution A[\/reveal-answer]\r\n[hidden-answer a=\"207583\"]\r\n\r\nTo find the confidence interval, start by finding the point estimate: the sample mean.\r\n

                    [latex]\\displaystyle\\overline{x}[\/latex] = 1.024<\/p>\r\nNext, find the EBM<\/em>. Because you are creating a 98% confidence interval, CL<\/em> = 0.98.\r\n

                    \"Graph<\/p>\r\n

                    You need to find z<\/em>0.01<\/sub> having the property that the area under the normal density curve to the right of z<\/em>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 z<\/em>0.01 <\/sub>= 2.326.<\/p>\r\n

                    EBM = ([latex]\\displaystyle{z}_{0.01}\\frac{{\\sigma}}{{\\sqrt{n}}}=(2.236)\\frac{{0.337}}{{\\sqrt{30}}}=0.1431[\/latex]<\/p>\r\n

                    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\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"935802\"]Show Solution B[\/reveal-answer]\r\n[hidden-answer a=\"935802\"]\r\n\r\n

                    \r\n

                    USING THE TI-83, 83+, 84, 84+ CALCULATOR<\/span><\/h4>\r\n<\/header>
                    \r\n
                    <\/div>\r\n<\/section>\r\n
                      \r\n \t
                    • Press STAT and arrow over to TESTS.<\/li>\r\n \t
                    • Arrow down to 7: ZInterval.<\/li>\r\n \t
                    • Press ENTER.<\/li>\r\n \t
                    • Arrow to Stats and press ENTER.<\/li>\r\n \t
                    • Arrow down and enter the following values:\r\n
                        \r\n \t
                      • \u03c3<\/em>: 0.337<\/li>\r\n \t
                      • [latex]\\displaystyle\\overline{x}[\/latex]:\u00a01.024<\/li>\r\n \t
                      • n<\/em>: 30<\/li>\r\n \t
                      • C<\/em>-level: 0.98<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
                      • Arrow down to Calculate and press ENTER.<\/li>\r\n \t
                      • The confidence interval is (to three decimal places) (0.881, 1.167).<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
                        \r\n

                        try it 3<\/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 \u03c3<\/em> = 0.337.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                        Phone Model<\/th>\r\nSAR<\/th>\r\nPhone Model<\/th>\r\nSAR<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
                        Blackberry Pearl 8120<\/td>\r\n1.48<\/td>\r\nNokia E71x<\/td>\r\n1.53<\/td>\r\n<\/tr>\r\n
                        HTC Evo Design 4G<\/td>\r\n0.8<\/td>\r\nNokia N75<\/td>\r\n0.68<\/td>\r\n<\/tr>\r\n
                        HTC Freestyle<\/td>\r\n1.15<\/td>\r\nNokia N79<\/td>\r\n1.4<\/td>\r\n<\/tr>\r\n
                        LG Ally<\/td>\r\n1.36<\/td>\r\nSagem Puma<\/td>\r\n1.24<\/td>\r\n<\/tr>\r\n
                        LG Fathom<\/td>\r\n0.77<\/td>\r\nSamsung Fascinate<\/td>\r\n0.57<\/td>\r\n<\/tr>\r\n
                        LG Optimus Vu<\/td>\r\n0.462<\/td>\r\nSamsung Infuse 4G<\/td>\r\n0.2<\/td>\r\n<\/tr>\r\n
                        Motorola Cliq XT<\/td>\r\n1.36<\/td>\r\nSamsung Nexus S<\/td>\r\n0.51<\/td>\r\n<\/tr>\r\n
                        Motorola Droid Pro<\/td>\r\n1.39<\/td>\r\nSamsung Replenish<\/td>\r\n0.3<\/td>\r\n<\/tr>\r\n
                        Motorola Droid Razr M<\/td>\r\n1.3<\/td>\r\nSony W518a Walkman<\/td>\r\n0.73<\/td>\r\n<\/tr>\r\n
                        Nokia 7705 Twist<\/td>\r\n0.7<\/td>\r\nZTE C79<\/td>\r\n0.869<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"457012\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"457012\"]\r\n

                        [latex]\\displaystyle\\overline{x}[\/latex] = 0.940<\/p>\r\n

                        [latex]\\displaystyle\\frac{{\\alpha}}{{2}}=\\frac{{1-CL}}{{2}}=\\frac{{1-0.93}}{{2}}[\/latex] = 0.035<\/p>\r\n

                        [latex]\\displaystyle{z}_{0.05}[\/latex] = 1.812<\/p>\r\n

                        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[\/hidden-answer]\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.","rendered":"

                        \n

                        Learning Outcomes<\/h3>\n
                        \n
                          \n
                        • Given a sample mean and error bound, create a confidence interval<\/li>\n
                        • Using the formula for creating a confidence interval or technology, construct a confidence interval for a population mean based on a normal distribution<\/li>\n
                        • Interpret a confidence interval for a population mean in context<\/li>\n<\/ul>\n<\/section>\n<\/div>\n
                          \n

                          Recall: INTERVAL NOTATION & DEFINITION<\/h3>\n\n\n\n\n\n
                          Inequality<\/th>\nWords<\/th>\nInterval Notation<\/th>\n<\/tr>\n<\/thead>\n
                          [latex]{a}\\lt{x}\\lt{ b}[\/latex]<\/td>\nall real numbers between\u00a0a<\/em> and b<\/em>, not including a and b<\/td>\n[latex]\\left(a,b\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                          The main concept to remember is that parentheses represent solutions greater than or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. In statistics we use interval notation when we are writing a confidence interval. A confidence interval is an estimate of a population parameter. Since we are estimating the lower and upper limits we use parentheses to enclose the interval. A confidence interval can also be called an interval estimator.<\/p>\n<\/div>\n

                          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 EBM<\/em> = 5.<\/p>\n

                          \n

                          Recall: Convert a PERCENT TO A DECIMAL<\/h3>\n
                            \n
                          1. Write the percent as a ratio with the denominator 100.<\/li>\n
                          2. Convert the fraction to a decimal by dividing the numerator by the denominator.<\/li>\n<\/ol>\n<\/div>\n

                            Calculating the Confidence Interval<\/h2>\n

                            To construct a confidence interval for a single unknown population mean\u00a0\u03bc<\/em>,\u00a0where the population standard deviation is known<\/strong>, we need [latex]\\overline{{x}}[\/latex]\u00a0as an estimate for\u00a0\u03bc<\/em>\u00a0and we need the margin of error. Here, the margin of error (EBM<\/em>) is called the\u00a0error bound for a population mean<\/span>\u00a0(abbreviated\u00a0EBM<\/em><\/strong>). The sample mean [latex]\\overline{{x}}[\/latex]\u00a0is the\u00a0point estimate<\/strong>\u00a0of the unknown population mean\u00a0\u03bc<\/em>.<\/p>\n

                            The confidence interval estimate will have the form:<\/strong><\/p>\n

                            (point estimate \u2013 error bound, point estimate + error bound) or, in symbols, [latex]\\displaystyle{(\\overline{{x}}-{E}{B}{M},\\overline{{x}}+{E}{B}{M})}[\/latex]<\/p>\n

                            The margin of error (EBM<\/em>) depends on the\u00a0confidence level<\/span>\u00a0(abbreviated\u00a0CL<\/em><\/strong>). 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 their conclusions.<\/p>\n

                            There is another probability called alpha (\u03b1<\/em>). \u03b1<\/em> is related to the confidence level, CL<\/em>. \u03b1<\/em> is the probability that the interval does not contain the unknown population parameter.<\/p>\n

                            Mathematically,\u00a0\u03b1<\/em> + CL<\/em> = 1.<\/p>\n