{"id":86,"date":"2022-06-13T19:51:01","date_gmt":"2022-06-13T19:51:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/nhti-introstats\/chapter\/answers-to-selected-exercises-8\/"},"modified":"2022-06-13T19:51:01","modified_gmt":"2022-06-13T19:51:01","slug":"answers-to-selected-exercises-8","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/nhti-introstats\/chapter\/answers-to-selected-exercises-8\/","title":{"raw":"Answers to Selected Exercises","rendered":"Answers to Selected Exercises"},"content":{"raw":"\n1. mean = 4 hours; standard deviation = 1.2 hours; sample size = 16\n\n3.\n\n1. Check student's solution.\n\n2. 3.5, 4.25, 0.2441\n\n5. The fact that the two distributions are different accounts for the different probabilities.\n\n7.\n
    \n \t
  1. \u03a7<\/em> = amount of change students carry<\/li>\n \t
  2. \u03a7<\/em> ~ E<\/em>(0.88, 0.88)<\/li>\n \t
  3. X<\/span>\u23af\u23af\u23af<\/span><\/span><\/span><\/span><\/span><\/span><\/span> = average amount of change carried by a sample of 25 sstudents.<\/li>\n \t
  4. X<\/span>\u23af\u23af\u23af<\/span><\/span><\/span><\/span><\/span><\/span><\/span> ~ N<\/em>(0.88, 0.176)<\/li>\n \t
  5. 0.0819<\/li>\n \t
  6. 0.1882<\/li>\n \t
  7. The distributions are different. Part a is exponential and part b is normal.<\/li>\n<\/ol>\n9.\n
      \n \t
    1. length of time for an individual to complete IRS form 1040, in hours.<\/li>\n \t
    2. mean length of time for a sample of 36 taxpayers to complete IRS form 1040, in hours.<\/li>\n \t
    3. N<\/em>(<\/span>10<\/span>.53, <\/span>1<\/span>3<\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n \t
    4. Yes. I would be surprised, because the probability is almost 0.<\/li>\n \t
    5. No. I would not be totally surprised because the probability is 0.2312<\/li>\n<\/ol>\n11.\n
      \n
      \n
        \n \t
      1. the length of a song, in minutes, in the collection<\/li>\n \t
      2. U<\/em>(2, 3.5)<\/li>\n \t
      3. the average length, in minutes, of the songs from a sample of five albums from the collection<\/li>\n \t
      4. N<\/em>(2.75, 0.0220)<\/li>\n \t
      5. 2.74 minutes<\/li>\n \t
      6. 0.03 minutes<\/li>\n<\/ol>\n<\/section><\/div>\n<\/section><\/div>\n
        <\/div>\n
        <\/div>\n
        \n\n13.\n
          \n \t
        1. True. The mean of a sampling distribution of the means is approximately the mean of the data distribution.<\/li>\n \t
        2. True. According to the Central Limit Theorem, the larger the sample, the closer the sampling distribution of the means becomes normal.<\/li>\n \t
        3. The standard deviation of the sampling distribution of the means will decrease making it approximately the same as the standard deviation of X as the sample size increases.<\/li>\n<\/ol>\n15.\n
            \n \t
          1. X<\/em> = the yearly income of someone in a third world country<\/li>\n \t
          2. the average salary from samples of 1,000 residents of a third world country<\/li>\n \t
          3. [latex]\\overline{X}[\/latex] \u223c N<\/em>[latex]\\left(2000,\\frac{{8000}}{{\\sqrt{1000}}}\\right)[\/latex]<\/li>\n \t
          4. Very wide differences in data values can have averages smaller than standard deviations.<\/li>\n \t
          5. The distribution of the sample mean will have higher probabilities closer to the population mean.\n
            P<\/em>(2000 < [latex]\\overline{X}[\/latex]< 2100) = 0.1537<\/div>\n
            P<\/em>(2100 < [latex]\\overline{X}[\/latex]< 2200) = 0.1317<\/div><\/li>\n<\/ol>\n17. [latex]\\displaystyle\\overline{{X}}[\/latex] ~ N<\/em>(4.59, [latex]\\frac{{0.10}}{{\\sqrt{16}}}[\/latex])\n\n<\/div>\n

            The Central Limit Theorem for Sums<\/h2>\n18. 0.3345\n\n20. 7,833.92\n\n22. 0.0089\n\n24. 7,326.49\n\n26. 77.45%\n\n28. 0.4207\n\n30. 3,888.5\n\n32. 0.8186\n\n34. 5\n\n36. 0.9772\n\n38. The sample size, n<\/em>, gets larger.\n\n40. 49\n\n42. 26.00\n\n44. 0.1587\n\n46. 1,000\n\n49.\n
              \n \t
            1. the total length of time for nine criminal trials<\/li>\n \t
            2. N<\/em>(189, 21)<\/li>\n \t
            3. 0.0432<\/li>\n \t
            4. 162.09; ninety percent of the total nine trials of this type will last 162 days or more.<\/li>\n<\/ol>\n51.\n
                \n \t
              1. X<\/em> = the salary of one elementary school teacher in the district<\/li>\n \t
              2. X<\/em> ~ N<\/em>(44,000, 6,500)<\/li>\n \t
              3. \u03a3X<\/em> ~ sum of the salaries of ten elementary school teachers in the sample<\/li>\n \t
              4. \u03a3X<\/em> ~ N<\/em>(44000, 20554.80)<\/li>\n \t
              5. 0.9742<\/li>\n \t
              6. $52,330.09<\/li>\n \t
              7. 466,342.04<\/li>\n \t
              8. Sampling 70 teachers instead of ten would cause the distribution to be more spread out. It would be a more symmetrical normal curve.<\/li>\n \t
              9. If every teacher received a $3,000 raise, the distribution of X<\/em> would shift to the right by $3,000. In other words, it would have a mean of $47,000.<\/li>\n<\/ol>\n

                Using the Central Limit Theorem<\/h2>\n53. N<\/em>(25, 0.0577)\n\n55. 0.0003\n\n57. 25.07\n\n59.\n
                  \n \t
                1. N<\/em>(2,500, 5.7735)<\/li>\n \t
                2. 0<\/li>\n<\/ol>\n61. 10\n\n63. N[latex]\\left(10,\\frac{{10}}{{8}}\\right)[\/latex]\n\n65. 0.7799\n\n67. 1.69\n\n69. 0.0072\n\n74.\n
                    \n \t
                  1. \n
                      \n \t
                    1. X<\/em> = the closing stock prices for U.S. semiconductor manufacturers<\/li>\n \t
                    2. i. $20.71; ii. $17.31; iii. 35<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n \n
                        \n \t
                      1. Exponential distribution, \u03a7 ~ Exp<\/em>[latex]\\left(\\frac{{1}}{{20.71}}\\right)[\/latex]<\/li>\n \t
                      2. Answers will vary.<\/li>\n \t
                      3. i. $20.71; ii. $11.14<\/li>\n \t
                      4. Answers will vary.<\/li>\n \t
                      5. Answers will vary.<\/li>\n \t
                      6. Answers will vary.<\/li>\n \t
                      7. N[latex\\left(20.71,\\frac{{17.31}}{{\\sqrt{5}}}\\right)<\/li>\n<\/ol>\n76. two hours.\n\n78. 40.3\n\n80. almost zero\n\n82.\n
                          \n \t
                        1. 0<\/li>\n \t
                        2. 0.1123<\/li>\n \t
                        3. 0.0162<\/li>\n \t
                        4. 0.0003<\/li>\n \t
                        5. 0.0268<\/li>\n<\/ol>\n85.\n
                            \n \t
                          1. Check student\u2019s solution.<\/li>\n \t
                          2. [latex]\\overline{X}~\\left(60,\\frac{{9}}{{\\sqrt{25}}}\\right)[\/latex]<\/li>\n \t
                          3. 0.5000<\/li>\n \t
                          4. 59.06<\/li>\n \t
                          5. 0.8536<\/li>\n \t
                          6. 0.1333<\/li>\n \t
                          7. N<\/em>(1500, 45)<\/li>\n \t
                          8. 1530.35<\/li>\n \t
                          9. 0.6877<\/li>\n<\/ol>\n87.\n
                              \n \t
                            1. $52,330<\/li>\n \t
                            2. $46,634<\/li>\n<\/ol>\n89. We have\n