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               242     CHAPTER 7 POINT ESTIMATION OF PARAMETERS


                                 combinations of independent normal random variables follow a normal distribution (see
                                 Equation 5-41), we can say that the sampling distribution of X   X 2  is normal with mean
                                                                                   1

                                                        X 1  X 2  X 1  X 2  1    2                     (7-7)
                                 and variance

                                                                            2    2
                                                         2       2    2      1    2
                                                          X 1  X
                                                            2    X 1  X 2  n 1  n 2                    (7-8)
                                 If the two populations are not normally distributed and if both sample sizes n and n are
                                                                                                       2
                                                                                                 1
                                 greater than 30, we may use the central limit theorem and assume that  X 1  and  X 2  follow
                                 approximately independent normal distributions. Therefore, the sampling distribution of
                                 X   X 2  is approximately normal with mean and variance given by Equations 7-7 and 7-8,
                                  1
                                                                                                X  will still be
                                                   1
                                 respectively. If either n or n is less than 30, the sampling distribution of X 1  2
                                                       2
                                 approximately normal with mean and variance given by Equations 7-7 and 7-8, provided that
                                 the population from which the small sample is taken is not dramatically different from the nor-
                                 mal. We may summarize this with the following definition.
                       Definition
                                                                                                   2
                                    If we have two independent populations with means   1  and   2  and variances   and
                                                                                                   2
                                      2
                                      and if X 1  and X 2  are the sample means of two independent random samples of
                                      2
                                    sizes n and n from these populations, then the sampling distribution of
                                               2
                                          1
                                                             X   X   1     2
                                                              1
                                                                   2
                                                                         1
                                                                              2
                                                         Z                                          (7-9)
                                                                          2
                                                                   2
                                                               2  1 n    2 n 2
                                                                     1
                                    is approximately standard normal, if the conditions of the central limit theorem
                                    apply. If the two populations are normal, the sampling distribution of Z is exactly
                                    standard normal.
               EXAMPLE 7-15      The effective life of a component used in a jet-turbine aircraft engine is a random variable
                                 with mean 5000 hours and standard deviation 40 hours. The distribution of effective life is
                                 fairly close to a normal distribution. The engine manufacturer introduces an improvement
                                 into the manufacturing process for this component that increases the mean life to 5050 hours
                                 and decreases the standard deviation to 30 hours. Suppose that a random sample of n 1   16
                                 components is selected from the “old” process and a random sample of n 2   25 components
                                 is selected from the “improved” process. What is the probability that the difference in the two
                                 sample means X   X 1  is at least 25 hours? Assume that the old and improved processes can
                                              2
                                 be regarded as independent populations.
                                    To solve this problem, we first note that the distribution of  X 1  is normal with mean
                                                                  1
                                   1   5000 hours and standard deviation   1n   40  116   10  hours, and the distribution
                                                                       1
                                                                                          2
                                 of X 2  is normal with mean   2    5050 hours and standard deviation   1n   30  125
                                                                                               2
                                 6 hours. Now the distribution of X   X 1  is normal with mean        5050   5000
                                                                                           1
                                                                                      2
                                                             2
                                                            2
                                                                    2
                                                                           2
                                                    2
                                                                                       2
                                 50 hours and variance   2 n    1 n   162   1102   136  hours . This sampling distribu-
                                                       2
                                                              1
                                                                            X   25  is the shaded portion of the
                                 tion is shown in Fig. 7-9. The probability that  X 2  1
                                 normal distribution in this figure.
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