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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.