Applied Statistics and Probability for Engineers

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 is normal with mean

(7-7)

and variance

(7-8)

If the two populations are not normally distributed and if both sample sizes n 1 and n 2 are

greater than 30, we may use the central limit theorem and assume that and follow

approximately independent normal distributions. Therefore, the sampling distribution of

is approximately normal with mean and variance given by Equations 7-7 and 7-8,

respectively. If either n 1 or n 2 is less than 30, the sampling distribution of will still be

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.

X 1 X 2

X 1 X 2

X 1 X 2



2

X 1 X 2 

2

X 1 

2

X 2 

^21

n 1 

^22

n 2

X 1 X 2 X 1 X 2  1  2

X 1 X 2

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 is at least 25 hours? Assume that the old and improved processes can

be regarded as independent populations.

To solve this problem, we first note that the distribution of is normal with mean

 1 5000 hours and standard deviation hours, and the distribution

of is normal with mean  2 5050 hours and standard deviation 

6 hours. Now the distribution of is normal with mean 

50 hours and variance hours^2. This sampling distribu-

tion is shown in Fig. 7-9. The probability that is the shaded portion of the

normal distribution in this figure.

X 2 X 1  25

^22 n 2 ^21 n 1  1622  11022  136

X 2 X 1  2  1  5050  5000

X 2  2  1 n 2  (^30)  125
 1  1 n 1  (^40)  116  10
X 1
X 2 X 1
If we have two independent populations with means and and variances ^22 and
^22 and if and are the sample means of two independent random samples of
sizes n 1 and n 2 from these populations, then the sampling distribution of
(7-9)
is approximately standard normal, if the conditions of the central limit theorem
apply. If the two populations are normal, the sampling distribution of Zis exactly
standard normal.
Z
X 1 X 2  1  1  22
2 ^21 n 1 ^22 n 2
X 1 X 2
 1  2
Definition
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