Applied Statistics and Probability for Engineers

7-5 SAMPLING DISTRIBUTIONS OF MEANS 241

shape of the population. If n 30, the central limit theorem will work if the distribution of the

population is not severely nonnormal.

EXAMPLE 7-13 An electronics company manufactures resistors that have a mean resistance of 100 ohms and

a standard deviation of 10 ohms. The distribution of resistance is normal. Find the probability

that a random sample of n25 resistors will have an average resistance less than 95 ohms.

Note that the sampling distribution of is normal, with mean and a

standard deviation of

Therefore, the desired probability corresponds to the shaded area in Fig. 7-7. Standardizing

the point in Fig. 7-7, we find that

and therefore,

The following example makes use of the central limit theorem.

EXAMPLE 7-14 Suppose that a random variable Xhas a continuous uniform distribution

Find the distribution of the sample mean of a random sample of size n40.

The mean and variance of Xare 5 and. The central limit

theorem indicates that the distribution of is approximately normal with mean and

variance. The distributions of Xand are shown in Fig. 7-8.

Now consider the case in which we have two independent populations. Let the first pop-

ulation have mean  1 and variance and the second population have mean  2 and variance

^22. Suppose that both populations are normally distributed. Then, using the fact that linear

^21

X^2 ^2 n (^1)  3314024  (^1)  120 X
X X 5
^2  16  (^422)  12  (^1)  3
f 1 x 2 e
(^1) 2, 4x 6
0, otherwise
0.0062
P 1 X 952 P 1 Z2.5 2
z
95  100
2
2.5
X 95
X

1 n

10
125
 2
X X100 ohms
(^95100) x
σX = 2
Figure 7-8 The distributions of Xand
Xfor Example 7-14.
Figure 7-7 Probability for Example 7-13.
4 56 x
σX = 1/120
465 x
2
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