Applied Statistics and Probability for Engineers

240 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

If X 1 , X 2 ,, Xnis a random sample of size ntaken from a population (either finite

or infinite) with mean and finite variance ^2 , and if is the sample mean, the lim-

iting form of the distribution of

(7-6)

as nS, is the standard normal distribution.

Z

X

 1 n

X

p

Theorem 7-2:

The Central

Limit Theorem

The normal approximation for depends on the sample size n. Figure 7-6(a) shows the

distribution obtained for throws of a single, six-sided true die. The probabilities are equal

(16) for all the values obtained, 1, 2, 3, 4, 5, or 6. Figure 7-6(b) shows the distribution of the

average score obtained when tossing two dice, and Fig. 7-6(c), 7-6(d), and 7-6(e) show the

distributions of average scores obtained when tossing three, five, and ten dice, respectively.

Notice that, while the population (one die) is relatively far from normal, the distribution of

averages is approximated reasonably well by the normal distribution for sample sizes as small

as five. (The dice throw distributions are discrete, however, while the normal is continuous).

Although the central limit theorem will work well for small samples (n4, 5) in most cases,

particularly where the population is continuous, unimodal, and symmetric, larger samples will

be required in other situations, depending on the shape of the population. In many cases of

practical interest, if n 30, the normal approximation will be satisfactory regardless of the

X

123456 x

(a) One die

123456 x

(b) Two dice

123456 x

(c) Three dice

123456 x

(d) Five dice

123456 x

(e) Ten dice

Figure 7-6

Distributions of average

scores from throwing

dice. [Adapted with

permission from Box,

Hunter, and Hunter

(1978).]

variance , if the sample size nis large. This is one of the most useful theorems in statis-

tics, called the central limit theorem.The statement is as follows:

^2 n

c07.qxd 5/15/02 10:18 M Page 240 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files: