Applied Statistics and Probability for Engineers

254 CHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE

8-2.5 A Large-Sample Confidence Interval for 

We have assumed that the population distribution is normal with unknown mean and known

standard deviation . We now present a large-sample CIand that does not require these as-

sumptions. Let X 1 , X 2 , p, Xnbe a random sample from a population with unknown mean

and variance. Now if the sample size nis large, the central limit theorem implies that

has approximately a normal distribution with mean and variance ^2 n. Therefore

has approximately a standard normal distribution. This ratio could be

used as a pivotal quantity and manipulated as in Section 8-2.1 to produce an approximate CI

for . However, the standard deviation is unknown. It turns out that when nis large, replac-

ing by the sample standard deviation Shas little effect on the distribution of Z. This leads to

the following useful result.

Z 1 X 2 1  1 n 2

^2 X

When nis large, the quantity

has an approximate standard normal distribution. Consequently,

(8-13)

is alarge sample confidence interval for , with confidence level of approximately

100(1  )%.

xz    2

s

1 n

xz     2

s

1 n

X

S   1 n

Definition

1.230

1.330

0.040

0.044

1.200

0.270

0.490

0.190

0.830

0.810

0.710

0.500

0.490

1.160

0.050

0.150

0.190

0.770

1.080

0.980

0.630

0.560

0.410

0.730

0.590

0.340

0.340

0.840

0.500

0.340

0.280

0.340

0.750

0.870

0.560

0.170

0.180

0.190

0.040

0.490

1.100

0.160

0.100

0.210

0.860

0.520

0.650

0.270

0.940

0.400

0.430

0.250

0.270

Equation 8-13 holds regardless of the shape of the population distribution. Generally nshould

be at least 40 to use this result reliably. The central limit theorem generally holds for n 30,

but the larger sample size is recommended here because replacing by Sin Zresults in addi-

tional variability.

EXAMPLE 8-3 An article in the 1993 volume of the Transactions of the American Fisheries Societyreports

the results of a study to investigate the mercury contamination in largemouth bass. A sample

of fish was selected from 53 Florida lakes and mercury concentration in the muscle tissue was

measured (ppm). The mercury concentration values are

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