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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 CI and that does not require these as-
sumptions. Let X 1 , X 2 , p , X n be a random sample from a population with unknown mean
and variance 2 . Now if the sample size n is large, the central limit theorem implies that X
2
has approximately a normal distribution with mean and variance n. Therefore
Z 1X 2 1 1n2 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 n is large, replac-
ing by the sample standard deviation S has little effect on the distribution of Z. This leads to
the following useful result.

Definition
When n is large, the quantity

X
S 1n

has an approximate standard normal distribution. Consequently,

s s
x z 2 x z 2 (8-13)
1n 1n

is a large sample confidence interval for , with confidence level of approximately
100(1 )%.

Equation 8-13 holds regardless of the shape of the population distribution. Generally n should
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 S in Z results in addi-
tional variability.

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

1.230 0.490 0.490 1.080 0.590 0.280 0.180 0.100 0.940
1.330 0.190 1.160 0.980 0.340 0.340 0.190 0.210 0.400
0.040 0.830 0.050 0.630 0.340 0.750 0.040 0.860 0.430
0.044 0.810 0.150 0.560 0.840 0.870 0.490 0.520 0.250
1.200 0.710 0.190 0.410 0.500 0.560 1.100 0.650 0.270
0.270 0.500 0.770 0.730 0.340 0.170 0.160 0.270