## N V N

(iv) If the sample size is large,s the random variable

is approximately distributed as standard normal.

Notice that for fixed p, the standard error of the sample proportion decreases as the sample size increases. This implies that for increasing sample size, the distribution of px becomes more concentrated about its mean, as illustrated in Figure 6.4. The implication is that for any particular population proportion, the probability that the sample and population proportions will differ by any fixed amount decreases as the number of sample members increases. In other words, if we take a bigger sample from a population, our inference about the proportion of population members that possess some particular characteristic becomes more firm.

When the sample size is large, the normal approximation to the binomial distribution provides a very convenient procedure for calculating the probability that a sample proportion lies in some given range. This is illustrated in the following examples.

A random sample of 250 homes was taken from a large population of older homes to estimate the proportion of such homes in which the electric wiring was unsafe. Suppose that, in fact, 30% of all homes in this population have unsafe wiring. Find the probability that the proportion of homes in the sample with unsafe wiring is between .25 and .35.

We have p = .30 n = 250 Denote by px the proportion of homes in the sample with unsafe wiring. We then require

5 In general, the approximation is satisfactory for samples of fifty or more observations. The quality of the approximation depends also on p: ideally, we should have np( 1 — p) > 9.

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