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EXAMPLE 12 Determining whether the Central Limit Theorem for Proportions applies

In Example 11, we learned that color blindness linked to the X chromosome afflicts 8% of men. Determine the approximate normality of the sampling distribution of ˆp, the proportion of men who have color blindness linked to the X chromosome, for samples of size (a) 50 and (b) 100.

Solution

We need to check both conditions to find whether the sampling distribution of ˆp is approximately normal.

  1. We are given that p=0.08 and n=50

    n·p=50·0.08=4andn·q=50·(0.92)=46

    Because 4 is not ≥ 5, the first condition is not satisfied. The Central Limit Theorem for Proportions cannot be used. We cannot conclude that the sampling distribution of ˆp is approximately normal.

  2. Here, p=0.08 and n=100.

    n·p=100·0.08=8andn·q=100·(0.92)=92

    Because both 8 and 92 are ≥ 5, both conditions are satisfied. The Central Limit Theorem for Proportions applies, and we can conclude that the sampling distribution of ˆp is approximately normal. From Example 11, we have μˆp=0.08 and σˆp=0.02713. Thus, the sampling distribution of ˆp is approximately normal with μˆp=0.08 and σˆp=0.02713

NOW YOU CAN DO

Exercises 7–18.

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