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The National Institutes of Health reported that color blindness linked to the X chromosome afflicts 8% of men. Determine the approximate normality of the sampling distribution of \( p \), the proportion of men who have color blindness linked to the X chromosome, for samples of size (a) 50 and (b) 100. Let \( p \) be the sample proportion of men who are color blind.

(a) Is it appropriate to approximate the \( p \) distribution with a normal distribution for a sample of \( n = 50 \) men? Explain.

(b) Is it appropriate to approximate the \( p \) distribution with a normal distribution for a sample of \( n = 100 \) men? For a sample size of \( n = 100 \), what are the values of \( \mu_p \) and \( \sigma_p \)?

Answer :

It is not appropriate to approximate the sampling distribution of p with a normal distribution for a sample size of 50 men.

(a) To determine the approximate normality of the sampling distribution of p for a sample size of 50, we need to check whether the sample size is large enough for the normal approximation to hold. The sample size is sufficiently large if both np and n(1-p) are greater than or equal to 10, where n is the sample size and p is the true proportion of men who have color blindness linked to the X chromosome. In this case, np = 50(0.08) = 4 and n(1-p) = 50(0.92) = 46.24, which are not both greater than or equal to 10. Therefore, it is not appropriate to approximate the sampling distribution of p with a normal distribution for a sample size of 50 men.

(b) For a sample size of 100 men, np = 100(0.08) = 8 and n(1-p) = 100(0.92) = 92, which are both greater than or equal to 10. Therefore, it is appropriate to approximate the sampling distribution of p with a normal distribution for a sample size of 100 men.

The values of Hp and Op for a sample size of 100 are:

Hp (hypothesized value of p) = 0.08 (given in the problem)

Op (expected value of p under the null hypothesis) = 0.08 (since the null hypothesis assumes that the true proportion of men with color blindness linked to the X chromosome is 0.08)

Know more about chromosome here: https://brainly.com/question/30993611

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