Question

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

In a random sample of 504 judges, it was found that 291 were introverts.

(a) Let p represent the proportion of all judges who are introverts. Find a point estimate for p. (Round your answer to four decimal places.)


(b) Find a 99% confidence interval for p. (Round your answers to two decimal places.)

lower limit
upper limit

Give a brief interpretation of the meaning of the confidence interval you have found.

We are 1% confident that the true proportion of judges who are introverts falls above this interval.We are 99% confident that the true proportion of judges who are introverts falls outside this interval.    We are 1% confident that the true proportion of judges who are introverts falls within this interval.We are 99% confident that the true proportion of judges who are introverts falls within this interval.

(c) Do you think the conditions np > 5 and nq > 5 are satisfied in this problem? Explain why this would be an important consideration.

No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately binomial.No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately normal.    Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately binomial.

Answer 1

a)

Point estimate of = 291 / 504 = 0.577

b)

p̂ = X / n = 291/504 = 0.577
p̂ ± Z(α/2) √( (p * q) / n)
0.577 ± Z(0.01/2) √( (0.5774 * 0.4226) / 504)
Z(α/2) = Z(0.01/2) = 2.576
Lower Limit = 0.577 - Z(0.01) √( (0.5774 * 0.4226) / 504) = 0.521
upper Limit = 0.577 + Z(0.01) √( (0.5774 * 0.4226) / 504) = 0.634

We are 99% confident that the true proportion of judges who are introverts falls within this interval.

c)

n = 0.577 * 504 = 291 > 5

n = ( 1 - 0.577) * 504 = 213

Yes, the conditions are satisfied. This is important because it allows us to say that is approximately normal.