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 66 professional actors, it was found that 42 were extroverts. (a) Let p represent the proportion of all actors who are extroverts. Find a point estimate for p. (Round your answer to four decimal places.) Incorrect: Your answer is incorrect. (b) Find a 95% 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 5% confident that the true proportion of actors who are extroverts falls above this interval. We are 5% confident that the true proportion of actors who are extroverts falls within this interval. We are 95% confident that the true proportion of actors who are extroverts falls outside this interval. We are 95% confident that the true proportion of actors who are extroverts 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

solution:-
given that n = 66 , x = 42
(a)point estimate p = x/n = 42/66
= 0.6364

(b)95% confidence for z is 1.96
confidence interval formula
p +/- z * sqrt(p*(1-p)/n)
=> 0.6364 +/- 1.96 * sqrt(0.6364 *(1-0.6364)/66)
=> 0.6364 +/- 0.1161
=> 0.52 , 0.75
lower limit = 0.52
upper limit = 0.75

Give a brief interpretation of the meaning of the confidence interval you have found
=> We are 95% confident that the true proportion of actors who are extroverts falls within this interval

Do you think the conditions np > 5 and nq > 5 are satisfied in this problem
Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal