Question

or 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 63 professional actors, it was found that 40 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.)


(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 within this interval.We are 5% confident that the true proportion of actors who are extroverts falls above 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.

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

Answer 1

a) The sample proportion is the best estimate for the point estimate which is computed here as:

p = 40/63 = 0.6349

Therefore 0.6349 is the required point estimate here.

b) From standard normal tables, we have:
P(-1.96 < Z < 1.96) = 0.95

Therefore the confidence interval here is obtained as:

This is the required confidence interval here.

The interpretation of this is that We are 95% confident that the true proportion of actors who are extroverts falls within this interval.

c) np = 40 and n(1-p) = 63 - 40 = 23 both are greater than 5 and so yes these conditions are satisfied here. This helps us to approximate it as a normal distribution using which we have obtained the confidence interval above.