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 61 professional actors, it was found that 45 were extroverts. (a) Let p represent the proportion of all actors who are extroverts. Find point estimates for p and q. (Round your answer to four decimal places.) p̂ = q̂ = (b) Find a 95% confidence interval for p. (Round your answers to two decimal places.) Find the maximal margin of error. (Round your answer to two decimal places.) E = Report the bounds from the 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 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. We are 5% 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 normal. No, the conditions are not 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 binomial. Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately binomial.

Answer 1

a) = 45/61 = 0.7377

   = 1 - 0.7377 = 0.2623

b) At 95% confidence level, the critical value is z_0.025 = 1.96

The 95% confidence interval is

The margin of error is

  

Lower bound = 0.63

Upper bound = 0.85

We are 95% confident that the true population of actors who are extroverts falls within this interval.

c) n= 61 * 0.7377 = 44.9997 > 5

n = 61 * 0.2623 = 16.0003 > 5

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