Question

In a random sample of 61 professional actors, it was found that 39 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 95% 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 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. (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 binomial. Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.

Answer 1

(a) Point estimate of p = = 39/61 = 0.6393

(b) the Z critical values(2 tail) at = 0.05 is 1.96

The Confidence Interval is given by:

Where the ME =

The Lower Limit = 0.6393 - 0.1205 = 0.5188       0.52 (Rounding to 2 decimal places)

The Upper Limit = 0.6393 + 0.1205 = 0.7598     0.76 (Rounding to 2 decimal places)

Interpretation: Option 1: We are 95% confident that the true proportion of actors who are extroverts are within this interval.

(c) Conditions: Option 4: Yes, the conditions are satisfied. This is important because allows us to say that p is approximately normal.

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