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 68 professional actors, it was found that 37
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 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 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 n·p > 5 and n·q > 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 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. No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately binomial.
(a)
Here point estimate of p = 37/68 = 0.5441
(b)
95% confidence interval= p +- z95% sqrt [p * (1-p)/n]
= 0.544 +- 1.96 * sqrt [0.544 * (1 - 0.544)/69]
= 0.544 +- 0.118
= (0.426, 0.662)
Lower Limit = 0.426
Upper Limit = 0.662
We are 95% confident that the true proportion of actors who are extroverts falls within this interval.
(c) Here Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.
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