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 70 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.)
1

(b)

Find a 95% confidence interval for p. (Round your answers to two decimal places.)
lower limit         2
upper limit         3

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 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 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.

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.     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.

Answer 1

Solution :

Given that,

n = 70

x = 42

a) Point estimate = sample proportion = = x / n = 42 / 70 = 0.6000

1 - = 1 - 0.6000 = 0.4000

At 95% confidence level

= 1 - 95%

=1 - 0.95 =0.05

/2 = 0.025

Z/2 = Z_{0.025 = 1.960}

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.96 (((0.6000 * 0.4000) / 70)

= 0.11

A 95% confidence interval for population proportion p is ,

± E

= 0.6000 ± 0.11

= ( 0.49, 0.71 )

lower limit = 0.49

upper limit = 0.71

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

c) Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.