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 63 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 5% 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 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 normal.    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 binomial.

Answer 1

(a).

Number of professional actors in the random sample : n=63

Number of extroverts  professional actors in the  random sample : x = 39

Proportion of all actors who are extroverts : p = Number of extroverts  professional actors in the  random sample / Number of professional actors in the  random sample = x/n = 0.6190

Sample Proportion of all actors who are extroverts : = 0.6190

(b). Formula for confidence interval for single population proportion : p

Given
Number of professional actors in the  random sample : n	63
Number of extroverts professional actors in the random sample: x	39
Sample Propotion : x/n	0.619
Confidence Level	95%
(= 100-95/100=5/100 ) = 0.05	0.05
/2 (=0.05/2=0.025)	0.025
	1.96

95% confidence interval for p

lower limit	0.50
upper limit	0.74

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.

(c)

n = 63; p =0.619;

np = 63 x 0.619 = 39 > 5 ( as p = x/n ; np =x)

q = 1-p = 1-0.619 = 0.381

n=63 , q= 0.381

nq = 63 x 0.381 = 24 > 5

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