In a random sample of 65 professional actors it was found th

In a random sample of 65 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.)

(b) Find a 95% confidence interval for p. (Round your answers to two decimal places.)

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.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 p? is approximately binomial.

No, the conditions are not satisfied. This is important because it allows us to say that p? is approximately normal.    

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

No, the conditions are not satisfied. This is important because it allows us to say that p? is approximately binomial.

Solution

a)

point estimate for p = 42/65 = 0.6462

margin of error = z * sqrt(p(1-p)/n)

= 1.96 * sqrt(42/65 * 13/65 * 1/120)

= 0.0643

confidence interval = mean +/- margin of error

so confidence interval is

0.58 < u < 0.71

c)

we are 95% confident that the true proportion of actors who are extroverts falls within the interval

d)

np = 42 , nq = 13

yes, the conditions are satisfied . this is important because it allows us to say that p is approximate normal