Question

1. 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 64 professional actors, it was found that 45 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        

(c) 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 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.

(d) 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.

What is the minimal sample size needed for a 95% confidence interval to have a maximal margin of error of 0.1 in the following scenarios? (For each answer, enter a number. Round your answers up the nearest whole number.)

(a)a preliminary estimate for p is 0.16

(b)there is no preliminary estimate for p

Answer 1

Solution:- Given that n = 64, x = 45
p = x/n = 45/64 = 0.7031
q = 1-p = 1-0.7031 = 0.2969
95% confidence interval
The critical value corresponding to the given situation is obtained as Z(α/2) = 1.96
1 - α = 1 - 0.95
α = 0.05
α/2 = 0.05/2 = 0.025
Z0.05 = 1.96

(a) point estimat for p = 0.7031

(b) 95% Confidence interval for the p = (0.59 , 0.82)
=> p +/- Z*sqrt(pq/n)
= 0.7031 +/- 1.96*sqrt(0.7031*0.2969/64)
= (0.5912 , 0.8150)
= (0.59 , 0.82)

(c) option B.

(d) option B.
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Solution:-
Given that E = 0.1, 95% Confidence interval for Z = 1.96

(a) p = 0.16

n = (Z/E)^2*p*q = (1.96/0.1)^2*0.16*0.84 = 51.63

n = 52

(b) no prior assume p = 0.5

n = (Z/E)^2*p*q = (1.96/0.1)^2*0.5*0.5 = 96.04

n = 96