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 530 judges, it was found that 289 were introverts.

(a) Let p represent the proportion of all judges who are introverts. Find a point estimate for p. (Round your answer to four decimal places.)

(b) Find a 99% 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 99% confident that the true proportion of judges who are introverts falls outside this interval?.

We are 99% confident that the true proportion of judges who are introverts falls within this interval?.    

We are 1% confident that the true proportion of judges who are introverts falls above this interval.?

We are 1% confident that the true proportion of judges who are introverts falls within this interval?.

(d) 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 normal?

.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 binomial.?

Answer 1

The statistical software output for this problem is:

One sample proportion summary confidence interval:

p : Proportion of successes
Method: Standard-Wald

99% confidence interval results:

Proportion	Count	Total	Sample Prop.	Std. Err.	L. Limit	U. Limit
p	289	530	0.54528302	0.021629358	0.48956948	0.60099655

Hence,

a) Point estimate = 0.5453

b) 99% confidence interval:

Lower limit = 0.49

Upper limit = 0.60

c) We are 99% confident that the true proportion of judges who are introverts falls within this interval.

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