In a random sample of 530 judges, it was found that 287 were introverts.

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 287 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:

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 1% 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 99% confident that the true proportion of judges who are introverts 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.

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. 

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

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

Answer 1

(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.)

This question is an inference for a binomial proportions question.

ρhat = x/n

ρhat = 287/530 = 0.5415

(b)

Find a 99% confidence interval for p. (Round your answers to two decimal places.)

Step 1. The number of successes and the number of failures must be greater than 5

X>= 5 and (n - x) >=5

x= 287 >5 = true

n=530

530 - 287 = 300 > 5 = true

Step 2. Determine the Confidence interval:

99% Confidence interval = a=.01, or .005 on both sides of the binomial curve. 

Using the statistical appendix table. (B,2 A)

99% Confidence interval = Z(1-(a/2)) = 2.576

Confidence interval Estimate formula

 ρhat ± Z(1-(a/2)) √Sqrt(ρhat (1-ρhat )/n)

 0.5415 ± 2.576 * √(0.5415(1-0.5415)/530)

 0.5415 ± 2.576 * √(0.5415(0.4585)/530)

 0.5415 ± 2.576 * √(0.000468)

 0.5415 ± 2.576 * 0.02163

Addition and subtraction

 0.5415 + 0.05571 = 0.59721  

0.5415 - 0.05571 = 0.48579

lower limit: 0.48579

upper limit: 0.59721  

Phat Estimate = 0.5415

CI interval (0.48579, 0.59721) 

Give a brief interpretation of the meaning of the confidence interval you have found.

Correct Answer: We are 99% confident that the true proportion of judges who are introverts 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!,np > 5 and nq > 5 are satisfied in this problem (alternatively, can also use X>= 5 and (n - x) >=5 to satisfy).

To derive a confidence interval, we must make an assumption of a distribution (Binomial distribution, Normal distribution, etc). X>= 5 and (n - x) >=5 is the criterion that provides evidence of a valid assumption. That is, that Phat is approximately binomial. If this criterion is not fulfilled, then this causes skewness, or deviation away from the assumed distribution, which biases your confidence interval estimates and Phat estimates.

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