For this problem, carry at least four digits after the decimal in your calculati

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 524 judges, it was found that 277 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 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 outside 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 n1 > 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 p is approximately binomial. Yes, the conditions are satisfied. This is important because it allows us to say that p is approximately binomial. 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 normal.

Explanation / Answer

a. Here x=277 and n=524

Now point estimate of p=phat=x/n=277/524=0.53

b. Now here np=277>5 so we will use normal distribution, so z value for 99% CI is 2.58 because P(-2.58<z<2.58)=0.99

Further Margin of Error=z*SE(p)

Where SE(p)=sqrt(p(1-p)/n)=0.022

So CI=p+/=E=0.53+/-0.022=(0.508,0.552)

We are 99% confident that the true proportion of judges who are introverts falls within (0.508,0.552) interval

c. Here np=277 and nq=246.28>5

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

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