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 63 professional actors, it was found that 39 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 within this interval. We are 95% confident that the true proportion of actors who are extroverts falls outside this interval.     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 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 p is approximately normal. 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 binomial. Yes, the conditions are satisfied. This is In order to use a normal distribution to compute confidence intervals for p, what conditions on np and nq need to be satisfied?

Case studies showed that out of 10,256 convicts who escaped from certain prisons, only 7817 were recaptured.

(a) Let p represent the proportion of all escaped convicts who will eventually be recaptured. 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 three decimal places.)

Give a brief statement of the meaning of the confidence interval.

1% of the confidence intervals created using this method would include the true proportion of recaptured escaped convicts. 1% of all confidence intervals would include the true proportion of recaptured escaped convicts.     99% of all confidence intervals would include the true proportion of recaptured escaped convicts. 99% of the confidence intervals created using this method would include the true proportion of recaptured escaped convicts.

(c) Is use of the normal approximation to the binomial justified in this problem? Explain.

Yes; np > 5 and nq > 5. No; np > 5 and nq < 5.     No; np < 5 and nq > 5. Yes; np < 5 and nq < 5.

Explanation / Answer

1)

a) point estimate p=39/63 =0.6190

b)

std error =(p(1-p)/n)1/2 =0.0612

for 95% CI; z=1.96

hence lower limit =estimated proportion - z*std error =0.50

upper limit =estimated proportion + z*std error =0.74

We are 95% confident that the true proportion of actors who are extroverts falls within this interval.

c)

Yes, the conditions are satisfied. This is In order to use a normal distribution to compute confidence intervals for p, what conditions on np and nq need to be satisfied?

2)

a) point estimate for p =7817/10256=0.7622

b)

std error =(p(1-p)/n)1/2 =0.0042

for 95% CI; z=2.5758

hence lower limit =estimated proportion - z*std error =0.751

upper limit =estimated proportion + z*std error =0.773

99% of the confidence intervals created using this method would include the true proportion of recaptured escaped convicts.

c) Yes; np > 5 and nq > 5.

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