Statistics Question

Question

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

A random sample of 5508 physicians in Colorado showed that 3224 provided at least some charity care (i.e., treated poor people at no cost)

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(a) Let p represent the proportion of all Colorado physicians who provide some charity care. 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.)

lower limit
upper limit

Give a brief explanation of the meaning of your answer in the context of this problem.

We are 1% confident that the true proportion of Colorado physicians providing at least some charity care falls above this interval.We are 99% confident that the true proportion of Colorado physicians providing at least some charity care falls outside this interval.    We are 1% confident that the true proportion of Colorado physicians providing at least some charity care falls within this interval.We are 99% confident that the true proportion of Colorado physicians providing at least some charity care falls within this interval.

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

Yes; np > 5 and nq > 5.

No; np > 5 and nq < 5.    

Yes; np < 5 and nq < 5.

No; np < 5 and nq > 5.

Answer 1

point estimate for population proportion p is p^ = 3224/5508 = 0.585

z-critical for 99% confidence interval is 2.576

standard error = sqrt(p^* (1-p^)/n) = sqrt(0.585 * 0.415/5508) = 0.0066

99% Confidence interval = p^ +/- z*standard error

0.585 +/- 2.576 * 0.0066 = 0.585 +/- .017

= 0.568, 0.602

We are 99% confident that the true proportion of Colorado physicians providing at least some charity care falls between 0.568 and 0.602.

If np and n(1-p) > 5, then the normal approximation is appropriate and that is the case here

np = 3224, n(1-p) = 2284

Answer 2

point estimate phat = 3224/5508 = 0.5853

z-critical for 99% CI is 2.58

standard error = sqrt(phat* (1-phat)/5508) = sqrt(0.5853 * 0.4147/5508) = 0.0066

99% CI = 0.5853 +/- 2.58 * 0.0066 = (0.568,0.602)

We are 99% confident that the true proportion of Colorado physicians providing at least some charity care falls within this interval.

Normal approximation is appropriate here because np (3224) and nq (2284) are > 5.