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 5832 physicians in Colorado showed that 2954 provided at least some charity care (i.e., treated poor people at no cost).

(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.

99% of the confidence intervals created using this method would include the true proportion of Colorado physicians providing at least some charity care.99% of all confidence intervals would include the true proportion of Colorado physicians providing at least some charity care.    1% of all confidence intervals would include the true proportion of Colorado physicians providing at least some charity care.1% of the confidence intervals created using this method would include the true proportion of Colorado physicians providing at least some charity care.

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

No; np > 5 and nq < 5.

Yes; np > 5 and nq > 5.   

No; np < 5 and nq > 5.

Yes; np < 5 and nq < 5.

Answer 1

Solution :

Given that,

n = 5832

x = 2954

Point estimate = sample proportion = = x / n = 2954 / 5832 = 0.5065

1 - = 1 - 0.5065 = 0.4935

At 99% confidence level

= 1 - 99%

=1 - 0.99 =0.01

/2 = 0.005

Z/2 = Z_{0.005 = 2.576}

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 2.576 (((0.5065 * 0.4935) / 5832)

= 0.0169

A 99% confidence interval for population proportion p is ,

± E  

= 0.5065 ± 0.0169

= ( 0.490, 0.524 )

lower limit = 0.490

upper limit = 0.524

99% of the confidence intervals created using this method would include the true proportion of Colorado physicians providing at least some charity care

c) Yes; np > 5 and nq > 5