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 5427 physicians in Colorado showed that 3251 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.1% 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.

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

Answer 1

a)

Number of Items of Interest,x =3251

Sample Size,n = 5427

point estimate , p̂ = x/n = 0.5990

b)

Level of Significance,α = 0.01

z -value =Zα/2 = 2.576 [excel formula =NORMSINV(α/2)]

  

Standard Error , SE = √[p̂(1-p̂)/n] = 0.0067

margin of error , E = Z*SE = 2.576*0.0067=0.0171

  

99%Confidence Interval is

Interval Lower Limit = p̂ - E = 0.599-0.0171=0.5819

Interval Upper Limit = p̂ + E =0.599+0.0171=0.6162

  

99% confidence interval is (0.582< p < 0.616)

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