Question

3.In the study of 315 patients, 241 have health insurance. Assume the patients are unrelated, so that health insurance coverage of any two individuals is independent of each other.

a.What proportion of the patients have health insurance?

b.Report a 95% confidence interval for the proportion of patients like those in this sample who have health insurance.

c.Briefly interpret the interval in (b).

d.How many subjects should be included in a study to produce a margin of error of 4% for a 95% confidence interval?

Answer 1

Solution :

a ) Given that

n =315

x = 241

= x / n =241 / 315 = 0.765

1 - = 1 - 0.765 = 0.235

b ) At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.960

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

= 1.960 * (((0.765* 0.235) / 315 )

= 0.047

A 95 % confidence interval for population proportion p is ,

- E < P < + E

0.765 - 0.047 < p < 0.765 + 0.047

0.718 < p < 0.812

c ) Given that,

= 0.765

1 - = 1 - 0.765 = 0.235

margin of error = E = 4% = 0.04

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.960

Sample size = n = ((Z / 2) / E)2 * * (1 - )

= (1.960 / 0.04)2 * 0.765 * 0.235

= 431.62

= 432

n = sample size = 432