Question

Out of 480 respondents in a recent health survey 42 reported a history of diabetes.

a. Estimate the true proportion of people with a history of diabetes with 95% confidence.

b. Using the survey data above, what should the sample size be if the researchers wanted to be accurate to within 2% of the true proportion?

c. What should the sample size be if the researchers wanted to be accurate to within 2% of the true proportion, assuming they had no previous data?

Answer 1

Solution :

Given that,

n = 480

x = 42

a)

Point estimate = sample proportion = = x / n = 0.088

1 - = 0.912

b)

margin of error = E = 0.02

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.96

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

= (1.96 / 0.02)2 * 0.088 * 0.912

= 770.78

sample size = 771

c)

Given that,

= 0.5

1 - = 0.5

margin of error = E =

At 99% confidence level the z is ,

= 1 - 99% = 1 - 0.99 = 0.01

/ 2 = 0.01 / 2 = 0.005

Z/2 = Z0.005 = 2.576

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

= (2.576 / 0.02)2 * 0.5 * 0.5

= 2401

sample size = 2401