Question

27. Call me: A sociologist wants to construct a 95% confidence interval for the proportion of children aged 8–10 living in New York who own a cell phone.

a. A survey by the National Consumers League estimated the nationwide proportion to be 0.32. Using this estimate, what sample size is needed so that the confidence interval will have a margin of error of 0.02?

b. Estimate the sample size needed if no estimate of p is available.

c. If the sociologist wanted to estimate the proportion in the entire United States rather than in New York, would the necessary sample size be larger, smaller, or about the same? Explain.

Answer 1

Solution,

Given that,

a) = 0.32

1 - = 1 - 0.32 = 0.68

margin of error = E = 0.02

At 95% confidence level

= 1 - 95%

= 1 - 0.95 =0.05

/2 = 0.025

Z/2 = Z0.025  = 1.96

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

= (1.96 /0.02 )2 * 0.32 * 0.68

= 2089.83

sample size = n = 2090

b) =  1 - =   0.5

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

= (1.96 / 0.02)2 * 0.5 * 0.5

sample size = n = 2401

c) smaller