Question

Is college worth it? Among a simple random sample of 305 American adults who do not have a four-year college degree and are not currently enrolled in school, 144 said they decided not to go to college because they could not afford school.

1. Calculate a 99% confidence interval for the proportion of Americans who decide to not go to college because they cannot afford it, and interpret the interval in context. Round to 4 decimal places.

( ,  )

2. Suppose we wanted the margin of error for the 99% confidence level to be about 1.25%. What is the smallest sample size we could take to achieve this? Note: For consistency's sake, round your z* value to 3 decimal places before calculating the necessary sample size.

Choose n =

Answer 1

a)

Sample proportion = 144 / 305 = 0.472

99% confidence interval for p is

- Z/2 * sqrt [ (1 - ) / n ] < p < + Z/2 * sqrt [ (1 - ) / n ]

0.472 - 2.5758 * Sqrt [ 0.472 ( 1 - 0.472) / 305] < p < 0.472 + 2.5758 * Sqrt [ 0.472 ( 1 - 0.472) / 305]

0.398 < p < 0.546

99% CI is ( 0.398 , 0.546 )

b)

Sample size n = Z²/2 * P( 1 - p) / E²

= 2.576² * 0.472 ( 1 - 0.472) / 0.0125²

= 10584​​​​​​​ (Rounded up to nearest integer)