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 =
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)
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