Question

Is college worth it? Among a simple random sample of 341 American adults who do not have a four-year college degree and are not currently enrolled in school, 154 said they decided not to go to college because they could not afford school.

1. Calculate a 95% 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 95% 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

Given that, sample size ( n ) = 341 and x = 154

sample proportion = x/n = 154/341 = 0.4516

1) A 95% confidence level has significance level = 0.05 and critical value is,

The 95% C.I. for population proportion ( p ) is,

Therefore, a 95% confidence interval for the proportion of Americans who decide to not go to college because they cannot afford it is ( 0.3988, 0.5044)

That means, we are 95% confident that the proportion of Americans who decide to not to go to college because they cannot afford it is between 0.3988 and 0.5044

2) margin of error ( E ) = 1.25% = 0.0125

we want to find, the sample size ( n ),

Case 1) if we use above proportion p = 0.4516 then required sample size is,

Case 2) if we don't use above proportion then we assume p = 0.5 then required sample size is,