Question

Among a simple random sample of 326 American adults who do not have a four-year college degree and are not currently enrolled in school, 48% said they decided not to go to college because they could not afford school. Suppose an earlier hypothesis test determined that the data do not provide strong evidence that less than half of American adults who decide not to go to college make this decision because they cannot afford college.

(a)

Calculate a 90% 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 your answers to one decimal place.)

We are 90% confident that (?) % to (?) % of all Americans who decide not to go to college do so because they cannot afford it.

(b)

Suppose we wanted the margin of error for the 90% confidence level to be about 1.5%. How large of a survey would you recommend? (Round your answer up to the nearest whole number.)

(?) people

Answer 1

a)
sample proportion, = 0.48
sample size, n = 326
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.48 * (1 - 0.48)/326) = 0.0277

Given CI level is 90%, hence α = 1 - 0.9 = 0.1
α/2 = 0.1/2 = 0.05, Zc = Z(α/2) = 1.64

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.48 - 1.64 * 0.0277 , 0.48 + 1.64 * 0.0277)
CI = (0.435 , 0.525)

We are 90% confident that 43.5% to 52.5% of all Americans who decide not to go to college do so because they cannot afford it.

b)
The following information is provided,
Significance Level, α = 0.1, Margin of Error, E = 0.015

The provided estimate of proportion p is, p = 0.48
The critical value for significance level, α = 0.1 is 1.64.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.48*(1 - 0.48)*(1.64/0.015)^2
n = 2983.66

Therefore, the sample size needed to satisfy the condition n >= 2983.66 and it must be an integer number, we conclude that the minimum required sample size is n = 2984
Ans : Sample size, n = 2984