Question

Fewer young people are driving. In year A, 62.9% of people under 20 years old who were eligible had a driver's license. Twenty years later in year B that percentage had dropped to 46.7%. Suppose these results are based on a random sample of 1,300 people under 20 years old who were eligible to have a driver's license in year A and again in year B.

(a) At 95% confidence, what is the margin of error of the number of eligible people under 20 years old who had a driver's license in year A? (Round your answer to four decimal places.)

At 95% confidence, what is the interval estimate of the number of eligible people under 20 years old who had a driver's license in year A? (Round your answers to four decimal places.)

(b) At 95% confidence, what is the margin of error of the number of eligible people under 20 years old who had a driver's license in year B? (Round your answer to four decimal places.)

At 95% confidence, what is the interval estimate of the number of eligible people under 20 years old who had a driver's license in year B? (Round your answers to four decimal places.)

(c) Is the margin of error the same in parts (a) and (b)? Why or why not?

The margin of error in part (a) is (smaller/larger) than the margin of error in part (b). This is because the sample proportion of eligible people under 20 years old who had a driver's license in year B is (closer to 0/closer to 0.5/ closer to 1) than the sample proportion of eligible people under 20 years old who had a driver's license in year A. This leads to a (smaller/larger) interval estimate in part (b).

Answer 1

a)
sample proportion, = 0.629
sample size, n = 1300
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.629 * (1 - 0.629)/1300) = 0.0134

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

Margin of Error, ME = zc * SE
ME = 1.96 * 0.0134
ME = 0.0263

b)
sample proportion, = 0.467
sample size, n = 1300
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.467 * (1 - 0.467)/1300) = 0.0138

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

Margin of Error, ME = zc * SE
ME = 1.96 * 0.0138
ME = 0.0270

c)

The margin of error in part (a) is (smaller) than the margin of error in part (b). This is because the sample proportion of eligible people under 20 years old who had a driver's license in year B is (closer to 0.5/) than the sample proportion of eligible people under 20 years old who had a driver's license in year A. This leads to a (larger) interval estimate in part (b).