Question

Fewer young people are driving. In year A, 67.9% 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 44.7%. Suppose these results are based on a random sample of 1,700 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.)

to

(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.)

to

(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  ---Select--- 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  ---Select--- 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  ---Select--- smaller larger interval estimate in part (b).

Answer 1

a)
sample proportion, = 0.679
sample size, n = 1700
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.679 * (1 - 0.679)/1700) = 0.0113

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 * 0.0113
ME = 0.0221

b)

sample proportion, = 0.447
sample size, n = 1700
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.447 * (1 - 0.447)/1700) = 0.0121

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 * 0.0121
ME = 0.0237

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).