Question

Commuting. there are two routes he could take to work. Aneighbor who has lived there a long time tells him Route Awill average 5 minutes faster than Route B. The man decides to experiment. Each day he flips a coin to determine which way to go, driving each route 20 days. He finds that Route Atakes an average of 40 minutes, with stan- dard deviation 3 minutes, and Route B takes an average of 43 minutes, with standard deviation 2 minutes. His- tograms of travel times for the routes are roughly sym- metric and show no outliers. a.Find a 95% confidence interval for the difference in average commuting time for the two routes. b.Should the man believe the old-timer s claim that he can save an average of 5 minutes a day by always driving Route A

Answer 1

a)We are 95% confident that the average commuting time for route A is between 1.3577 and 4.6423 minutes shorter than the average committing time for rout B.

(b) No, because the confidence internal does not contain —5, which corresponds with an average of 5 minutes shorter for route A.

Explanation:

Given:

n_1 = 20

x_1= 40

s_1 = 3

n_2 = 20

x_2= 43

s_2 = 2

d_f = 33.1

c = 95%. 0.95

(a) Determine the t-value by looking in the row starting with degrees of freedom df = 33.1 > 32 and in the column with c = 95% in the Student's t distribution table in the appendix:

t
`\alpha`/2 = 2.037

The margin of error is then:

E = t
`\alpha`/2 *√s_1^2/n_1+s_2^2/n_2

E = 2.037 *√3^2/20+s_2^2/20

= 1.64

The endpoints of the confidence interval for u_1 — u_2 are:

(x_1 — x_2) — E = (40 — 43) — 1.6423 = —3 — 1.6423= —4.6423

(x_1 - x_2) + E = (40 — 43) + 1.6423 = —3 + 1.6423= —1.3577

a)We are 95% confident that the average commuting time for route A is between 1.3577 and 4.6423 minutes shorter than the average committing time for rout B.

(b) No, because the confidence internal does not contain —5, which corresponds with an average of 5 minutes shorter for route A.