Question

An individual can take either a scenic route to work or a nonscenic route. She decides that use of the nonscenic route can be justified only if it reduces true average travel time by more than \(10 \mathrm{~min}\). a. If \(\mu_{1}\) refers to the scenic route and \(\mu_{2}\) to the nonscenic route, what hypotheses should be tested? b. If \(\mu_{1}\) refers to the nonscenic route and \(\mu_{2}\) to the scenic route, what hypotheses should be tested?

Short answer

The hypotheses to be tested are: a. \(H_0 : \mu_{1} - \mu_{2} <= 10\) (nonscenic route doesn't reduce travel time or reduces it by up to 10 minutes) and \(H_1: \mu_{1} - \mu_{2} > 10\) (nonscenic route reduces travel time by more than 10 minutes). b. \(H_0 : \mu_{2} - \mu_{1} <= 10\) (nonscenic route doesn't reduce travel time or reduces it by up to 10 minutes) and \(H_1: \mu_{2} - \mu_{1} > 10\) (nonscenic route reduces travel time by more than 10 minutes).

Step-by-step solution

Hypothesis for the nonscenic route

Given that the use of nonscenic route (\(\mu_{2}\)) can only be justified if it can reduce the true average travel time by more than 10 minutes compared to the scenic route (\(\mu_{1}\)): The Null Hypothesis (\(H_0\)) assumes that the difference between \(\mu_{1}\) and \(\mu_{2}\) is less than or equal to 10 minutes while The Alternate Hypothesis (\(H_1\)) assumes the opposite. This can be represented as: \(H_0 : \mu_{1} - \mu_{2} <= 10\) and \(H_1: \mu_{1} - \mu_{2} > 10\)

Hypothesis for the scenic route

If \(\mu_{1}\) refers to the nonscenic route and \(\mu_{2}\) to the scenic route: To justify the use of nonscenic route (\(\mu_{1}\)), it should reduce the average travel time by more than 10 minutes when compared to the scenic route (\(\mu_{2}\)). This can be represented as: Null Hypothesis (\(H_0\)): \(\mu_{2} - \mu_{1} <= 10\) and Alternate Hypothesis (\(H_1\)): \(\mu_{2} - \mu_{1} > 10\)