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 the mean travel time by more than 10 minutes. a. If \(\mu_{1}\) refers to the mean travel time for scenic route and \(\mu_{2}\) to the mean travel time for nonscenic route, what hypotheses should be tested? b. If \(\mu_{1}\) refers to the mean travel time for nonscenic route and \(\mu_{2}\) to the mean travel time for scenic route, what hypotheses should be tested?

Short answer

For \(\mu_{1}\) scenic and \(\mu_{2}\) nonscenic, the hypotheses are: \(H_{0}: \mu_{1} - \mu_{2} \leq 10\) and \(H_{1}: \mu_{1} - \mu_{2} > 10\). For \(\mu_{1}\) nonscenic and \(\mu_{2}\) scenic, the hypotheses are: \(H_{0}: \mu_{2} - \mu_{1} \leq 10\) and \(H_{1}: \mu_{2} - \mu_{1} > 10\).

Step-by-step solution

Hypotheses when \(\mu_{1}\) is for scenic and \(\mu_{2}\) for nonscenic

We want to know if the nonscenic route (\(\mu_{2}\)) saves more than 10 minutes compared to the scenic route (\(\mu_{1}\)). The null hypothesis (\(H_{0}\)) should thus represent that the time saved is less than or equal to 10 minutes, and the alternative hypothesis (\(H_{1}\)) should correspond to the time saved being more than 10 minutes. Therefore, we get: \(H_{0}: \mu_{1} - \mu_{2} \leq 10\) (Does not save time or saves up to 10 mins.) \(H_{1}: \mu_{1} - \mu_{2} > 10\) (Saves more than 10 mins.)

Hypotheses when \(\mu_{1}\) is for nonscenic and \(\mu_{2}\) for scenic

This time we want to know if the nonscenic route (\(\mu_{1}\)) saves more than 10 minutes compared to the scenic route (\(\mu_{2}\)). The null hypothesis (\(H_{0}\)) should thus represent that the time saved is less than or equal to 10 minutes, and the alternative hypothesis (\(H_{1}\)) should correspond to the time saved being more than 10 minutes. Therefore, we get: \(H_{0}: \mu_{2} - \mu_{1} \leq 10\) (Does not save time or saves up to 10 mins.) \(H_{1}: \mu_{2} - \mu_{1} > 10\) (Saves more than 10 mins.)