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Chapter 13: Problem 61

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

Expert verified
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

01

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.)
02

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.)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis
In hypothesis testing, the null hypothesis acts like the status quo that we initially assume to be true. It represents a sort of "no difference" or "no effect" stance. In the context of our exercise about comparing scenic and nonscenic routes, the null hypothesis is that any potential time savings using the nonscenic route do not exceed 10 minutes.
To understand this better, let's break it down:
  • If \( \mu_{1} \) represents the mean travel time for the scenic route and \( \mu_{2} \) for the nonscenic route, the null hypothesis \( (H_{0}) \) is: \( \mu_{1} - \mu_{2} \leq 10 \) minutes.
  • If \( \mu_{1} \) represents the mean travel time for the nonscenic route and \( \mu_{2} \) for the scenic route, then the null hypothesis \( (H_{0}) \) becomes: \( \mu_{2} - \mu_{1} \leq 10 \) minutes.
The null hypothesis is typically what we attempt to "reject" through our testing. Yet, it's essential to remember that not rejecting \( H_{0} \) doesn't prove it true 鈥� it only indicates insufficient evidence against it.
Alternative Hypothesis
The alternative hypothesis reflects what you suspect might actually be true 鈥� it's the statement we want to find evidence for. When we're testing hypotheses about travel times, the alternative hypothesis suggests savings beyond 10 minutes.
The specifics involve:
  • When looking at \( \mu_{1} \) as the mean travel time for the scenic route and \( \mu_{2} \) for the nonscenic, the alternative hypothesis \( (H_{1}) \) claims: \( \mu_{1} - \mu_{2} > 10 \) minutes. This means the nonscenic route saves more than 10 minutes.
  • Reversing roles, if \( \mu_{1} \) is for the nonscenic and \( \mu_{2} \) for the scenic, then \( (H_{1}) \) becomes: \( \mu_{2} - \mu_{1} > 10 \). This indicates the nonscenic route still saves over 10 minutes.
In practical terms, when statistical evidence suggests we can reject the null hypothesis in favor of the alternative, it offers support for savings greater than what \( H_{0} \) assumes.
Mean Difference
The concept of mean difference is crucial in understanding hypothesis testing, especially when comparing two different sets like travel routes. The mean difference refers to the average amount by which one group's results (or means) differ from another's.
In our exercise, mean difference helps to quantify exactly how much time one route saves over the other.
  • When considering \( \mu_{1} \) as the scenic route and \( \mu_{2} \) as the nonscenic, the mean difference formula is presented as: \( \mu_{1} - \mu_{2} \).
  • Similarly, if \( \mu_{1} \) is the nonscenic and \( \mu_{2} \) the scenic, it is expressed as: \( \mu_{2} - \mu_{1} \).
This difference becomes central to statistical tests used to decide which hypothesis to support. The greater the calculated mean difference beyond a defined threshold (like 10 minutes here), the stronger the evidence against the null hypothesis.

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Most popular questions from this chapter

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