To determine the appropriate alternative hypothesis for the student's question, we need to understand what she is trying to test. Here is a step-by-step breakdown:
1. Understanding the Problem:
- The student is trying to compare her running times before and after starting a new training regimen.
- Initially, her average time to run a mile was 8 minutes.
2. Formulating Hypotheses:
- Null Hypothesis (H₀): This is the hypothesis that there is no effect or no difference. In this case, the null hypothesis would state that the average time to run a mile is still 8 minutes.
[tex]\[
H_0: \mu = 8
\][/tex]
- Alternative Hypothesis (H₁ or Ha): This is what the student seeks to test. The student wants to know if the new training regimen has changed her running times, either making them faster or slower. Hence, she is looking to see if there is any difference (i.e., the mean time is no longer 8 minutes).
3. Choosing the Alternative Hypothesis:
- If the student's goal was to test specifically if her times have improved (i.e., the mean time is less than 8 minutes), the alternative hypothesis would be:
[tex]\[
\mu < 8
\][/tex]
This corresponds to option B.
- If the student's goal was to test specifically if her times have worsened (i.e., the mean time is greater than 8 minutes), the alternative hypothesis would be:
[tex]\[
\mu > 8
\][/tex]
This corresponds to option A.
- However, the student's goal here is to determine if there is any difference in her running times, which could mean either an increase or a decrease. Therefore, the appropriate alternative hypothesis should reflect that the mean time is not equal to 8 minutes:
[tex]\[
\mu \neq 8
\][/tex]
This corresponds to option C.
4. Verifying the Answer:
- The alternative hypothesis should match the purpose of the test, which in this case is to check for any difference, not specifically an increase or decrease. Hence, the correct choice is indeed:
[tex]\[
\mu \neq 8
\][/tex]
Therefore, the appropriate alternative hypothesis is:
C. [tex]\(\mu \neq 8\)[/tex]
