Let's analyze the given problem step by step to determine whether we should accept or reject the null hypothesis using the provided sample data.
### 1. Setting up the Hypotheses
The auto maker claims that the transmissions can last over 150,000 miles before failing.
- Null Hypothesis (\( H_0 \)): The mean number of miles before failure is 150,000 miles or less.
[tex]\[ H_0: \mu \leq 150,000 \][/tex]
- Alternative Hypothesis (\( H_1 \)): The mean number of miles before failure is greater than 150,000 miles.
[tex]\[ H_1: \mu > 150,000 \][/tex]
### 2. Test Statistics and Critical Values
From the given data:
- Sample Mean (\( \bar{x} \)) = 150.7
- Hypothesized Mean (\( \mu_0 \)) = 150
- Sample Size (n) = 42
- Sample Variance = 4.551
- Degrees of Freedom (df) = 41
- t Stat = 2.17
- \( P(T \leq t) \) one-tail = 0.018
- t Critical one-tail = 1.683
- \( P(T \leq t) \) two-tail = 0.036
- t Critical two-tail = 2.02
### 3. One-Tailed Testing Decision
For a one-tailed test, we're interested in determining if the mean number of miles is greater than 150,000. Here are the rules for our decision-making process:
- If \( t \) Stat > t Critical one-tail
- If \( P \) value one-tail < 0.05
Given our data:
- t Stat = 2.17
- t Critical one-tail = 1.683
- \( P(T \leq t) \) one-tail = 0.018
#### Comparison:
- Since 2.17 > 1.683, we can reject the null hypothesis.
- Since 0.018 < 0.05, we can also reject the null hypothesis.
### 4. Two-Tailed Testing Decision
For completeness, let's also consider the two-tailed test, even though the claim was specifically for over 150,000 miles.
For a two-tailed test:
- If \( t \) Stat > t Critical two-tail
- If \( P \) value two-tail < 0.05
Given our data:
- t Stat = 2.17
- t Critical two-tail = 2.02
- \( P(T \leq t) \) two-tail = 0.036
#### Comparison:
- Since 2.17 > 2.02, we can reject the null hypothesis.
- Since 0.036 < 0.05, we can also reject the null hypothesis.
### 5. Conclusion
Based on the above analysis for both one-tailed and two-tailed testing, we have:
#### One-Tailed:
- Decision: Reject the null hypothesis.
#### Two-Tailed:
- Decision: Reject the null hypothesis.
### Final Decision:
The evidence from the sample data supports the claim that the transmissions can indeed last over 150,000 miles before failing. Therefore, based on both one-tailed and two-tailed tests, we reject the null hypothesis.
