Alright, let’s analyze the given data and solve the problem step-by-step.
### Problem Interpretation
The auto maker wants to determine statistically whether the transmissions last longer than 150,000 miles on average. To do this, a sample is tested, and results are given.
### Step-by-Step Solution
1. Formulate the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The mean mileage before failure is not greater than 150,000 miles (i.e., [tex]\(\mu \leq 150\)[/tex]).
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): The mean mileage before failure is greater than 150,000 miles (i.e., [tex]\(\mu > 150\)[/tex]).
This is a one-tailed test.
2. Sample Statistics:
- Sample mean ([tex]\(\bar{x}\)[/tex]): 150.7 thousand miles
- Sample variance ([tex]\(s^2\)[/tex]): 4.551 thousand miles
- Sample size ([tex]\(n\)[/tex]): 42
- Hypothesized population mean ([tex]\(\mu_0\)[/tex]): 150 thousand miles
3. Test Statistic:
- Given the [tex]\(t\)[/tex] statistic: 2.17
4. Degrees of Freedom (df):
- [tex]\(df = n - 1 = 42 - 1 = 41\)[/tex]
5. Critical [tex]\(t\)[/tex] Value for One-Tailed Test:
- Given the critical [tex]\(t\)[/tex] value for a one-tailed test at the chosen significance level: 1.683
6. P-Value for One-Tailed Test:
- Given the [tex]\(p\)[/tex] value for one-tailed test: 0.018
### Decision Rule
- If the [tex]\(t\)[/tex] statistic is greater than the critical [tex]\(t\)[/tex] value (1.683), we reject the null hypothesis.
- Alternatively, if the p-value is less than the significance level ([tex]\(\alpha\)[/tex]), we reject the null hypothesis. Typically, [tex]\(\alpha = 0.05\)[/tex] for a 95% confidence level.
### Analysis
- Our [tex]\(t\)[/tex] statistic (2.17) is greater than the critical [tex]\(t\)[/tex] value (1.683).
- The p-value for the one-tailed test (0.018) is less than the typical significance level of 0.05.
### Conclusion
Given that the [tex]\(t\)[/tex] statistic is greater than the critical [tex]\(t\)[/tex] value and the p-value is less than the significance level, we reject the null hypothesis. There is sufficient evidence at the 95% confidence level to support the claim that the transmissions last more than 150,000 miles on average.
### Additional Information
- Two-Tailed Test Information:
- Critical [tex]\(t\)[/tex] value for two-tailed test: 2.02
- p-value for two-tailed test: 0.036
- Confidence Interval:
- The margin for the 95% confidence interval was given as 0.665 thousand miles.
- Although our primary analysis was for a one-tailed test, provided values for a two-tailed test and confidence intervals can be useful for other related inferences or checks.
This comprehensive step-by-step solution concludes that the transmissions are statistically shown to last more than 150,000 miles before failure.
