This problem involves testing the standard deviation of a population based on a sample, which can be done using a chi-squared test for the population variance.

Step-by-Step Solution:

1. Formulate the Hypotheses:

   • Null Hypothesis ($H_0$): $\sigma^2 \geq 4000^2 = 16,000,000$ (the standard deviation is at least 4000 km).
   • Alternative Hypothesis ($H_a$): $\sigma^2 < 4000^2 = 16,000,000$ (the standard deviation is less than 4000 km).

2. Test Statistic: The test statistic for a population variance is: $\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}$ where:

   • $n = 16$ (sample size),
   • $s = 3645.94$ (sample standard deviation),
   • $\sigma_0^2 = 4000^2 = 16,000,000$ (hypothesized population variance).

   Compute $s^2$: $s^2 = 3645.94^2 = 13,299,854.44$

   Substituting values: $\chi^2 = \frac{(16-1)(13,299,854.44)}{16,000,000} = \frac{15 \times 13,299,854.44}{16,000,000} \approx 12.47$

3. Degrees of Freedom: The degrees of freedom for this test are $df = n - 1 = 16 - 1 = 15$.

4. Critical Value: This is a one-tailed test with $\alpha = 0.05$. From a chi-squared table or calculator, the critical value for $\chi^2_{0.05, 15}$ is: $\chi^2_{\text{critical}} = 24.996$

5. Decision Rule:

   • If $\chi^2 \leq \chi^2_{\text{critical}}$, reject $H_0$.
   • If $\chi^2 > \chi^2_{\text{critical}}$, fail to reject $H_0$.

6. Compare the Test Statistic to the Critical Value: $\chi^2 = 12.47 < 24.996$

7. Conclusion: Since the test statistic is less than the critical value, we reject the null hypothesis. At $\alpha = 0.05$, there is sufficient evidence to conclude that the standard deviation of tire life is less than 4000 kilometers.

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Do you have any questions or would you like me to explain any part in more detail?

Related Questions:

1. What assumptions must be satisfied for the chi-squared test to be valid?
2. How does increasing the sample size affect the power of this test?
3. What would happen if the significance level $\alpha$ was changed to 0.01?
4. Can you explain how to use a chi-squared distribution table?
5. What are other applications of the chi-squared test for variance in engineering?

Tip:

For chi-squared tests, always ensure that the sample data comes from a normally distributed population, as the test is sensitive to deviations from normality.