To test the claim that the mean time of car ownership for all cars is less than 6.5 years, we can perform a one-sample t-test. Here's the step-by-step process:

Step 1: State the Hypotheses

• Null Hypothesis (H₀): $\mu \geq 6.5$ years (The mean time of ownership is 6.5 years or more).
• Alternative Hypothesis (H₁): $\mu < 6.5$ years (The mean time of ownership is less than 6.5 years).

Step 2: Significance Level

• The significance level (α) is 0.05.

Step 3: Test Statistic

The test statistic for a one-sample t-test is calculated using the formula:

$t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$

Where:

• $\bar{x} = 6.36$ (sample mean)
• $\mu_0 = 6.5$ (population mean under the null hypothesis)
• $s = 3.61$ (sample standard deviation)
• $n = 100$ (sample size)

Substituting the given values:

$t = \frac{6.36 - 6.5}{\frac{3.61}{\sqrt{100}}} = \frac{-0.14}{\frac{3.61}{10}} = \frac{-0.14}{0.361} \approx -0.387$

Step 4: Determine the Critical Value and Decision Rule

For a one-tailed t-test with $n - 1 = 99$ degrees of freedom at a significance level of 0.05, we can find the critical value from a t-distribution table or using statistical software.

The critical t-value for $df = 99$ at $α = 0.05$ is approximately $-1.660$ (since it's a left-tailed test).

Step 5: Compare the Test Statistic with the Critical Value

• The calculated t-value is $-0.387$.
• The critical t-value is $-1.660$.

Step 6: Conclusion

Since $-0.387 > -1.660$, we fail to reject the null hypothesis.

Conclusion: There is not enough evidence to support the claim that the mean time of car ownership for all cars is less than 6.5 years at the 0.05 significance level.

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Would you like more details or have any questions? Here are some related questions that might interest you:

1. How is the t-distribution different from the normal distribution?
2. What are the assumptions required for conducting a t-test?
3. How do degrees of freedom affect the critical value in a t-test?
4. What is the effect of sample size on the power of a hypothesis test?
5. How can you perform this t-test using statistical software like Excel or Python?

Tip: When conducting hypothesis tests, always ensure that the assumptions of the test (like normality and independence) are reasonably met.