A poll used a sample of 100 randomly selected car owners. Within the​ sample, the mean time of ownership for a single car was 6.36 years with a standard deviation of 3.61 years. Test the claim by the owner of a large dealership that the mean time of ownership for all cars is less than 6.5 years. Use a 0.05 significance level.

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.

---

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.

Learn how to perform a one-sample t-test to test the claim that the mean time of car ownership for all cars is less than 6.5 years. This problem involves calculating a test statistic using sample mean, population mean, sample standard deviation, and sample size. Suitable for advanced high school students and beyond.

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