This problem involves hypothesis testing to determine if the tire lifespan is lower than the manufacturer’s claim of 79,000 km. Let us solve step by step.

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1. Determine the appropriate test: z-test or t-test?

Since the population standard deviation ($\sigma = 12,800$) is known and the sample size is 32 ($n > 30$), we use a z-test.

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2. State the hypotheses

• $H_0$: The mean lifespan of the tires is greater than or equal to 79,000 km ($\mu \geq 79,000$).
• $H_1$: The mean lifespan of the tires is less than 79,000 km ($\mu < 79,000$).

Mathematically: $H_0: \mu \geq 79,000 \quad \text{and} \quad H_1: \mu < 79,000$

This is a one-tailed test (left-tailed) because we are testing if the mileage is lower.

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3. Test statistic formula

For a z-test, the test statistic is: $z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

Where:

• $\bar{x} = 76,500$ km (sample mean),
• $\mu = 79,000$ km (claimed mean),
• $\sigma = 12,800$ km (population standard deviation),
• $n = 32$ (sample size).

Substitute the values: $z = \frac{76,500 - 79,000}{12,800 / \sqrt{32}} = \frac{-2,500}{12,800 / 5.6569} = \frac{-2,500}{2,263.13} \approx -1.10$

4. Critical value

At a significance level of $\alpha = 0.02$ for a one-tailed (left-tailed) test, the critical z-value is: $z_{\text{critical}} = -2.05 \quad \text{(from z-tables or statistical software)}.$

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5. Decision rule

• If $z \leq z_{\text{critical}}$, reject $H_0$.
• If $z > z_{\text{critical}}$, fail to reject $H_0$.

Here: $z = -1.10 \quad \text{and} \quad z_{\text{critical}} = -2.05.$ Since $-1.10 > -2.05$, we fail to reject $H_0$.

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6. Conclusion

The correct decision is:

• Fail to reject the null hypothesis.

The correct conclusion is:

• There is not enough evidence to conclude that the tires last less than the kilometers claimed.

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Summary of Results

1. Test Type: z-test
2. Hypotheses:
   $H_0: \mu \geq 79,000 \quad H_1: \mu < 79,000$
3. Test Statistic: $z = -1.10$
4. Critical Value: $z_{\text{critical}} = -2.05$
5. Decision: Fail to reject $H_0$.
6. Conclusion: There is not enough evidence to conclude that the tires last less than the kilometers claimed.

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Would you like to explore related concepts or clarify any steps?
Here are 5 related questions for practice:

1. What changes in the conclusion if the sample mean was 75,000 km instead of 76,500 km?
2. How does increasing the sample size impact the test statistic?
3. What would the decision be if $\alpha = 5\%$ instead of $2\%$?
4. What is the relationship between the significance level and the probability of a Type I error?
5. How would a two-tailed test differ in hypothesis and critical values?

Tip: When conducting hypothesis tests, always sketch a bell curve to visualize the critical regions and test statistic placement.