To assess whether the data is inconsistent with the tire manufacturer’s claim (that the average tire lifespan is at least 50,000 miles), we can perform a hypothesis test.

Step 1: Set up hypotheses

The hypotheses are:

• Null Hypothesis $(H_0)$: $\mu \geq 50,000$
• Alternative Hypothesis $(H_a)$: $\mu < 50,000$

This is a left-tailed test, as we are testing if the mean lifespan is significantly less than 50,000 miles.

Step 2: Identify known values

• Population standard deviation $\sigma = 8,000$
• Sample size $n = 28$
• Sample mean $\bar{x} = 46,500$
• Population mean claimed by the manufacturer $\mu_0 = 50,000$
• Significance level $\alpha = 0.05$

Since the population standard deviation is known, we can use a z-test.

Step 3: Calculate the test statistic

The formula for the z-test statistic is: $z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$

Substitute the values: $z = \frac{46,500 - 50,000}{\frac{8,000}{\sqrt{28}}}$

1. Calculate the standard error (SE): $SE = \frac{8,000}{\sqrt{28}} \approx 1,511.86$

2. Calculate the z-value: $z = \frac{46,500 - 50,000}{1,511.86} \approx \frac{-3,500}{1,511.86} \approx -2.31$

Step 4: Determine the critical value

For a significance level of $\alpha = 0.05$ in a one-tailed test, the critical value for $z$ is approximately $-1.645$.

Step 5: Make a decision

• If $z < -1.645$, we reject the null hypothesis.
• Calculated z-value = $-2.31$, which is less than $-1.645$.

Conclusion

Since $z = -2.31$ is less than $-1.645$, we reject the null hypothesis. This indicates that the data is highly inconsistent with the manufacturer's claim. There is sufficient evidence at the 0.05 significance level to conclude that the average lifespan of the tires is likely less than 50,000 miles.

Would you like further details on any part of this solution?

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1. What is a z-test and when is it used in hypothesis testing?
2. Why do we use the population standard deviation in this test instead of the sample standard deviation?
3. What does it mean if the test statistic is less than the critical value?
4. How does the sample size affect the outcome of the hypothesis test?
5. What would change in the solution if the significance level were 0.01 instead of 0.05?

Tip: In hypothesis testing, carefully distinguish between population and sample standard deviations—using the wrong one can lead to incorrect conclusions.