A particular brand of tires claims that its deluxe tire averages at least 50,000 miles before it needs to be replaced. From past studies of this tire, the standard deviation is known to be 8,000. A survey of owners of that tire design is conducted. From the 28 tires surveyed, the mean lifespan was 46,500 miles with a standard deviation of 9,800 miles. Using alpha = 0.05, is the data highly inconsistent with the claim?

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?

---

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.

Explore the hypothesis testing process for a tire brand's claim of an average lifespan of at least 50,000 miles. A sample of 28 tires showed a mean lifespan of 46,500 miles. This page details the statistical methods, including z-tests and significance levels, relevant to this analysis.

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