Question

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?

Short answer

No, the data is not highly inconsistent with the claim.

Step-by-step solution

Given Information

Null hypothesis $H_0$: $\mu \ge 50000$

Alternate Hypothesis $\left(H\alpha \right):\mu <50000$,it is left-tailed test

Explanation

Since the p-value $(0.0130)<$the alpha value$(0.05)$, the $H_0$ value is rejected.

The Confidence Interval (CI) for population mean $\left(\mu \right)$is:

sample mean

$=46700\pm 1.96\times \frac{800}{\sqrt{32}}$

$=46700\pm 2772$

Conclusion

Lower boundary is $46700-2772=43928$

Upper boundary is $46700+2772=49472$

The CI is $(43928,49472)$

Since the claimed average of $50000$does not fall within boundaries of the confidence interval , the inference that the data does not support the claim of $5\%$level could be drawn.