Question

HW #44. 5

The manufacturer of a particular brand of tires claims they average at least 50,000 miles before needing to be replaced. From past studies of this tire, it is known that the population standard deviation is 8,000 miles.

A survey of tire owners was conducted. From the 23 tires surveyed, the mean lifespan was 45500 miles. Using alpha = 0.05, can we prove that the data in inconsistent with the manufacturers claim?

We should use a ? t z  test.

What are the correct hypotheses?

H₀: Select an answer s μ s² x̄ p̂ σ p σ²  ? = < ≠ ≥ ≤ >  

H_A: Select an answer σ p μ p̂ x̂ σ² s s²  ? ≤ = < ≠ > ≥  

Based on the hypotheses, find the following:

Test Statistic=

p-value=

The correct decision is to Select an answer Fail to reject the null hypothesis Reject the null hypothesis Accept the alternative hypotheis Accept the null hypothesis  .

The correct conclusion would be: Select an answer There is not enough evidence to conclude that the tires last fewer miles than claimed There is enough evidence to conclude that the tires last fewer miles than claimed There is enough evidence to conclude that the tires do not last fewer miles than claimed There is not enough evidence to conclude that the tires do not last fewer miles than claimed  .

Answer 1

sample mean is Xˉ=45500 and the known population standard deviation is σ=8000, and the sample size is n=23.

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ≥50000

Ha: μ<50000

This corresponds to a left-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a left-tailed test is zc=−1.64.

The rejection region for this left-tailed test is R={z:z<−1.64}

Decision about the null hypothesis

Since it is observed that z=−2.698<zc​=−1.64, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0035, and since p=0.0035<0.05, it is concluded that the null hypothesis is rejected.

Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is less than 50000, at the 0.05 significance level.