A tire company claims that the lifetimes of its tires average 50,000 miles. The standard deviation of tire lifetimes is known to be $\sigma = 5,000$ miles. You sample 100 tires and will test the hypothesis $H _ { 0 } : \mu = 50,000$ versus $H _ { 1 } : \mu < 50,000$ at the $\alpha = 0.01$ level of significance.
Find the power of the test against the alternative $\mu _ { 1 } = 49,000$

Question

A tire company claims that the lifetimes of its tires average 50,000 miles. The standard deviation of tire lifetimes is known to be $\sigma = 5,000$ miles. You sample 100 tires and will test the hypothesis $H _ { 0 } : \mu = 50,000$ versus $H _ { 1 } : \mu < 50,000$ at the $\alpha = 0.01$ level of significance.
Find the power of the test against the alternative $\mu _ { 1 } = 49,000$
A) 2.326
B) 0.8531
C) 0.5205
D) 0.3707

Quizplus · Accepted Answer

The power of the test is the probability of correctly rejecting the null hypothesis when the true mean is 49,000 miles. To find this, we calculate the z-score for the alternative mean under the distribution assuming the null hypothesis is true, and then find the probability of observing a value less than this z-score. The calculation involves finding the z-score for the difference between the hypothesized mean (50,000) and the alternative mean (49,000), divided by the standard error ($\sigma / \sqrt{n}$), where $\sigma = 5,000$ and $n = 100$. This gives us a z-score for the alternative hypothesis. We then find the area to the left of this z-score in the standard normal distribution, which gives us the power of the test. The correct calculation leads to the power being closest to the value provided in option D, 0.3707.

A Tire Company Claims That the Lifetimes of Its Tires σ=5,000\sigma = 5,000σ=5,000

A Tire Company Claims That the Lifetimes of Its Tires $\sigma = 5,000$