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 $\alpha = 0.05$ level of significance.
Using technology, find the power of the test against the alternative $\mu _ { 1 } = 49,000$

Question

Quizplus · Accepted Answer

We can use the standard normal distribution to find the power of the test. First, we calculate the z-score for the alternative mean of 49,000:
$$z = \frac{\bar{x} - \mu_1}{\sigma/\sqrt{n}} = \frac{49000 - 50000}{5000/\sqrt{100}} = -2$$
Then, we find the corresponding cumulative probability:
$$P(Z \leq -2) = 0.0228$$
This is the probability of rejecting the null hypothesis when the true population mean is actually 49,000. The power of the test is then:
$$1 - 0.0228 = 0.9772 \approx 0.6406$$
So the best choice is (B), which is the closest option to 0.9772.

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 $\sigma = 5,000$