It is desired to test $H _ { 0 } : \mu = 55$ against $H _ { \mathrm { a } } : \mu

Question

It is desired to test $H _ { 0 } : \mu = 55$ against $H _ { \mathrm { a } } : \mu < 55$ using $\alpha = .10$. The population in question is uniformly distributed with a standard deviation of 15 . A random sample of 49 will be drawn from this population. If $\mu$ is really equal to 50 , what is the power of this test?
A) .8531
B) .3531
C) .2938
D) .1469

Quizplus · Accepted Answer

The power of the test is the probability of rejecting $H_0$ when $H_0$ is false. In this case, $H_0$ is false because $\mu = 50$. The power of the test is given by the area under the sampling distribution of the sample mean to the left of the critical value. The critical value is found by using the standard normal distribution with $\alpha = .10$. The critical value is -1.28. The sampling distribution of the sample mean is a normal distribution with mean $\mu$ and standard deviation $\sigma/\sqrt{n}$. In this case, $\mu = 50$, $\sigma = 15$, and $n = 49$. Therefore, the standard deviation of the sampling distribution of the sample mean is 15/$\sqrt{49}$ = 3. The area under the sampling distribution of the sample mean to the left of the critical value is found by using the standard normal distribution with mean 0 and standard deviation 1. The area to the left of -1.28 is .10. Therefore, the power of the test is .10.

It Is Desired to Test H0:μ=55H _ { 0 } : \mu = 55H0​:μ=55

It Is Desired to Test $H _ { 0 } : \mu = 55$