It is desired to test $H _ { 0 } : \mu = 40$ against $H _ { \mathrm { a } } : \mu

Question

It is desired to test $H _ { 0 } : \mu = 40$ against $H _ { \mathrm { a } } : \mu < 40$ using $\alpha = .10$. The population in question is uniformly distributed with a standard deviation of 10 . A random sample of 36 will be drawn from this population. If $\mu$ is really equal to 35 , what is the probability that the hypothesis test would lead the investigator to commit a Type II error?
A) $.4573$
B) $.9573$
C) $.0854$
D) $.0427$

Quizplus · Accepted Answer

The probability of a Type II error is the probability of failing to reject the null hypothesis when it is false. In this case, the null hypothesis is that the mean is 40, and the alternative hypothesis is that the mean is less than 40. The population standard deviation is 10, and a sample of 36 will be drawn from the population. If the mean is really equal to 35, then the probability of a Type II error is the probability that the sample mean will be greater than or equal to 40. This probability can be calculated using the normal distribution with a mean of 35 and a standard deviation of 10/sqrt(36) = 1.67. The probability that the sample mean will be greater than or equal to 40 is 0.0427.

It Is Desired to Test H0:μ=40H _ { 0 } : \mu = 40H0​:μ=40

It Is Desired to Test $H _ { 0 } : \mu = 40$