It is desired to test $\mathrm { H } _ { 0 } : \mu = 50$ against $\mathrm { H } _ { \mathrm { A } } : \mu \neq 50$ using $\alpha = 0.10$. 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 45 , what is the power of the test?
A) $0.1041$
B) $0.7544$
C) $0.2456$
D) $0.8959$

Question

Quizplus · Accepted Answer

The power of the test is the probability of correctly rejecting the null hypothesis when it is false. Given $\mu = 45$, standard deviation $\sigma = 15$, sample size $n = 49$, and $\alpha = 0.10$, we calculate the Z-scores for the critical values and then find the probability of observing a sample mean that leads to rejecting $H_0$. The calculation involves finding the Z-score for the actual mean ($\mu = 45$) under the sampling distribution assuming $H_0$ is true ($\mu = 50$), and then using the standard normal distribution to find the area under the curve that corresponds to rejecting $H_0$. This area represents the power of the test, which in this case is $0.7544$, indicating a high likelihood of correctly rejecting $H_0$ when $\mu$ is indeed 45.

It Is Desired to Test H0:μ=50\mathrm { H } _ { 0 } : \mu = 50H0​:μ=50

It Is Desired to Test $\mathrm { H } _ { 0 } : \mu = 50$