Suppose you want to test H0: = 500 against Ha: 7 500 using

Chapter 7, Problem 97E

(choose chapter or problem)

QUESTION:

Suppose you want to test $H_0: \mu\ =\ 500$ against $H_a: \mu\ >\ 500$ using $\alpha\ =\ .05$. The population in question is normally distributed with standard deviation 100. A random sample of size n = 25 will be used.

a. Sketch the sampling distribution of $\bar{x}$ assuming that $H_0$ is true.

b. Find the value of $\bar{x}_0$, that value of $\bar{x}$ above which the null hypothesis will be rejected. Indicate the rejection region on your graph of part a. Shade the area above the rejection region and label it $\alpha$.

c. On your graph of part a, sketch the sampling distribution of $\bar{x}$ if $\mu\ =\ 550$. Shade the area under this distribution that corresponds to the probability that $\bar{x}$ falls in the nonrejection region when $\mu\ =\ 550$. Label this area $\beta$.

d. Find $\beta$.

e. Compute the power of this test for detecting the alternative $H_a:\mu\ =\ 550$.

Questions & Answers

QUESTION:

a. Sketch the sampling distribution of $\bar{x}$ assuming that $H_0$ is true.

d. Find $\beta$.

e. Compute the power of this test for detecting the alternative $H_a:\mu\ =\ 550$.

ANSWER:

Solution

Given that we are suppose to test $H{\ }_{0}:\mu{}=500\ against\ H{\ }_{1}:\mu{}>500$

Here this is right tailed test and $\alpha{}=0.05$

And population standard deviation $\sigma{}=100$ and n=25

Step 1 of 5

a) We have to sketch the sampling distribution of $\overline{x}$ assuming that $H{\ }_{0}$ is true

From the central limit theorem the sampling distribution of $\overline{x}$ is approximately follows the normal distribution with $\mu{}{\ }_{\overline{x}}=\mu{}=500$