# Determining which test statistic is better under two different scenarios

Problem: I have 9 independent normal observations from a normal dist. with $\sigma=10$. And we want to test $H_0:\mu=150$vs $H_A:\mu\neq 150$. Also $\alpha=0.05$ which uses the test statistic $M_1=|\bar{x}-150|$ with a critical value of $6.53$ and for a second test statistic $M_2=|\bar{x}-152|$ with a critical value of $7.55$.

Under Test statistic 1 when I calculate the power of test against the alternative $\mu=151$ I get a power $1$ of $.0604$ and when the alternative is $\mu=149$ I get a power $2$ of $.0604$.

Under Test statistic 2 for the same conditions I get a power $1$ of $.0298866$ and when the alternative is $\mu=149$ I get a power $2$ of $.08698$.

Now according to this information I have to determine whats the best test. Since we do not know what the actual mean that is $\mu$ and since 150 and 152 are not that far from one another we can compare their power. Since test statistic 2 has a bigger power in both scenarios then test statistic 1 then id say that is the better test since higher power means that the chances of a type 1 error would decrease. Would this be a correct line of reasoning? Or am I forgetting to take alpha into consideration? I was thinking that Under test statistic 2 the alpha is $.023512$ which is lower than $.05$. So I think it still works out to be test stat 2.

$\alpha$ is a defined quantity and specifies your chances of a Type I error.

Power, on the other hand, is more subjective since it depends on how wrong your null hypothesis is. Note that $power = 1-P(\text{Type II error })$.

In general, you want the test that gives you the highest power for a given $\alpha$. In many cases, the power of one test is uniformly higher then the power of another test, regardless of the degree to which the null hypothesis is wrong. In this case, the higher power test is called uniformly more powerful than the lower test.

In some cases, you can actually identify the uniformly most powerful test (UMPT). Which is obviously the ideal test when it is applicable.

You need to compare the power of Statistic 1 vs. Statistic 2 under a range of "true" values, holding $\alpha$ constant. Hopefully one will have a consistently higher power and you can choose that one.

• I'll just have to graph the power functions and see which one has a higher power as I get at what your saying. Jan 14 '15 at 20:56
• @user60887 yep...that should help you make your decision. Also, you can form risk function from your Type I and Type Ii risks minimize the expected loss. en.wikipedia.org/wiki/Admissible_decision_rule
– user76844
Jan 14 '15 at 21:07