# Finding type 1 error probability from the power of test.

Using $$n=16$$ observations from normally distributed population $$H_0: \mu = 30$$ is tested against $$H_A : \mu> 30$$. If power of the test, $$1-β = 0.85$$ when $$\mu_A = 34$$, what is the probability of making Type 1 Error? Assume that $$σ = 9$$.

We have $$Power = P(\text{Rejecting } H_0 \text{ when } \mu = 34) = 0.85$$

Let $$a$$ be the point that we reject $$H_0$$ if $$\bar{x} \gt a$$. Then,

$$P(\text{Rejecting } H_0 \text{ when } \mu = 34) = P(\bar{x} \gt a |\mu=34)= P(Z \gt \frac{a-\mu}{\sigma/\sqrt{n}}) = P(Z \gt \frac{a-34}{9/4}) = 0.85 => P(Z \lt \frac{a-34}{9/4}) = 0.15$$

Then, from z-table I find $$\frac{a-34}{9/4} \approx -2.17 => a \approx 29.11$$.

What makes me wonder about what I did until here is the value of $$a$$. According to the way I defined $$a$$, a sample with mean $$29.5$$ would cause me to reject $$H_0:\mu=30$$ and accept $$H_A:\mu \gt 30$$.

I feel like I made some mistakes, can I get some help please?

Your only problem is in saying the $$15\%$$ quantile of the normal distribution is $$-2.17.$$ Not sure what table you looked it up in but it's very wrong. It should be $$-1.04,$$ and then we get a threshold of $$31.67$$.
I see the error you made... $$-2.17$$ is the $$1.5\%$$ quantile, so you just accidentally typed $$.015$$ instead of $$.15$$ somewhere. Note that if we actually demanded this very high $$98.5\%$$ power, we would get a threshold for rejection below $$30$$ as you have calculated. This is fine... if we want to reject the null hypothesis a lot, we need to set a threshold where we reject it a lot and there's no hard-fast rule that says this needs to be above our null hypothesis. However, notice there's a trade-off here and our type one error will be very bad here... well over 50%. (Not that it was great even at $$85\%$$ power, as you will calculate.)
• It is sad to make suck a stupid mistake. I even checked the z-table several times to make sure. Also thanks for the further explanation about $98.5%$ power. Commented Dec 28, 2019 at 21:28