# Finding the probability of a Type II error or the power, as specified

For the given significance test, determine the probability of a Type II error or the power, as specified.

Suppose we wish to test $H_{0}: p = 0.5$ against $H_{1}: p < 0.4$ using $α = 0.05$. If $p$ is actually equal to $0.4$, what is the probability of a type II error assuming $n = 150$?

How do i find the probability of a Type II error?

I've only managed to find the z-score to be -2.45, and its corresponding p-value to be 0.0071. Thanks.

• Please double-check the question you quote. Is the alternative supposed to be $p< 0.40$ rather than $p<40$? Commented Nov 7, 2017 at 16:52
• @Glen_b I was given exactly $H_{1}: p < 40$, but i wouldn't rule out a typo in the question, and it actually does make sense. Are you able to derive the answer from these information? Commented Nov 7, 2017 at 16:57
• If it is really 40 (meaning that $p$ is not a proportion as it appears to be), we'd need to to know more about the circumstances (what's the model here?). If it's $0.4$ and $p$ is a proportion then yes, it's a reasonably straightforward binomial calculation. Commented Nov 7, 2017 at 22:58

In order to calculate the power of an alternative hypothesis, one should first formulate it as a simple hypothesis: For example, $H_1:p<40$ is a complex hypothesis as it refers to multiple possible values of $p$. A simple hypothesis would be something like $H_1:p=39.9$. The process of calculating the power (for simple hypotheses) is as follows:

1. Identifying the discussed probability (preferably normal or approximated to normal)
2. Building the Likelihood Ratio Test
3. Extracting the test statistic $T(x)$ from the LRT
4. Formulating the test shape, something like $\delta(x):T(x)>c$ or $\delta(x):T(x)<c$
5. Solving the equation $P_{H_0}(\delta(x))=\alpha$ in order to find the critical value $c_\alpha$, which will make the equation $P_{H_0}(T(x)\leq c_\alpha)=1-\alpha$ hold (this is just an example for the case of $\delta(x):T(x)>c$).
6. Finding the power by solving the equation $\pi=1-\beta=P_{H_1}(T(x)>c_\alpha)$

Please note that $P_{H_0}$ refers to the distribution under the null assumption whereas $P_{H_1}$ refers to the distribution under the specified alternative.

• i haven't reach that high of a level, and I believe my question requires simple calculation. Is there a more direct way to find the probability of the type II error? Commented Nov 7, 2017 at 16:25
• A type II error (true negative) requires specifying a simple alternative and of course knowing the underlying distribution. Commented Nov 7, 2017 at 16:40
• ah i see, thanks. From the method given above, is it possible to derive the value from only the information I gave in the question? (because that's what i'm given) Commented Nov 7, 2017 at 16:47
• If we assume the question discusses binomial distribution and (as noted before) $H_1:p<0.4$ (which makes more sense) then the answer is yes, using the normal approximation. Commented Nov 7, 2017 at 17:07
• i've did some research and they requires the standard error to find the test II error. Mind sharing how you would approach this question? Commented Nov 7, 2017 at 17:27