2
$\begingroup$

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.

Improve this question
$\endgroup$
3
  • $\begingroup$ Please double-check the question you quote. Is the alternative supposed to be $p< 0.40$ rather than $p<40$? $\endgroup$
    – Glen_b
    Commented Nov 7, 2017 at 16:52
  • $\begingroup$ @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? $\endgroup$
    – idolo
    Commented Nov 7, 2017 at 16:57
  • $\begingroup$ 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. $\endgroup$
    – Glen_b
    Commented Nov 7, 2017 at 22:58

1 Answer 1

Reset to default
1
$\begingroup$

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.

Improve this answer
$\endgroup$
6
  • $\begingroup$ 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? $\endgroup$
    – idolo
    Commented Nov 7, 2017 at 16:25
  • $\begingroup$ A type II error (true negative) requires specifying a simple alternative and of course knowing the underlying distribution. $\endgroup$
    – Spätzle
    Commented Nov 7, 2017 at 16:40
  • $\begingroup$ 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) $\endgroup$
    – idolo
    Commented Nov 7, 2017 at 16:47
  • $\begingroup$ 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. $\endgroup$
    – Spätzle
    Commented Nov 7, 2017 at 17:07
  • $\begingroup$ 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? $\endgroup$
    – idolo
    Commented Nov 7, 2017 at 17:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.