Can someone explain to me the mechanics of how Type II error is reduced and the Power of a Test is increased as the size of samples increases?
Explain the intuition of how this exactly works, if that is possible.
Thanks!
If you want intuition here is what I can offer without giving (more) formulas…
Type II errors (the probability of which is Beta) and the power of a test (1-beta) pertain to the world assuming the null hypothesis is false.
A type II error is when you gather a sample and conduct a test of hypothesis, but you sample tells you the null is “true” (you don’t have evidence to reject the null). In other words, you’ve made an incorrect conclusion about the world (since the null is false in actuality).
The other outcome (again, assuming the null is actually false), is that your sample and test tell you the null can be rejected (i.e. it is likely false). Here, you conclude that the null is false and this aligns with the truth. The probability that this occurs is (1-beta)-- this is the power of the test-- the chance that you reject the null hypothesis if it is actually false.
The easiest and least technical way to explain this (from my perspective)…Ideally, we would like to have the entire population because we could ascertain the truth. However, we have finite resources and must sample. Assuming we take a representative sample, we will be closer to the truth with a larger sample rather than a smaller sample. With a larger sample, we are more likely to reject the null if it is actually false(more likely to see the truth)–we have increased the power of the test. On the other side, we are less likely to “accept” the null (Fail to reject), even though it is false. You can see this by remembering that [power = 1 - Beta]. If one goes up, the other must go down.
If you like probability notation:
beta = P(Type II error) = P(Accept Ho|Ho False); read as “probability of accepting Ho, given/assuming that Ho is false.”
power = P(Reject Ho|Ho False); read as “probability of rejecting Ho, given/assuming Ho is false.”
Hope this helps!
A Type II error is failing to reject a false null hypothesis.
As the sample size increases, the standard error decreases, so the acceptance region narrows. This makes it more likely that you will reject the null hypothesis, and, consequently, more likely that you will reject a false null hypothesis, avoiding a Type II error.
The power of the test is:
1 − P(Type II error)
So, as P(Type II error) decreases, the power of the test increases.