# the null hypothesis when it is in reality true

Recall that when we fail to reject H_{0} in a test of hypothesis that either the null hypothesis is true (here the mean expenditures in 2005 are the same as those in 2002 and equal to $3,302) or we committed a Type II error (i.e., we failed to reject H_{0} when in fact it is false). In summarizing this test, we conclude that we do not have sufficient evidence to reject H_{0}. We do not conclude that H_{0} is true, because there may be a moderate to high probability that we committed a Type II error. It is possible that the sample size is not large enough to detect a difference in mean expenditures.

## We reject the Null Hypothesis when in reality, ..

### Type I and Type II Errors - Statistics Lectures

One-tailed tests have more power than two-tailed tests, given that you have specified the correct tail. If you specify the wrong tail, power is essentially 0, because there is no way to correctly reject the null hypothesis and interpret it correctly. For this reason, most researchers always conduct two-tailed tests. Even though they are less powerful, they prevent the awkwardness of having to retain the null hypothesis even when the mean difference is huge but in the opposite direction of what was expected.

### So, we reject the null hypothesis and say that the ..

Usually, the null hypothesis is boring and the alternative hypothesis is interesting. For example, let's say you feed chocolate to a bunch of chickens, then look at the sex ratio in their offspring. If you get more females than males, it would be a tremendously exciting discovery: it would be a fundamental discovery about the mechanism of sex determination, female chickens are more valuable than male chickens in egg-laying breeds, and you'd be able to publish your result in *Science* or *Nature*. Lots of people have spent a lot of time and money trying to change the sex ratio in chickens, and if you're successful, you'll be rich and famous. But if the chocolate doesn't change the sex ratio, it would be an extremely boring result, and you'd have a hard time getting it published in the *Eastern Delaware Journal of Chickenology*. It's therefore tempting to look for patterns in your data that support the exciting alternative hypothesis. For example, you might look at 48 offspring of chocolate-fed chickens and see 31 females and only 17 males. This looks promising, but before you get all happy and start buying formal wear for the Nobel Prize ceremony, you need to ask "What's the probability of getting a deviation from the null expectation that large, just by chance, if the boring null hypothesis is really true?" Only when that probability is low can you reject the null hypothesis. The goal of statistical hypothesis testing is to estimate the probability of getting your observed results under the null hypothesis.