# Significance Tests / Hypothesis Testing

Just as when we test hypotheses concerning two population means, we take a random sample from each population, of sizes *n*_{1} and *n*_{2}, and compute the sample standard deviations *s*_{1} and *s*_{2}. In this context the samples are always independent. The populations themselves must be normally distributed.

## In hypothesis testing does a hypothesis not get rejected despite

### Hypothesis Testing One Sample T Test TutorTeddy com

Now instead of testing 1000 plant extracts, imagine that you are testing just one. If you are testing it to see if it kills beetle larvae, you know (based on everything you know about plant and beetle biology) there's a pretty good chance it will work, so you can be pretty sure that a *P* value less than 0.05 is a true positive. But if you are testing that one plant extract to see if it grows hair, which you know is very unlikely (based on everything you know about plants and hair), a *P* value less than 0.05 is almost certainly a false positive. In other words, *if you expect that the null hypothesis is probably true, a statistically significant result is probably a false positive.* This is sad; the most exciting, amazing, unexpected results in your experiments are probably just your data trying to make you jump to ridiculous conclusions. You should require a much lower *P* value to reject a null hypothesis that you think is probably true.

### Next section: to Inferential statistics (testing hypotheses)

Now imagine that you are testing those extracts from 1000 different tropical plants to try to find one that will make hair grow. The reality (which you don't know) is that one of the extracts makes hair grow, and the other 999 don't. You do the 1000 experiments and do the 1000 frequentist statistical tests, and you use the traditional significance level of *P**P**P**P* values less than 0.05, but almost all of them are false positives.

## Hypothesis testing is the subject of this chapter.

The 0.95 quantile for the standard normal distribution is 1.645. Thiscorresponds to the critical value for the usual upper-tailed z-testwith a significance level of 0.05. In our case, we are using thesample mean, not z, as the test statistic. The 0.95 quantile for thesampling distribution of the mean under the null hypothesis is 0 +1.645*(1/sqrt(5)) = 0.736, i.e., the critical value is the null valueplus the z-value times the standard error of the mean. We rejectH_{o} if the sample mean is greater than 0.736.