## Significance Tests / Hypothesis Testing

Before actually conducting a hypothesis test, you have to put two possible hypotheses on the table — the null hypothesis is one of them. But, if the null hypothesis is rejected (that is, there was sufficient evidence against it), what’s your alternative going to be? Actually, three possibilities exist for the second (or alternative) hypothesis, denoted H_{a}. Here they are, along with their shorthand notations in the context of the pie example:

In all tests of hypothesis, there are two types of errors that can be committed. The first is called a Type I error and refers to the situation where we incorrectly reject H_{0} when in fact it is true. This is also called a false positive result (as we incorrectly conclude that the research hypothesis is true when in fact it is not). When we run a test of hypothesis and decide to reject H_{0} (e.g., because the test statistic exceeds the critical value in an upper tailed test) then either we make a correct decision because the research hypothesis is true or we commit a Type I error. The different conclusions are summarized in the table below. Note that we will never know whether the null hypothesis is really true or false (i.e., we will never know which row of the following table reflects reality).

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

Here we want to assess whether the sample mean of 200.3 in the Framingham sample is statistically significantly different from 203 (i.e., beyond what we would expect by chance). We will run the test using the five-step approach.

## failing to reject the null hypothesis when it is false.

which we get by inserting the hypothesized value of the population mean difference (0) for the population_quantity. If or (that is, ), we say the data are not consistent with a population mean difference of 0 (because does not have the sort of value we expect to see when the population value is 0) or "we **reject the hypothesis that the population mean difference is 0**". If t were -3.7 or 2.6, we would reject the hypothesis that the population mean difference is 0 because we've observed a value of t that is unusual if the hypothesis were true.

## rejecting the null hypothesis when it is true.

If (that is, ), we say the data are consistent with a population mean difference of 0 (because has the sort of value we expect to see when the population value is 0) or "we **fail to reject the hypothesis that the population mean difference is 0**". For example, if t were 0.76, we would fail reject the hypothesis that the population mean difference is 0 because we've observed a value of t that is unremarkable if the hypothesis were true.

## rejecting the null hypothesis when it is false.

Depending on how you want to "summarize" the exam performances will determine how you might want to write a more specific null and alternative hypothesis. For example, you could compare the **mean** exam performance of each group (i.e., the "seminar" group and the "lectures-only" group). This is what we will demonstrate here, but other options include comparing the **distributions**, **medians**, amongst other things. As such, we can state: