## Example 11.2. Hypotheses with One Sample of One Categorical Variable

### State Null and Alternative Hypotheses

When considering whether we reject the null hypothesis and accept the alternative hypothesis, we need to consider the direction of the alternative hypothesis statement. For example, the alternative hypothesis that was stated earlier is:

### Example 11.4. Hypotheses with Two Samples of One Categorical Variable

The alternative hypothesis tells us two things. First, what **predictions** did we make about the effect of the independent variable(s) on the dependent variable(s)? Second, what was the predicted **direction** of this effect? Let's use our example to highlight these two points.

Sarah predicted that her teaching method (independent variable: teaching method), whereby she not only required her students to attend lectures, but also seminars, would have a positive effect (that is, increased) students' performance (dependent variable: exam marks). If an alternative hypothesis has a direction (and this is how you want to test it), the hypothesis is one-tailed. That is, it predicts direction of the effect. If the alternative hypothesis has stated that the effect was expected to be negative, this is also a one-tailed hypothesis.

## State Null and Alternative Hypotheses

If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above the cut-off value, we fail to reject the null hypothesis and cannot accept the alternative hypothesis. You should note that you cannot accept the null hypothesis, but only find evidence against it.

## State Null and Alternative Hypotheses

The main purpose of statistics is to test a hypothesis. For example, you might run an experiment and find that a certain drug is effective at treating headaches. But if you can’t repeat that experiment, no one will take your results seriously. A good example of this was the discovery, which petered into obscurity because no one was able to duplicate the results.

## State Null and Alternative Hypotheses

The **level of statistical significance** is often expressed as the so-called **p****-value**. Depending on the statistical test you have chosen, you will calculate a probability (i.e., the *p*-value) of observing your sample results (or more extreme) **given that the null hypothesis is true**. Another way of phrasing this is to consider the probability that a difference in a mean score (or other statistic) could have arisen based on the assumption that there really is no difference. Let us consider this statement with respect to our example where we are interested in the difference in mean exam performance between two different teaching methods. If there really is no difference between the two teaching methods in the population (i.e., given that the null hypothesis is true), how likely would it be to see a difference in the mean exam performance between the two teaching methods as large as (or larger than) that which has been observed in your sample?

## State Null and Alternative Hypotheses

A hypothesis is an **educated guess** about something in the world around you. It should be testable, either by experiment or observation. For example: