A small (p)-valueis an indication that the null hypothesis is false.

To conduct a hypothesis test we will compare our sample to the theoretical distribution described by the null hypothesis (the hypothesis of "no difference" or "no effect"). To accomplish this, we need to describe a theoretical idea called the sampling distribution. Let's say that we have 10,000 people in our population. We want to measure the effects of our new wonder IQ drug. Our hypothesis is that it will INCREASE IQ. What type of hypothesis was this?( We can't test all 10,000, but we can take 100 and test them, then find their Mean. To be more accurate, we are going to do this again with another 100, and then another 100, and so on. If we were to take all these means, we have created a sampling distribution. This is where sampling error comes in. The more samples we take, the more the mean of our sampling distribution will look like the population mean. Each separate sample mean, however, will vary from the population mean. Sound a bit like standard deviation doesn't it? In fact, it is very similar and this is our modification. We are going to replace with the standard deviation of the sampling distribution.

The null and alternative hypotheses are:

The answer is yes since the hypothesized value o is 0.25 and we can check that:

which is equivalent to rejecting the null hypothesis:

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:

which is equivalent to rejecting the null hypothesis:

Now that you have identified the null and alternative hypotheses, you need to find evidence and develop a strategy for declaring your "support" for either the null or alternative hypothesis. We can do this using some statistical theory and some arbitrary cut-off points. Both these issues are dealt with next.

Let's return finally to the question of whether we reject or fail to reject the null hypothesis.

not rejecting the null hypothesis when the alternative is true.

Step 5. Check to see if the value of the test statistic falls in the rejection region. If it does, then reject o (and conclude a). If it does not fall in the rejection region, do not reject o.

the null hypothesis is rejected when it is true.

What if, after receiving the citron, all the subjects in the magistrate's study died anyway? What would our decision be? Now we would have made the decision to fail to reject the null hypothesis. When we fail to reject the null hypothesis we are stating that there was not a significant difference. What? Why can't we just say that we accept the null hypothesis?

Step 5. Since -value = 0.2938  0.05 (the  value), we cannot reject the null hypothesis.

One can never prove the truth of a statistical (null) hypothesis.

The decision rule is a statement that tells under what circumstances to reject the null hypothesis. The decision rule is based on specific values of the test statistic (e.g., reject H0 if Z > 1.645). The decision rule for a specific test depends on 3 factors: the research or alternative hypothesis, the test statistic and the level of significance. Each is discussed below.

If the P-value is small, that is, if P ≤ α, then we reject the null hypothesis H0.

failing to reject the null hypothesis when it is true.

That's the same as saying that we should reject the null hypothesis H0 if p0 is not in the (1−α)100% confidence interval!

the probability of rejecting the null hypothesis when the null hypothesis is true.

rejecting the null hypothesis when it is false.

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.