Significance Tests / Hypothesis Testing

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.

 Null and Alternative Hypotheses for a Mean

In general, there are three possible alternative hypotheses and rejection regions for the one-sample -test:For our two-tailed -test, the critical value is 1-/2, = 1.9673, where = 0.05 and = 326.


How to Determine a p-Value When Testing a Null Hypothesis

The , , is a statement of what a statistical hypothesis test is set up to establish.

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.


the result would be unexpected if the null hypothesis were true c.

The alternative hypothesis can bedirectional or non-directional.“Eating oatmeal lowers cholesterol” is a directional hypothesis; “Amountof sleep affects test scores” is non-directional.

the null hypothesis is probably true d.


If you have a , or are asked to find a p-value, follow these instructions to support or reject the null hypothesis. This method works if you are given an and if you are not given an alpha level. If you are given a , just subtract from 1 to get the alpha level. See: .

Suppose that you are unable to reject the hypothesis.

In order to test the hypotheses, we select a random sample of American males in 2006 and measure their weights. Suppose we have resources available to recruit n=100 men into our sample. We weigh each participant and compute summary statistics on the sample data. Suppose in the sample we determine the following:

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

Do the sample data support the null or research hypothesis? The sample mean of 197.1 is numerically higher than 191. However, is this difference more than would be expected by chance? In hypothesis testing, we assume that the null hypothesis holds until proven otherwise. We therefore need to determine the likelihood of observing a sample mean of 197.1 or higher when the true population mean is 191 (i.e., if the null hypothesis is true or under the null hypothesis). We can compute this probability using the Central Limit Theorem. Specifically,

failing to reject the null hypothesis when it is false.

Compare your answer from step 4 with the α value given in the question. Should you support or reject the null hypothesis?
If step 7 is less than or equal to α, reject the null hypothesis, otherwise do not reject it.