 # Null and Alternative Hypotheses

In our example, the population mean on the VAST test for people with no ACE training is 500. Suppose we randomly sample 25 graduates of ACE training and observe a sample mean of 550. We can compute how likely it is to observe a sample mean of 550 or larger if the population mean for ACE graduates is really 500. If that probability is very small, say less than .05, we may reject the hypothesis that the population mean is 500 and conclude that the population mean for ACE graduates is larger than 500. In this case, we can also reject the hypothesis that the population mean is any value less than 500 because observing a sample mean of 550 is even less likely if the population mean is less than 500.

The null model that we will test is that the average score for the population of all ACE graduates is 500, equal to the average score for non-ACE graduates. The alternative hypothesis is that the average test score for the population of ACE graduates is greater than 500, which can be stated as H1: m > 500. If the alternative hypothesis is not true, then the mean is equal to 500 or less than 500. This is the null hypothesis which can be stated as H0: m ≤ 500. The null model (m = 500) is part of the null hypothesis.

If our sample mean is much larger than 500, the null model can be rejected and we conclude that the alternative hypothesis is probably true, that our sample came from a population with a mean larger than 500. In this case we can reject not only the hypothesis that the population mean is equal to 500 but also all values less than 500. Formally, we would reject H0: m ≤ 500 which leaves us with the alternative, H1:m > 500.

Failure to reject the null hypothesis H0: m ≤ 500 does not imply that the null hypothesis is necessarily true. Rather, such findings are ambiguous – the mean may be less than 500 but it also could be somewhat greater than 500.

### One-tailed vs. Two-tailed Hypotheses

To conclude that ACE graduates have greater VAST scores, we would need to reject all other possibilities – these other possibilities are expressed as the null hypothesis, which is that ACE graduates have equal or lower scores than nongraduates. We will be persuaded that the mean is greater for the population of ACE graduates only if the sample mean is sufficiently larger than 500, on the upper tail of the distribution. This is called a one-tailed test.

In another situation we may be interested in whether a population has a mean that differs from 500, either larger or smaller. In that case the null hypothesis would be that the mean is equal to 500 and the alternative hypothesis is that the mean is not equal to 500. Symbolically, H0: m = 500 and H1: m ≠ 500. We may reject the null hypothesis if the sample mean is sufficiently far from 500 in either tail of the sampling distribution. Hence, this is called a two-tailed test.

In practice, the decision to use a one-tailed test must be made before looking at the data. If we would be interested in a departure from the null population mean in either direction, then we should use a two-tailed test. A one-tailed test is used only when we are willing, before we look at the data, to ignore a large difference from the null mean in one of the two directions. (Another example of one- vs. two-tailed tests.)

### Question F: Forming the Null Hypothesis

Suppose we are interested in deciding whether ACE graduates have greater VAST scores than the population of VAST test takers who have not taken ACE training. A random sample of 30 ACE graduates yields a sample mean of 510 (i.e., X = 510) and standard deviation of 90 (S = 90). The population mean for untrained test takers is 500 (m = 500) with a standard deviation of 100 (s = 100).

If we were interested in testing whether ACE training program graduates on average score better than 500, what form would our null hypothesis take?

#### H0: Population Mean > 500

Incorrect! This is the alternate hypothesis, the hypothesis that we accept if we reject the null hypothesis.

#### H0: Population Mean < 500

Incorrect! This hypothesis does not include the null effect of m = 500.

#### H0: Population Mean = 500

Incorrect! This would be the null hypothesis for a two-tailed test, in which we would be interested in whether the sample mean is different from 500 in either direction (upper or lower tail of the distribution). Because we are interested in whether the mean for the population of ACE graduates is greater than 500 (upper tail only), the correct test would be one-tailed. It is true that we will test the null model that m = 500 but the null hypothesis for a one-tailed test also includes values less than 500.

#### H0: Sample Mean = 510

Incorrect! This is the observed sample mean. The null hypothesis refers to population values.

#### H0: Sample Mean ≤ 500

Incorrect! The null hypothesis refers to population values, not sample values.

#### None of the above (What is correct?)

Correct! The correct statement of the null hypothesis is H0: Population Mean ≤ 500. If we are able to reject this null hypothesis, we can conclude that the population mean is greater than 500 for the population represented by our sample. 