4.4 One-Sided and Two-Sided Tests

In the preceding section, you may have had some trouble when you were determining whether a research hypothesis is a null hypothesis or an alternative hypothesis. The research hypothesis stating that average media literacy is below 5.5 in the population, for example, represents the alternative hypothesis because it does not fix the hypothesized population value to one number. The accompanying null hypothesis must cover all other options, so it must state that the population mean is 5.5 or higher. But this null hypothesis does not specify one value as it should, right?

This null hypothesis is slightly different from the ones we have encountered so far, which equated the population value to a single value. If the null hypothesis equates a parameter to a single value, the null hypothesis can be rejected if the sample statistic is either too high or too low. There are two ways of rejecting the null hypothesis, so this type of hypothesis and test are called two-sided or two-tailed.

By contrast, the null hypothesis stating that the population mean is 5.5 or higher is a one-sided or one-tailed hypothesis. It can only be rejected if the sample statistic is at one side of the spectrum: only below (left-sided) or only above (right-sided) the hypothesized population value. In the media literacy example, the null hypothesis is only rejected if the sample mean is well below the hypothesized population value. A test of a one-sided null hypothesis is called a one-sided test.

Figure 4.3: One-sided and two-sided tests of a null hypothesis.

In a left-sided test of the media literacy hypothesis, the researcher is not interested in demonstrating that average media literacy among children can be larger than 5.5. She only wants to test if it is below 5.5, perhaps because an average score below 5.5 is alarming and requires an intervention, or because prior knowledge about the world has convinced her that average media literacy among children can only be lower than 5.5 on average in the population.

If it is deemed important to note values well over 5.5 as well as values well below 5.5, the research and null hypotheses should be two-sided. Then, a sample average well above 5.5 would also have resulted in a rejection of the null hypothesis. In a left-sided test, however, a high sample outcome cannot reject the null hypothesis.

4.4.1 Boundary value as hypothesized population value

Figure 4.4: Sampling distribution of average media literacy.

You may wonder how a one-sided null hypothesis equates the parameter of interest with one value as it should. The special value here is 5.5. If we can reject the null hypothesis stating that the population mean is 5.5 because our sample mean is sufficiently lower than 5.5, we can also reject any hypothesis involving population means higher than 5.5.

In other words, if you want to know if the value is not 5.5 or more, it is enough to find that it is less than 5.5. If it’s less than 5.5, then you know it’s also less than any number above 5.5. Therefore, we use the boundary value of a one-sided null hypothesis as the hypothesized value for the population in a one-sided test.

4.4.2 One-sided – two-sided distinction is not always relevant

Note that the difference between one-sided and two-sided tests is only useful if we test a statistic against one particular value or if we test the difference between two groups.

In the first situation, for example, if we test the null hypothesis that average media literacy is 5.5 in the population, we may only be interested in showing that the population value is lower than the hypothesized value. Another example is a test on a regression coefficient or correlation coefficient. According to the null hypothesis, the coefficient is zero in the population. If we only want to use a brand advertisement if exposure to the advertisement increases brand awareness among consumers, we apply a right-sided test to the coefficient for the effect of exposure on brand awareness because we are only interested in a positive effect (larger than the zero).

In the second situation, we compare the scores of two groups on a dependent variable. If we compare average media literacy after an intervention to media literacy before the intervention (paired-samples t test), we must demonstrate an increase in media literacy before we are going to use the intervention on a large scale. Again, a one-sided test can be applied.

In contrast, we cannot meaningfully formulate a one-sided null hypothesis if we are comparing three groups or more. Even if we expect that Group A can only score higher than Group B and Group C, what about the difference between Group B and Group C? If we can’t have meaningful one-sided null hypotheses, we cannot meaningfully distinguish between one-sided and two-sided null hypotheses.

4.4.3 From one-sided to two-sided p values and back again

Statistical software like SPSS usually reports either one-sided or two-sided p values. What if a one-sided p value is reported but you need a two-sided p value or the other way around?

In Figure 4.5, the sample mean is 3.9 and we have .015 probability of finding a sample mean of 3.9 or less if the null hypothesis is true. This probability is the surface under the curve to the left of the red line representing the sample mean. It is the one-sided p value that we obtain if we only take into account the possibility that the population mean can be smaller than the hypothesized value. We are only interested in the left tail of the sampling distribution.

Figure 4.5: Halve a two-sided p value to obtain a one-sided p value, double a one-sided p value to obtain a two-sided p value.

In a two-sided test, we have to take into account two different types of outcomes. Our sample outcome can be smaller or larger than the hypothesized population value. As a consequence, the p value must cover samples at opposite sides of the sampling distribution. We should not only take into account sample means that are smaller than 5.5 but also sample means that are just as much larger than the hypothesized population value. So our two-sided p value must include both the probability of .015 for the left tail and for the right tail of the distribution in Figure 4.5. We must double the one-sided p value to obtain the two-sided p value.

In contrast, if our statistical software tells us the two-sided p value and we want to have the one-sided p value, we can simply halve the two-sided p value. The two-sided p value is divided equally between the left and right tails. If we are interested in just one tail, we can ignore the half of the p value that is situated in the other tail. Of course, this only makes sense if a one-sided test makes sense.

Be careful if you divide a two-sided p value to obtain a one-sided p value. If your left-sided test hypothesizes that average media literacy is below 5.5 but your sample mean is well above 5.5, the two-sided p value can be below .05. But your left-sided test can never be significant because a sample mean above 5.5 is fully in line with the null hypothesis. Check that the sample outcome is at the correct side of the hypothesized population value.