8.3 One-Sided and Two-Sided Tests

In the teatime experiment we would discard the null hypothesis only if the lady had classified all four cups correctly. We call this a one-sided test, because only observations on one side of the reference distribution are capable of rejecting the null hypothesis. Classifying four cups correctly is the only “correct” outcome that would occur with a probability of less than 5% under the null hypothesis. Getting none of the cups right is equally improbable but that outcome would, of course, support the null hypothesis.

If we were to test the assertion that plants grown from a batch of seeds reach a certain height at a certain time after sowing, we would be making a two-sided test. The null hypothesis would be that they do reach the specified height, allowing for some random variation. It would be discarded if the plants were either too low or too high. Assuming that the height is expected to follow a normal distribution, this means that observations in two regions under the tails of the normal curve would weaken the null hypothesis. Testing at the 95% confidence level the combined area of these two tail regions would be 5%, meaning that each corresponds to 2.5% probability.

The reason that it is important to be aware if you are making a one- or two-sided test is that most tables of distribution functions assume a one-sided test. We already encountered this problem when calculating confidence intervals in Chapter 7, as the table of the t-distribution in the Appendix is based on this assumption. It states that the 5% point (α = 0.05) for four degrees of freedom is t = 2.132. This means that there is 5% probability of finding a t-value greater than 2.132 (or smaller than −2.132, since the distribution is symmetric about the mean). For a two-sided test we would have to look for the α/2 point, which is 2.776. This is because the combined probability of finding a value that is either smaller than −2.776 or greater than 2.776 is 5%.

A word of caution is necessary at this point, as this book shows how to solve statistical problems in Microsoft Excel®. To add confusion to the situation, the Excel worksheet function TINV (which provides the probability points of the t-distribution) assumes that we are making a two-sided test. Table 8.1 is provided to help alleviate this confusion. It gives the probability points for the t-distribution with four degrees of freedom as returned from the table in the Appendix and from the TINV function. The α-values in the first column are the entry values of the table and the function. The italicized t-values are correct for a two-sided test at the 95% confidence level, whereas the shaded values are correct for a one-sided test.

Table 8.1 One- and two-sided probabilities from the t-distribution. Italicized t-values are correct for a two-sided test at the 95% confidence level, shaded values are correct for a one-sided test.