    Chapter 6: Analysing the Data Part III: Common Statistical Tests # The One-Sample t Test

We are going to use an example for the one sample t-test about whether the academic staff at the UNE psychology department publish differently than the national average of 8 publications per staff member between 1974 and 1996. A random sample of size 5 was taken from the 17 members of academic staff here at UNE. The data obtained were : 3, 21, 1, 15, and 7. The sample mean is = 9.4 and the sample standard deviation, s, is 8.41. The question is whether this sample mean of 9.4 gives us reason to believe that UNE academic staff publish differently than the national average of 8. Descriptively it might seem so, but had a different random sample of size 5 been collected, the sample mean would have been different. What we need is a way of determining whether the difference between 9.4 and 8 is a real difference or merely an apparent difference. If it can be considered a real difference we have evidence that to say that UNE academic staff have published more than the national average of 8 publications each. However, if the difference of 1.4 could easily be explained as a chance fluctuation then we cannot say that UNE staff publish more than the national average. The one-sample t test gives us a way of answering that question.

The data

 Staff Member Publications 1 7 2 3 3 44 4 4 5 21 6 11 7 10 8 3 9 0 10 6 11 0 12 1 13 2 14 15 15 15 16 0 17 28 µ 10 11.52

Figure 6.6 Data for one-sample t-test example.

First, we specify the null hypothesis to be tested: Ho: m = 8. The alternative hypothesis is non-directional or "two-tailed": H1: m 8. We've taken a random sample of size 5 from our population (whose mean we are pretending for this example we don't know) and computed the sample mean. Now, we ask a simple question: What is the probability of getting a sample mean of 9.4 or larger OR 6.6 or smaller if the population mean is actually 8? Before I suggest a way of answering this question, let me first talk about why we are interested in the "6.6 or smaller."

The alternative hypothesis is phrased non-directionally or "two-tailed." This means that we have no particular interest in whether the actual population mean is larger or smaller than the null hypothesised value of 8. What we are interested in is how probable it is to get a sample mean that deviates 1.4 units (from 9.4 minus 8) from the null hypothesised value of the population. Because we have no interest in the direction of the deviation, we consider 1.4 units in the opposite direction as well. This is the nature of a non-directional hypothesis test. Of interest is the probability of getting a deviation from the null hypothesised population mean this large or larger, regardless of the direction of the difference. In deciding whether or not the data are consistent with the null hypothesis, we need to consider sample mean deviations on either side of the null hypothesised population mean. We call this a two-tailed test. Had we phrased the alternative hypothesis directionally, because we were interested in a particular direction of the deviation, we'd conduct a one-tailed or directional test and only consider deviations in that direction of interest when computing the probability. For example, if we were interested in the alternative that UNE staff publish more than the national average, our alternative hypothesis would be H1: m > 8 and we'd be interested only in the probability of getting a sample mean of 9.4. We would ignore evidence in the other direction even if the deviation in the other direction is substantial. The issue of whether to use two-tailed tests or one-tailed tests is a thorny one. Experts around the world cannot agree. For our purposes, we will always be using two-tailed tests. In the School of Psychology at UNE, two-tailed tests are the standard approach even when the research hypotheses are phrased in a directional manner.

 © Copyright 2000 University of New England, Armidale, NSW, 2351. All rights reserved Maintained by Dr Ian Price Email: iprice@turing.une.edu.au