**Keith M Bower describes an example of the paired (ordependent) t-test procedure using a statistical software package.**

The paired t-test procedure is used to compare the mean difference between two populations when one believes that some dependency exists. For example, if one were to test pulse rates for a group of individuals prior to, then upon completion of, a fitness regime, it would be appropriate to compare the pulse rates for each individual.

What follows is an example of the paired (or dependent) t-test procedure using the statistical software package, Minitab. We may wish to test if the mean difference is significantly different from zero, ie test H0:µdifference=0, versus an alternative hypothesis such as H1:µdifference‚0.

The t-test was originally developed by the English statistician William Sealy Gosset (1876-1937), while working for a brewing company in Ireland. Gosset published under the pseudonym Student (1908)1 which is why it is frequently referred to as aStudent's' t-test.

For this example we shall consider a set of data from Stichler, Richey and Mandel (1952)2 that deal with measurements of tyre wear (thousmi).

Each tyre was subject to measurement by two different methods, the first based on weight loss, and the second based on groove wear.

An assumption for the paired t-test procedure is that the distribution in which the differences analysed come from is Normal (aka Gaussian). We may therefore wish to create a column for the differences between the two measurement systems, and investigate the distributional properties.

The Anderson-Darling (A-D) Normality Test illustrated in Fig.1 shows that we are unable to reject the null hypothesis, H0: data follow a Normal distribution vs. H1: data do not follow a Normal distribution, at thea=0.10 significance level.

This is because the p-value for the A-D test is 0.887, which is greater than 0.10, a frequently used level of significance for such a hypothesis test (as opposed to the more traditional 0.05 level). Note that the p-value is the probability of incorrectly rejecting the null hypothesis (iemaking a Type I error).

As is discussed by Hogg and Ledolter (1987)3, when the assumption of Normality is violated, the t-test procedure still works well if the underlying distribution (in this case for the differences) is symmetric, unimodal and continuous. If the values (in this case the differences) are seriously skewed, it may appropriate to use a nonparametric procedure, such as the one-sample ign test.

However, provided that the precaution of randomization is employed, use of the paired t-test procedure without concern as to the form of the underlying distribution is a valid approach. For more information, the interested reader is referred to Box, Hunter and Hunter (1978)4 .

We may now perform a paired t-test. As the output in Fig. 2 exhibits, we are able to reject the null hypothesis, µdifference = 0 at the a = 0.05 level of significance as the p-value is less than 0.05. In fact, the evidence strongly suggests that there is a difference between the two methods (the p-value is very low). From the data in this study we can conclude that, on average, the weight loss method gives higher measurements than the groove wear method.

It would be useful for us to have some ppreciation as to a reasonable range of values for what the true mean difference may be for the two types of measurement systems. Here we tested the null hypothesis using a significance level of a=0.05. A 100*(1-a)percent confidence interval, ie a 95percent confidence interval, is also reported in the Minitab output in Fig.2 and displayed in Fig.3. We may be 95percent confident that the true mean difference in tyre tread measurements is between 2.837 and 6.275. One should always keep in mind that though a difference may be statistically significant (as in this case), there may not be a practical difference.

In conclusion, one finds that the paired t-test can be used to investigate the mean difference between two dependent populations. If the means from two independent populations are to be investigated, the two-sample t-test may be used. For the comparison of means from two or more independent populations, the ANOVA procedure5 may be used.

Keith M. Bower is a Technical Training Specialist with Minitab, Inc, State College, PA, USA. " target="_blank">www.minitab.com

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