Samples
and
populations

Probability
theory and statistical inferrence

Inferring a population
mean

Tests of
statistical significance

Tests of
differences between
means

Coefficient
of
correlation

Caution
with
association

Chi square

Other
tests of
significance

Caution in
using
statistical
test results

Aids

Formulating the null and alternative hypotheses

For the t test, the null hypothesis takes the form that two population means, represented by µ₁for the mean of one population and µ₁for the mean of the other population, are the same. The hypothesis can be expressed as:

µ₁ = µ₂ or that  µ₁- µ₂= 0

We will work through a simple illustration of a t test. Suppose we asked a random sample of 10 employees in Office A and another 10 in Office B to rate their morale on a 10-point scale. Now suppose, in addition, that the manager of Office B had conducted a special program to improve morale. To find out whether his efforts improved morale, the manager persuaded the manager of Office A to collect similar ratings for morale so there would be a basis for testing the effect of the morale improvement program in Office B. In Table 19.3, the number of days absent are shown under the column headed X₁ for workers in Office A and under X₂ for workers in Office B. (In practice, we would not use a sample as small as 10. We are using a small number to keep the calculations simple for this illustration). For this test, the null hypothesis is that there is no difference in means for ratings of morale in the two offices. The alternative hypothesis is that the mean for Office B will be larger than the mean for Office A. This hypothesis is based on the assumption that the morale improving program would have a positive result. In addition, we need to set the level of significant to use in choosing between the two hypotheses. Let's say we select the .05 level.

Table 19.3. Illustrative calculations needed for a t test

Office A		Office B
X1	X12	X2	X22
4	16	7	49
8	64	5	25
5	25	6	36
2	4	7	49
4	16	6	36
6	36	7	49
5	25	5	25
7	49	4	16
8	64	7	49
5	25	8	64
54	324	62	398
N 10		10
M 5.4		6.2

At first glance, the manager of Office B would be pleased to see that the mean morale score for the sample of his workers was higher than that for the control group of workers. But we know that variations in scores and means occur among samples. The question therefore still remains: Is the difference of .8 on a 10-point scale enough to support rejection of the null hypothesis? The answer lies in the result of the t test, and for this we need to do some