Statistical Analysis for Education and Psychology Researchers

which gives the value df=17.4

Interpretation

The t′-statistic represents the size of the difference between two groups, the larger the t′-
value the greater the size of the difference. In this example, the alternative hypothesis is
non-directional, H 1 :μ 1 ≠μ 2 , and the rejection region is |t|>t 1 −α/2. To be statistically
significant the calculated t′-value needs to exceed the critical t-value, with 17.4 degrees
of freedom at the 5 per cent level. Notice that the degrees of freedom is not a whole
number and a critical value for 17.4 df is not shown in the t-table in Appendix A4 (Table
3). We can find the critical value by interpolating between the two nearest values namely
df=17 (critical t=2.110) and df=18 (critical t=2.101). The critical value is approximately
2.106 with df=17.4. Using the TINV function in SAS the exact critical value is also
found to be 2.106. See the following SAS code,

data a;

alpha=0.975; ** This is equivalent to a two-tailed test

at the 5%

level**;

tcrit=round (tinv (0.975,17.4),.001); ** Round

statement rounds

value to**;

put tcrit=; ** 3 decimal

places ** ;

run;

In this example the observed t′-value of −3.0338 exceeds the critical t-value of 2.106, and
we can conclude that the means are significantly different at the 5 per cent level with a
two-tailed test. The 7-year-old pupils would seem to have a significantly higher score
than the 6-year-old pupils (t′=−3.0338, df=17.4, p<0.05).

Pooled Variance Estimate

Three steps are involved in calculating a pooled (equal variance) estimate: i) Check that
the sample variances are homogeneous and find the difference in means between the two
groups; ii) Calculate the standard error of the difference in means; and iii) evaluate t
which is the ratio of the difference in means (i above) to the standard error of this
difference in means (ii above).

1 Difference in means and check for homogeneity

The first step is the same as for the separate variance estimate, the

difference in means is −6.3. Using the approximate rule of thumb for

homogeneity of variances, (5.072/4.170 is less than 2) the variances are

similar.

2 Standard error of the difference in means (pooled variance estimate)

Statistical analysis for education and psychology researchers 300