Statistical Analysis for Education and Psychology Researchers

This is evaluated using equation 8.15 for the pooled variance estimate.

3 t-ratio

As in the separate variance estimate t is the ratio of the difference in

means to the standard error of the difference,

t=−3.034

The associated degrees of freedom for a pooled variance estimate is simply n 1 + n 2
−2=df=18.

Interpretation

Assuming a two-tailed test because no specific difference was specified by the researcher
and a 5 per cent significance level, interpretation is the same as in the previous example
except that the degrees of freedom are a whole number and can therefore be looked up in
a table of the t-distribution. The p-value for a t of −3.034 is found by comparing the
observed t-ratio (t=−3.034) to a critical t-value with 18 df in Table 3 (Appendix A4). For
18 df and 5 per cent two-tailed the critical t is 2.101. Since the observed value exceeds
the critical value we can conclude that there is evidence of a significant age difference in
correct score for subtraction tasks (t=−3.034, df=18, p<0.05).
In these two worked examples the separate and pooled variance estimates of the t-
ratios are similar because the sample variances are not very different. The negative t-ratio
is attributable to the larger mean being subtracted from the smaller mean (in both
examples the mean for 7-year-olds was subtracted from the mean for 6-year-olds). The
response variable would be checked for normality in the usual way with a normal
probability plot.

Confidence Interval (CI) for Difference in Means

It is useful to present an estimate of the plausible range of population mean differences as
well as the result of a hypothesis test. The 95 per cent CI for the difference in means is
given by,

Confidence

interval

difference

between

means

Inferences involving continuous data 301