Statistical Analysis for Education and Psychology Researchers

Interpretation

With a significance level of 5 per cent, a two-tailed test, and df=8 (10–2), the critical t-
value shown in Table 3, Appendix A4 is 2.306. Since the calculated value falls beyond
this critical value, 5.716>2.306, we can reject the null hypothesis and conclude that a
simple linear model describes the predictive relationship between the variables MATHS
and SMATHS. With a two-parameter regression model, intercept and one explanatory
variable, the null hypothesis, H 0 : β 1 =0 and test statistic value, t=5.716 are equivalent to
the null hypothesis of model fit given by the F-statistic, because F=t^2. The t-value is
equal to 32.6800.5=5.717 which is the value obtained earlier (allowing for rounding
errors).
Use of the t-statistic for testing model fit when there is one explanatory variable is
preferable to the F-statistic because it allows for one-sided alternative hypotheses to be
tested, namely H 1 : β 1 >0 or H 1 : β 1 <0, where t has n−2 df. The rejection region for the two-
sided test is the absolute value of t>t 1 −α/2, where t 1 −α/2 is the t-value such that
α/2=p(t>t 1 −α/2). For a one sided test, t>t 1 −α where t 1 −α is the t-value such that α=p(t>t 1 −α).

Confidence interval for the regression slope

Whenever possible we should calculate a confidence interval as well as a p-value for a
test of significance. Confidence intervals for a regression slope, similar to the tests of
significance, are based on the sampling distribution of t (the slope estimate b 1 is
approximately normally distributed). The 95 per cent confidence interval for the
population value of the regression slope is evaluated using equation 8.2. For a 95 per cent
CI around the slope relating teachers’ estimate of maths ability to pupils’ maths

attainment, with the values alpha=0.05, df=n−2 (=8), b 1 = 3.7019, and
t 1 −α/2=t0.025=2.306, when we substitute these into equation 8.2 we obtain:
3.7019−(2.306×0.6475) to 3.7019+(2.306×0.6475)^

The 95 per cent confidence interval estimates are 2.2087 to 5.1950.

Interpretation

The confidence interval does not include zero and all the values are positive so it is
reasonable to conclude that we would be 95 per cent confident of finding a positive
relationship in the population between teacher estimated maths ability and pupils’ maths
attainment on the standardized test. (This conclusion would be reasonable if we had
obtained these results with a larger sample.) We would expect the mean standardized test
score to increase with every unit increase in teachers’ estimated ability scores, this
increase may range from 2.209 to 5.200 with the average unit increase being 3.70.
However, this is a rather large confidence interval width and is likely to be attributable to
the small sample size (n=10). With a larger sample the confidence interval width is likely
to be reduced.

Statistical analysis for education and psychology researchers 266