Question

A developmental psychologist is examining the development of language skills from age 2 to age 4. Three different groups of children are obtained, one for each age, with n = 18 children in each group. Each child is given a language-skills assessment test. The resulting data were analyzed with an ANOVA to test for mean differences between age groups. The results of the ANOVA are presented in the following table. Fill in all of missing values:

Source:   SS     df     MS    F

Between:  48                        -

Within:

– – –

Total    252               -         -

Find the critical F-value using an α = .01.

What can you conclude with respect to the null hypothesis?

Calculate η2 and state whether the effect is small, medium, or large.

Answer 1

The Hypothesis:

H0: There is no difference between the means of the three treatments.\

Ha: There is a significant difference between the means of at least 2 of the treatments.

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ANOVA Calculations

Total number of groups, k = 3

Total children = 18 * 3 = 54

df between = k -1 = 3 - 1 = 2

df within = N - k = 54 - 3 = 51

SS between = 48 (given)

MS between = SS between / df betwween = 48/2 = 24

SS within = SS total - SS between = 252 - 48 = 204

MS within = SS within / df within = 204 / 51 = 4

F test = MS between / MS within = 24/4 = 6

The ANOVA Table is as below:

Source	Sums of Squares	DF	Mean Squares	F
Between	48	2	24	6
Within	204	51	4
Total	252	53

The F critical: at = 0.01, df between = 2, df within = 51; Fcritical = 5.047

The Decision: Since F test is > F critical, we Reject H0.

The Conclusion: There is sufficient evidence at the = 0.01 to conclude that there is a significant difference between at least 2 of the treatments.

Effect Size: = SS between / SS total = 48 / 252 = 0.19

small are values close to 0.02

medium are values close to 0.13

large are values close to 0.26

Here we see that = 0.19 is greater than the medium effect size, and hence we can say that the effect is large.