Question

1. a researcher reports an f - ratio with ( d f_{\text {between }}=3 ) and ( d f_{\text {within }}=40 ) for an independent - measures anova.

a. how many treatment conditions were compared in the experiment?
b. how many subjects participated in the experiment?
c. use appendix b to find the critical value for f. use ( alpha=.05 ).
d. what is the critical region for ( alpha=.01 )?

Explanation:

Step1: Calculate number of treatment conditions

The formula for \(df_{between}\) in an independent - measures ANOVA is \(df_{between}=k - 1\), where \(k\) is the number of treatment conditions. Given \(df_{between}=3\), we solve the equation \(k-1 = 3\).

$$k=3 + 1$$

Step2: Calculate number of subjects

The formula for \(df_{within}=N - k\), where \(N\) is the total number of subjects. We know \(df_{within}=40\) and \(k = 4\) (from part a). Then \(N=df_{within}+k\)

$$N=40 + 4$$

Step3: Find critical value for \(\alpha = 0.05\)

For an \(F\) - distribution with \(df_{between}=3\) (numerator degrees of freedom) and \(df_{within}=40\) (denominator degrees of freedom), using the \(F\) - table (Appendix B).

Step4: Find critical region for \(\alpha=0.01\)

Using the \(F\) - table with \(df_{between}=3\) and \(df_{within}=40\) for \(\alpha = 0.01\)

Answer:

a. \(k = 4\) treatment conditions.
b. \(N=44\) subjects.
c. The critical value for \(F\) with \(\alpha = 0.05\), \(df_{1}=3\), \(df_{2}=40\) is \(F = 2.84\) (from \(F\) - table).
d. The critical value for \(F\) with \(\alpha=0.01\), \(df_{1}=3\), \(df_{2}=40\) is \(F = 4.31\). So the critical region is \(F\geq4.31\)