For the 2014 GSS, Table 12.32 shows sample means of political ideology (higher values being more conservative), classified by gender and by race, for those over 50 in age. For H0: no interaction, software reports F = 21.7, df1 = 1 and df2 = 1081, and P-value <0.001.

(a) Suppose that instead of the two-way ANOVA, you performed separate one-way ANOVAs for gender and for race. Suppose the ANOVA for gender does not show a significant effect. Explain how this could happen, even though the two-way ANOVA implies that the gender effect varies by race. (Hint: Will the overall sample means for females and males be more similar than they are for each race?)

(b) Summarize what you would learn about the gender effect from a two-way ANOVA that you would fail to learn from a one-way ANOVA.