- Random Numbers and Probability Distributions
- Casino Royale: Roll the Dice
- Normal Distribution
- The Student Who Taught Everyone Else
- Statistical Distributions in Action
- Hypothetically Yours
- The Mean and Kind Differences
- Worked-Out Examples of Hypothesis Testing
- Exercises for Comparison of Means
- Regression for Hypothesis Testing
- Analysis of Variance
- Significantly Correlated
Analysis of Variance
Analysis of variance, ANOVA, is the prescribed method of comparing means across groups of three or more. The null hypothesis in this case states that the average values do not differ across the groups. The alternative hypothesis states that at least one mean value is different from the rest.
I use the F-test for ANOVA. If the probability (p-value) associated with the F-test is greater than the threshold value, which is usually .05 for the 95% confidence level, we fail to reject the null hypothesis. In instances where the probability value for the F-test is less than .05, we reject the null hypothesis. In such instances, we conclude that at least one mean value differs from the rest.
I will repeat the comparison of means for the three age groups using the ANOVA test. The R code and the resulting output (see Figure 6.38) follow.
Figure 6.38 ANOVA output for influence of age on teaching evaluations
Note that the value reported under Pr(>F) is 0.0998, which is greater than 0.05. Thus, we fail to reject the null hypothesis and conclude that the teaching evaluations do not differ by age groups.
Let us test the average teaching evaluations for a discretized variable for beauty, which in raw form is a continuous variable. I convert the continuous variable into three categories namely: low beauty, average looking, and good looking. The R code and the resulting output (see Figure 6.39) follow.
x$f.beauty<-cut(x$beauty, breaks=3) x$f.beauty<-factor(x$f.beauty, labels=c("low beauty", "average looking", "good looking")) cbind(mean.eval=tapply(x$eval,x$f.beauty,mean), observations=table(x$f.beauty)) summary(aov(eval~f.beauty, data=x))
Figure 6.39 ANOVA output for influence of beauty on teaching evaluations
The probability value associated with the F-test is 0.0276, which is less than .05, our threshold value. I therefore reject the null hypothesis and conclude that teaching evaluations differ by students’ perception of instructors’ appearance.