An introduction to analysis of variance

Do prep classes effectively increase SAT scores? The following study cited in a major newspaper claims that they do (but not by a substantial amount): "A new study... shows that [SAT prep] classes are effective in boosting scoressomewhat. 'Private classes increased scores an average of 60 points, while less specialized high school courses added 30 points,' said Claudia Buchmann, coauthor of the study and associate professor of sociology at Ohio State University." (Source: Seema Mehta, "Students Believe in the SAT,"Los Angeles Times,March 1, 2008)

A researcher would like to compare the effects on students' SAT scores from the following three types of preparation: taking a private SAT prep class offered by ScoreExcel, a company offering exam reviews; taking a less specialized prep class offered by a local high school; and not taking any prep class.

Consider two statistical studies called Study A and Study B. In Study A, 120 high school seniors who had not taken the SAT were randomly divided into three groups of 40 students each. Students in the first group were assigned to take the prep class offered by ScoreExcel, those in the second group were assigned to take the prep class offered by the local high school, and those in the third group were not assigned to any prep class. At the end of the study, all the students took the SAT, and their scores were recorded.

In Study B, samples of students who had taken the SAT were randomly selected from populations defined by the three different types of preparation: those taking the prep class offered by ScoreExcel; those taking the prep class offered by the local high school; and those not taking any prep class. Each sample consisted of 40 high school seniors, and their SAT scores were recorded.

In both studies, the dependent variable is , and the independent variable is . Study A is an study, and Study B is an study.

Analysis of variance (ANOVA) is a statistical procedure to determine whether there are differences among the means of the populations defined by the treatments. In ANOVA, the null hypothesis is:

H: At least two of the population means are equal

H: All the population means are different

H: All the population means are equal

H: At least two of the population means are not equal

The alternative hypothesis is:

HaHa: All the population means are equal

HaHa: All the population means are different

HaHa: At least two of the population means are equal

HaHa: At least two of the population means are not equal

To perform ANOVA, you assume the following: for each population, the dependent variable has ____ distribution; the variance of the dependent variable ______ ; and the observations are _____ . The assumption of independence may be satisfied in an experimental study with ____ design or in an observational study with____ sampling.

Assume that the data collected for both Study A and Study B are identical. The following table summarizes the results. Use the information from the table to answer the remaining questions.

Treatment, j	Number of Observations, njj	Sample Mean,xjxj	Sample Variance,sj2sj2
Private Prep Class	40	690	500
High School Prep Class	40	680	520
No Prep Class	40	640	570

When the study has a balanced design, the between-treatments estimate of is given by _____ . Therefore, it is an estimate of based on the variability of the ____ . For the data above, the between-treatments estimate of is ____ .

When the study has a balanced design, the within-treatments estimate of is given by_____ . Therefore, it is an estimate of based on the variability of the _____ . For the data above, the within-treatments estimate of is _____ .

In ANOVA, the F test statistic is the ______ of the between-treatments estimate and the within-treatments estimate of the population variance . The value of the F test statistic for both studies is_______ .

When the null hypothesis is true, both the between-treatments and within-treatments estimates of ______ the population variance. When the null hypothesis is false, the between-treatments estimate of ______ , while the within-treatments estimateof ____ .

When the null hypothesis is true, the F test statistic is ______ . When the null hypothesis is false, the F test statistic is most likely______ . Hence, you should reject the null hypothesis for ______ .

In Study A, rejection of the null hypothesis _____ a cause-and-effect relationship between type of preparation and SAT scores. In Study B, rejection of the null hypothesis a cause-and-effect relationship between type of preparation and SAT scores.