Question

4. ANOVA calculations and rejection of the null hypothesis The following table summarizes the results of a study on SAT prep courses, comparing SAT scores of students in a private preparation dlass, a high school preparation dlass, and no preparation dass. Use the information from the table to answer the remaining questions. Number of Observations Treatment Sample Mean Sum of Squares (SS) Private prep dlass 40 610 97,500.00 High school prep class 40 600 101,400.00 No prep class 40 590 111,150.00 Using the data provided, complete the partial ANOVA summary table that follows. (Hint: T, the treatment total, can be calculated as the sample mean times the number of observations. G, the grand total, can be calculated from the values of T once you have calculated them.) Mean Square (MS) Sum of Squares (SS) df Source Between treatments Within treatments ANOVA summary tables typically have a Total" row not included in the partial table you just completed. Which of the following is a possible reason for including this row?

Answer 1

	Private	High school	No prep	Total
Count, n	40	40	40	120
Mean	610	600	590
Sum of square,SS	97500	101400	111150	0

Grand mean = (610+600+590)/3 = 600

Number of treatment, k = 3

Total sample Size, N =120

df(between) = k-1 =2

df(within) = N-k =117

df(total) = N-1 =119

SS(between) = (x̅1)²*n1 + (x̅2)²*n2 + (x̅3)²*n3 - (Grand Mean)²*N = 8000

SS(within) = SS1 + SS2 + SS3 = 310050

SS(total) = SS(between) + SS(within) =318050

MS(between) = SS(between)/df(between) =4000

MS(within) = SS(within)/df(within) =2650

Source of variation	SS	df	MS
Between Groups	8000	2	4000
Within Groups	310050	117	2650

The SStotal is sometimes easier to calculate than SSbetween . Since SSwithin + SSbetween = SStotal, you can use SStotal to calculate SS between.

In ANOVA, the F test statistic is the ration of the between-treatments variance and the within-treatments variance. The value the F test statistic is:

F = MS(between)/MS(within) = 1.5094

When the null hypothesis is true, The F test statistic is around 1.

When the null hypothesis is false, The F test statistic is most likely to be more than the F critical value at given significance value.

In general you should reject the null hypothesis for F value greater than the F critical value at given significance value.