Suppose you gave a skill test to 3 different PE classes and want to know if they differ in terms of their scores. The skill tests scores are as follows:

Class 1: 23, 21, 19, 22
Class 2: 18, 19, 21, 18
Class 3: 21, 20, 19, 22

Conduct an ANOVA and report the test statistic rounded to 2 decimal places.

To conduct an ANOVA (Analysis of Variance) and find the test statistic, you can follow these steps:

Step 1: Calculate the mean score for each class.
- For Class 1: (23 + 21 + 19 + 22) / 4 = 21.25
- For Class 2: (18 + 19 + 21 + 18) / 4 = 19
- For Class 3: (21 + 20 + 19 + 22) / 4 = 20.5

Step 2: Calculate the grand mean by averaging all the scores.
- Grand Mean: (21.25 + 19 + 20.5) / 3 = 20.58 (rounded to 2 decimal places)

Step 3: Calculate the sum of squares between groups (SSB).
- SSB measures the variability between group means.
- SSB = (n1 * (mean1 - grand mean)^2) + (n2 * (mean2 - grand mean)^2) + (n3 * (mean3 - grand mean)^2)
- SSB = (4 * (21.25 - 20.58)^2) + (4 * (19 - 20.58)^2) + (4 * (20.5 - 20.58)^2)
- SSB = 1.224 + 2.3056 + 0.0584 = 3.588 (rounded to 3 decimal places)

Step 4: Calculate the sum of squares within groups (SSW).
- SSW measures the variability within each group.
- SSW = (n1 - 1) * s1^2 + (n2 - 1) * s2^2 + (n3 - 1) * s3^2
where s1, s2, s3 are the sample variances within each group.
- To find the sample variance within each group, calculate the sum of squared deviations for each group.
- For Class 1: ((23 - 21.25)^2 + (21 - 21.25)^2 + (19 - 21.25)^2 + (22 - 21.25)^2) / (4 - 1) = 0.9167
- For Class 2: ((18 - 19)^2 + (19 - 19)^2 + (21 - 19)^2 + (18 - 19)^2) / (4 - 1) = 1
- For Class 3: ((21 - 20.5)^2 + (20 - 20.5)^2 + (19 - 20.5)^2 + (22 - 20.5)^2) / (4 - 1) = 0.75
- SSW = (4 - 1) * 0.9167 + (4 - 1) * 1 + (4 - 1) * 0.75
- SSW = 2.7501 + 3 + 2.25 = 7.0001 (rounded to 4 decimal places)

Step 5: Calculate the degrees of freedom (df).
- df_between = k - 1 = 3 - 1 = 2
(where k is the number of groups, which is 3 in this case)
- df_within = N - k = 12 - 3 = 9
(where N is the total number of observations, which is 12 in this case)

Step 6: Calculate the mean square between (MSB).
- MSB = SSB / df_between
- MSB = 3.588 / 2 = 1.794 (rounded to 3 decimal places)

Step 7: Calculate the mean square within (MSW).
- MSW = SSW / df_within
- MSW = 7.0001 / 9 = 0.7778 (rounded to 4 decimal places)

Step 8: Calculate the F-statistic.
- F = MSB / MSW
- F = 1.794 / 0.7778 = 2.309 (rounded to 3 decimal places)

So, the test statistic for the ANOVA is 2.309 (rounded to 2 decimal places).