A researcher has collected statistics pretest scores for students representing 3 different age groups

18-30 31-45 45 and up
3 5 10
7 10 2
9 9 8
6 8 8
4 6 5

Do hypothesis test using ANOVA to find if significant difference. Use alpha .05. All hypothesis steps and find degrees of freedom and critical values, calulate test statistic and effect size and write results in apa style.

Use Scheffe posttest if appropriate.

To perform an analysis of variance (ANOVA) to determine if there is a significant difference among the pretest scores of students in the three different age groups, follow these steps:

Step 1: Formulate Hypotheses
- Null Hypothesis (H0): There is no significant difference in the mean pretest scores for the three age groups.
- Alternative Hypothesis (Ha): There is a significant difference in the mean pretest scores for the three age groups.

Step 2: Choose a Significance Level
- In this case, the alpha level is given as 0.05, which means we set the significance level at 5%.

Step 3: Compute the ANOVA Test Statistic
- Calculate the sum of squares between groups (SSB), the sum of squares within groups (SSW), and the degrees of freedom for each.
- SSB represents the variation between groups, while SSW represents the variation within groups.

The following table summarizes the calculations:

age group | n | mean | SS | df | MS
--------------------------------------------------
18-30 | 5 | 5.4 | 8 | 2 | 4
31-45 | 5 | 6.8 | 4 | 2 | 2
45 and up | 5 | 7.6 | 6 | 2 | 3
Total | 15 | - | 18 | 12 | -

- Total sum of squares (SST) = Sum of squares between (SSB) + Sum of squares within (SSW)

Step 4: Calculate the F-Statistic
- Compute the F-statistic using the formula F = MSB / MSW, where MSB is the mean squares between and MSW is the mean squares within.

Using the provided data, MSB = 3 and MSW = 2.67.

Step 5: Determine the Critical Value
- Find the critical value by consulting the F-distribution table or using statistical software.
- The degrees of freedom for SSB and SSW are 2 and 12, respectively.

Using these degrees of freedom and an alpha level of 0.05, the critical value is 3.89.

Step 6: Make a Decision
- Compare the calculated F-statistic with the critical value.
- If the calculated F-statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

In this case, the calculated F-statistic (3/2.67) is approximately 1.12, which is less than the critical value of 3.89.

Thus, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant difference in the mean pretest scores among the three age groups.

Step 7: Calculate Effect Size
- One commonly used effect size measure for ANOVA is eta-squared (η²).

Using the formula η² = SSB / SST, where SSB is the sum of squares between groups and SST is the total sum of squares, we find η² = 18/18 = 1.

The effect size, represented by η², is 1, indicating a large effect.

Step 8: Scheffe Post Hoc Test (if appropriate)
- The Scheffe post hoc test can be used to compare the means of all possible pairs of groups if the ANOVA result is significant and you want to determine which specific groups differ significantly from each other.
- However, since the ANOVA result in this case is not significant, the Scheffe post hoc test is not applicable.

APA-Style Reporting:
After analyzing the data using ANOVA, it was determined that there was not a significant difference in mean pretest scores among the three age groups, F(2, 12) = 1.12, p > .05. The effect size, measured by eta-squared (η² = 1), indicated a large effect. Thus, the null hypothesis of no significant difference in pretest scores for the different age groups was not rejected.