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.
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To conduct the hypothesis test using ANOVA and determine if there is a significant difference between the pretest scores for the three age groups, follow these steps:
Step 1: State the null and alternative hypotheses:
- Null hypothesis (H0): There is no significant difference between the mean pretest scores of the three age groups.
- Alternative hypothesis (Ha): There is a significant difference between the mean pretest scores of the three age groups.
Step 2: Set the significance level (alpha):
- Given that the alpha level is 0.05 (or 5%), we will use this value to determine the critical region.
Step 3: Calculate the degrees of freedom:
- Between-groups degrees of freedom = k - 1 (where k is the number of groups)
- Within-groups degrees of freedom = N - k (where N is the total number of observations)
In this case, k = 3 (age groups) and N = 15 (total number of observations), so:
- Between-groups degrees of freedom = 3 - 1 = 2
- Within-groups degrees of freedom = 15 - 3 = 12
Step 4: Calculate the sum of squares:
- Total sum of squares (SST) = sum of squared deviations of all data from the grand mean
- Between-groups sum of squares (SSB) = sum of squared deviations of group means from the grand mean
- Within-groups sum of squares (SSW) = sum of squared deviations within each group
Step 5: Calculate the mean squares:
- Between-groups mean square (MSB) = SSB / between-groups degrees of freedom
- Within-groups mean square (MSW) = SSW / within-groups degrees of freedom
Step 6: Calculate the F-statistic:
- F-statistic = MSB / MSW
Step 7: Find the critical value:
- Look up the critical value for F with the appropriate degrees of freedom in an F-distribution table or use software/tools.
Step 8: Compare the F-statistic with the critical value:
- If the F-statistic is greater than the critical value, reject the null hypothesis and conclude that there is a significant difference between the mean pretest scores of the three age groups.
- If the F-statistic is not greater than the critical value, fail to reject the null hypothesis and conclude that there is no significant difference between the mean pretest scores of the three age groups.
Step 9: Calculate the effect size:
- One commonly used effect size measure for ANOVA is eta-squared (η²).
Step 10: Report the results in APA style:
- State the degrees of freedom (DF) for between-groups and within-groups.
- Report the value of the F-statistic with corresponding p-value.
- Provide the effect size measure (η²).
Now let's go through each step with the given data:
Step 1: State the null and alternative hypotheses:
H0: There is no significant difference between the mean pretest scores of the three age groups.
Ha: There is a significant difference between the mean pretest scores of the three age groups.
Step 2: Set the significance level (alpha):
Alpha = 0.05
Step 3: Calculate the degrees of freedom:
Between-groups degrees of freedom = k - 1 = 3 - 1 = 2
Within-groups degrees of freedom = N - k = 15 - 3 = 12
Step 4: Calculate the sum of squares:
First, calculate the grand mean (overall mean) of all the pretest scores:
Grand mean (M) = (sum of all scores) / (total number of scores) = (3+7+9+6+4+5+10+2+8+8+10+8+5) / 15 = 6.8
Total sum of squares (SST) = sum of squared deviations of all data from the grand mean:
SST = (3-6.8)² + (7-6.8)² + ... + (8-6.8)² + (5-6.8)²
Between-groups sum of squares (SSB) = sum of squared deviations of group means from the grand mean:
SSB = (mean_18_30 - grand mean)² + (mean_31_45 - grand mean)² + (mean_45_and_up - grand mean)²
Within-groups sum of squares (SSW) = sum of squared deviations within each group:
SSW = (3-mean_18_30)² + ... + (5-mean_45_and_up)²
Step 5: Calculate the mean squares:
MSB = SSB / between-groups degrees of freedom
MSW = SSW / within-groups degrees of freedom
Step 6: Calculate the F-statistic:
F-statistic = MSB / MSW
Step 7: Find the critical value:
Look up the critical value for F with 2 degrees of freedom for the numerator and 12 degrees of freedom for the denominator in an F-distribution table or use software/tools.
Step 8: Compare the 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.
Step 9: Calculate the effect size:
One commonly used effect size measure for ANOVA is eta-squared (η²). It is calculated by dividing SSB by SST.
Step 10: Report the results in APA style:
Write a formal APA-style results statement based on the findings, including the degrees of freedom, the F-statistic, the p-value, and the effect size measure (η²).