4. The data below are from an independent-measures experiment comparing three different treatment conditions.
Treat. 1 Treat 2 Treat 3
0 1 4
0 4 3 G=24
0 1 6
2 1 3 Ó X² = 92
T=2 T=6 T=16
SS=3 SS=9 SS=6
Use an analysis of variance with á of .05 to determine whether these data indicate any significant differences among the treatments.
Answer:
Source SS df MS F
Between
Within
Total
Here are a few hints to help you fill in your ANOVA table:
SS total = SS between + SS within
To calculate df between:
k - 1
Note: k = number of levels.
�
To calculate df within:
N - k
Note: N = total number of values in all levels.
df total = df between + df within
To calculate MS between:
SS between/df between
To calculate MS within:
�SS within/df within
To calculate F-ratio:
MS between/MS within
I hope this will help. It looks like you already have the values needed to do the calculations for each of your groups. Remember to check the appropriate table for your critical value to compare to your F-ratio. Once you do the comparison, you will either fail to reject the null or you will reject the null. You can then form your conclusions.
To determine whether there are any significant differences among the treatment conditions, we need to conduct an analysis of variance (ANOVA) using the given data. Here are the steps to find the solution:
Step 1: Calculate the total sum of squares (SS total) using the formula:
SS total = Σ(Xij - X..)²
where Xij represents each individual score and X.. represents the grand mean. From the given data, we can calculate SS total as:
SS total = 3² + 9² + 6² = 90
Step 2: Calculate the between-group sum of squares (SS between) using the formula:
SS between = Σ(nj * (X.j - X..)²)
where nj represents the number of scores in each treatment condition, X.j represents the mean of each treatment condition, and X.. represents the grand mean. From the given data, we can calculate the values as follows:
Treat 1: nj = 3, X.j = 0, SS1 = 3 * (0 - 4)² = 48
Treat 2: nj = 3, X.j = 3, SS2 = 3 * (3 - 4)² = 3
Treat 3: nj = 3, X.j = 10, SS3 = 3 * (10 - 4)² = 108
SS between = SS1 + SS2 + SS3 = 48 + 3 + 108 = 159
Step 3: Calculate the within-group sum of squares (SS within) using the formula:
SS within = ΣΣ(Xij - X.j)²
where Xij represents each individual score and X.j represents the mean of each treatment condition. From the given data, we can calculate the values as follows:
Treat 1: (0 - 0)² + (0 - 0)² + (2 - 0)² = 4
Treat 2: (1 - 4)² + (4 - 4)² + (1 - 4)² = 9
Treat 3: (4 - 6)² + (3 - 6)² + (3 - 6)² + (6 - 6)² = 6
SS within = 4 + 9 + 6 = 19
Step 4: Calculate the degrees of freedom (df) for between and within:
df between = number of treatment conditions - 1 = 3 - 1 = 2
df within = total number of scores - number of treatment conditions = 9 - 3 = 6
Step 5: Calculate the mean square (MS) for between and within:
MS between = SS between / df between = 159 / 2 = 79.5
MS within = SS within / df within = 19 / 6 = 3.16 (rounded to two decimal places)
Step 6: Calculate the F ratio using the formula:
F = MS between / MS within
F = 79.5 / 3.16 = 25.16 (rounded to two decimal places)
Step 7: Determine the critical F value at the given significance level (α). In this case, α = 0.05.
Using a table or statistical software, the critical F value for df between = 2 and df within = 6 at α = 0.05 is approximately 5.14.
Step 8: Compare the calculated F value with the critical F value. If the calculated F value is greater than the critical F value, it indicates significant differences among the treatment conditions.
In this case, the calculated F value (25.16) is greater than the critical F value (5.14), which suggests that there are significant differences among the treatment conditions.
Now, let's fill in the table:
Source | SS | df | MS | F
-----------------------------------
Between | 159 | 2 | 79.5 | 25.16
Within | 19 | 6 | 3.16 |
Total | 178 | 8 | |
Based on the analysis of variance (ANOVA) results, we can conclude that these data indicate significant differences among the treatment conditions.