To find the Sum of Squares Between Groups (SSB), we can follow these steps:

Step 1: Calculate the Mean of Each Group

• Store-Brand Group: 7, 4, 5, 2, 3 $\text{Mean}_{\text{Store-Brand}} = \frac{7 + 4 + 5 + 2 + 3}{5} = \frac{21}{5} = 4.2$

• Premium Group: 3, 2, 6, 5, 6 $\text{Mean}_{\text{Premium}} = \frac{3 + 2 + 6 + 5 + 6}{5} = \frac{22}{5} = 4.4$

• Super-Premium Group: 5, 7, 3, 5, 2 $\text{Mean}_{\text{Super-Premium}} = \frac{5 + 7 + 3 + 5 + 2}{5} = \frac{22}{5} = 4.4$

Step 2: Calculate the Grand Mean

This is the overall mean of all the values, combining all groups:

$\text{Grand Mean} = \frac{(7 + 4 + 5 + 2 + 3) + (3 + 2 + 6 + 5 + 6) + (5 + 7 + 3 + 5 + 2)}{15}$ $\text{Grand Mean} = \frac{21 + 22 + 22}{15} = \frac{65}{15} = 4.33$

Step 3: Calculate the Sum of Squares Between Groups (SSB)

$SSB = n_1 (\text{Mean}_{\text{Store-Brand}} - \text{Grand Mean})^2 + n_2 (\text{Mean}_{\text{Premium}} - \text{Grand Mean})^2 + n_3 (\text{Mean}_{\text{Super-Premium}} - \text{Grand Mean})^2$

Where $n_1$, $n_2$, and $n_3$ are the number of observations in each group (which is 5 for each group).

Substitute the values into the formula:

$SSB = 5 \left(4.2 - 4.33\right)^2 + 5 \left(4.4 - 4.33\right)^2 + 5 \left(4.4 - 4.33\right)^2$

$SSB = 5 \left( -0.13 \right)^2 + 5 \left( 0.07 \right)^2 + 5 \left( 0.07 \right)^2$

$SSB = 5 \times (0.0169) + 5 \times (0.0049) + 5 \times (0.0049)$

$SSB = 0.0845 + 0.0245 + 0.0245 = 0.1335$

Step 4: Round the Result

$SSB \approx 0.133$

Thus, the Sum of Squares Between Groups is approximately 0.133.

The correct answer is 0.133.

Would you like more details or clarification on this process?

Here are 5 follow-up questions you might find interesting:

1. What is the Sum of Squares Within Groups (SSW)?
2. How do you calculate the F-statistic for this data?
3. What does the Sum of Squares Between Groups tell you about the data?
4. How would you interpret the significance of the F-statistic in an ANOVA test for this data?
5. What is the formula for calculating the degrees of freedom in an ANOVA?

Tip: In ANOVA, the sum of squares between groups (SSB) represents the variability between the group means, and the sum of squares within groups (SSW) represents the variability within each group. Together, they help assess if the group means are significantly different.