To compute SS between (Sum of Squares Between Groups), we follow these steps:

Formula:

$SS_{\text{between}} = \sum_{k=1}^g n_k (\bar{X}_k - \bar{X}_{\text{total}})^2$

Where:

• $g$: Number of groups (3 in this case: Morning, Afternoon, Night)
• $n_k$: Number of participants in each group (5 each)
• $\bar{X}_k$: Mean of each group
• $\bar{X}_{\text{total}}$: Overall mean
• $SS_{\text{between}}$: Variance due to differences between groups

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Step-by-step Calculation:

1. Compute the group means ($\bar{X}_k$):

• Morning Mean ($\bar{X}_{\text{morning}}$): $\bar{X}_{\text{morning}} = \frac{10 + 8 + 7 + 5 + 9}{5} = \frac{39}{5} = 7.8$

• Afternoon Mean ($\bar{X}_{\text{afternoon}}$): $\bar{X}_{\text{afternoon}} = \frac{8 + 8 + 9 + 7 + 7}{5} = \frac{39}{5} = 7.8$

• Night Mean ($\bar{X}_{\text{night}}$): $\bar{X}_{\text{night}} = \frac{6 + 7 + 5 + 4 + 5}{5} = \frac{27}{5} = 5.4$

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2. Compute the overall mean ($\bar{X}_{\text{total}}$):

$\bar{X}_{\text{total}} = \frac{\text{Sum of all weights}}{\text{Total number of participants}}$

Sum of all weights: $10 + 8 + 7 + 5 + 9 + 8 + 8 + 9 + 7 + 7 + 6 + 7 + 5 + 4 + 5 = 105$

$\bar{X}_{\text{total}} = \frac{105}{15} = 7.0$

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3. Compute $SS_{\text{between}}$:

For each group: $SS_{\text{between}} = n_k (\bar{X}_k - \bar{X}_{\text{total}})^2$

• Morning: $SS_{\text{morning}} = 5 (7.8 - 7.0)^2 = 5 (0.8)^2 = 5 (0.64) = 3.2$

• Afternoon: $SS_{\text{afternoon}} = 5 (7.8 - 7.0)^2 = 5 (0.8)^2 = 5 (0.64) = 3.2$

• Night: $SS_{\text{night}} = 5 (5.4 - 7.0)^2 = 5 (-1.6)^2 = 5 (2.56) = 12.8$

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4. Add the components:

$SS_{\text{between}} = SS_{\text{morning}} + SS_{\text{afternoon}} + SS_{\text{night}}$

$SS_{\text{between}} = 3.2 + 3.2 + 12.8 = 19.2$

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Final Answer:

$SS_{\text{between}} = 19.2$

Would you like further details or clarification? Here are five follow-up questions and one tip:

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1. What are the differences between $SS_{\text{between}}$ and $SS_{\text{within}}$?
2. How is $SS_{\text{total}}$ related to $SS_{\text{between}}$ and $SS_{\text{within}}$?
3. Why do we square the deviations when computing SS values?
4. Can you calculate the F-ratio from the $SS_{\text{between}}$ and $SS_{\text{within}}$?
5. How would the $SS_{\text{between}}$ change if one group had a higher mean?

Tip: Always check that your calculations are consistent with the group sample sizes and overall totals to avoid errors.