To solve this, we use the relationship between the total variance in an ANOVA and the variance components associated with the different contrasts. The overall $F$-value is derived by partitioning the total between-group variance into two components: one related to the contrast (no pet vs. average of the three pet groups) and the other related to the differences among the pet groups.

Step-by-Step Solution:

1. Information Provided:

   • $F_{\text{contrast}} = 5.50$: No pet vs. average of three pet groups.
   • $F_{\text{omnibus}} = 4.50$: Differences among the three pet groups.
   • Same mean square within ($MSW$) for all calculations.

2. Formula for Total $F$-value: The total $F$-value for the overall between-group effect is related to the sum of the sums of squares (SS) for the two components, as follows: $F_{\text{total}} = \frac{\text{SS}_{\text{contrast}} + \text{SS}_{\text{omnibus}}}{MSW}$ Each $F$-value can be rewritten in terms of sums of squares: $F = \frac{\text{SS}}{MSW}$ Therefore, the total $F$-value can be expressed as: $F_{\text{total}} = F_{\text{contrast}} + F_{\text{omnibus}}$

3. Compute $F_{\text{total}}$: Using the given values: $F_{\text{total}} = 5.50 + 4.50 = 10.00$

Final Answer:

The overall $F$-value is: $\boxed{10.00 \, \text{(d)}}$

Let me know if you'd like further clarification or details on this calculation! 😊

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Related Questions:

1. What are the assumptions underlying ANOVA that are critical for valid results?
2. How does increasing the sample size ($n$) per group affect the $F$-value in this experiment?
3. What would happen to the $F_{\text{total}}$ if the groups were unequal in size?
4. How can we interpret $F_{\text{contrast}}$ and $F_{\text{omnibus}}$ in the context of real-world experimental design?
5. What are the post hoc tests used when differences are found among the three pet conditions?

Tip:

When partitioning variance in ANOVA, remember that the total $SS$ between groups is additive, so the corresponding $F$-values are also additive if they use the same $MSW$.