To study the effects of changes in nutrient cycles on bromeliads in Costa Rica, scientists designed an experiment to use with the one-way ANOVA procedure. Flowering plants growing in Costa Rica were randomly assigned to each of four treatment groups (nitrogen, phosphorus, both, and control) with eight flowering plants per group. You are going to complete the partial ANOVA table below in a series of questions (one part per question). Round all answers to 2 decimal places if necessary. What is the F test statistic?

Source of Variation Degrees of Freedom, DF Sum of Squares Mean Squares F test statistic
Treatment 2232
Error
Total 11761

Question

To study the effects of changes in nutrient cycles on bromeliads in Costa Rica, scientists designed an experiment to use with the one-way ANOVA procedure.  Flowering plants growing in Costa Rica were randomly assigned to each of four treatment groups (nitrogen, phosphorus, both, and control) with eight flowering plants per group.  You are going to complete the partial ANOVA table below in a series of questions (one part per question). Round all answers to 2 decimal places if necessary. What is the F test statistic?

Source of Variation	Degrees of Freedom, DF	Sum of Squares	Mean Squares	F test statistic
Treatment		2232		
Error				
Total		11761

BlackTom AI · Accepted Answer

For one-way ANOVA the F statistic is the ratio of the treatment mean square to the error mean square: F = MSTreatment / MSError.

1) Relevant knowledge points and concepts
- Degrees of freedom: df_treatment = k − 1 where k = number of groups (4), so df_treatment = 3. Total df = N − 1 where N = total observations (4 groups × 8 each = 32), so total df = 31. df_error = total df − df_treatment = 31 − 3 = 28.
- Sum of squares: SSTotal = 11761 (given). SSTreatment = 2232 (given). SSE = SSTotal − SSTreatment.
- Mean squares: MS = SS / df. F = MSTreatment / MSError.

2) Step-by-step calculations
- Compute SSE: SSE = 11761 − 2232 = 9529.
- Degrees of freedom: df_treatment = 3; df_error = 28; df_total = 31.
- Compute MSTreatment: MST = 2232 / 3 = 744.00.
- Compute MSError: MSE = 9529 / 28 = 340.3214286 → rounded to 2 decimals = 340.32.
- Compute F: F = 744.00 / 340.3214286 = 2.1862… → rounded to 2 decimals = 2.19.

3) Intermediate steps shown above include each division and subtraction.

4) Why the correct answer is correct
The F statistic measures treatment variance relative to within-group variance. Using the correct SS and df values yields MST = 744 and MSE ≈ 340.32, so F = 744/340.32 ≈ 2.19, matching the correct computation.

5) Why alternative calculations would be wrong
Common mistakes: using wrong df (e.g., dividing by 4 instead of 3 for treatment), using N instead of df when computing MS, or subtracting incorrectly from total SS. Any of those produce different (incorrect) F values.

6) Conclusion
The calculated F test statistic is 2.19, which matches the provided answer.

类似问题

Using the same dataset, Flaye now decides to investigate whether concert attendance differs by the day of the concert (Friday, Saturday, or Sunday). Assuming all assumptions are met, what statistical test would you suggest they run?

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