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 degrees of freedom total?

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 degrees of freedom total?

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 a one-way ANOVA the total degrees of freedom is N − 1, where N is the total number of observations.

Identify relevant values:
- Number of groups (k) = 4 (nitrogen, phosphorus, both, control).
- Observations per group = 8.
- Total observations N = k × 8 = 4 × 8 = 32.

Step-by-step reasoning and intermediate calculations:
1. Compute total sample size: N = 32.
2. Total degrees of freedom: DF_total = N − 1 = 32 − 1 = 31.
3. For completeness, the other ANOVA degrees of freedom are:
   - DF_treatment = k − 1 = 4 − 1 = 3.
   - DF_error (within) = N − k = 32 − 4 = 28.
4. Using the given sums of squares you can check consistency:
   - SS_treatment = 2232 (given).
   - SS_total = 11761 (given).
   - SS_error = SS_total − SS_treatment = 11761 − 2232 = 9529.
   - MS_treatment = SS_treatment / DF_treatment = 2232 / 3 = 744.00.
   - MS_error = SS_error / DF_error = 9529 / 28 = 340.32 (rounded).
   - F = MS_treatment / MS_error = 744.00 / 340.32 ≈ 2.19 (rounded).
   These supplementary calculations confirm the DF allocations are consistent.

Why 31 is correct and other values are incorrect:
- Some might mistakenly use N = 32 as DF_total; this is wrong because DF_total is N − 1.
- Using k − 1 (3) or N − k (28) corresponds to treatment or error DF, not total DF.
- The correct formula and the computed N lead uniquely to 31.

Conclusion: The total degrees of freedom is 31, which matches the provided answer.

类似问题

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单