The table provided seems to be part of an ANOVA (Analysis of Variance) problem. In this case, we're asked to calculate the sum of squares (SS) among treatments, which can be derived using the relationship between the mean square (MS), the degrees of freedom (DF), and the F-value.

Here’s the breakdown of the relevant formulas and steps:

Formula for F-statistic:

$F = \frac{MS_{treatments}}{MS_{error}}$ Where:

• $MS_{treatments}$ = Mean Square among treatments = $\frac{SS_{treatments}}{DF_{treatments}}$
• $MS_{error}$ is already given from the table (error term).

Steps:

1. From the table, we know:

   • $DF_{treatments} = 4$
   • $F = 1.56$
   • $SS_{error} = 3972$
   • $DF_{total} = 16$, so $DF_{error} = 16 - 4 = 12$ (since total DF = treatments DF + error DF)

2. First, calculate $MS_{error}$: $MS_{error} = \frac{SS_{error}}{DF_{error}} = \frac{3972}{12} = 331$

3. Next, using the F-statistic: $F = \frac{MS_{treatments}}{MS_{error}} \quad \Rightarrow \quad 1.56 = \frac{MS_{treatments}}{331}$ $MS_{treatments} = 1.56 \times 331 = 516.36$

4. Finally, calculate $SS_{treatments}$: $SS_{treatments} = MS_{treatments} \times DF_{treatments} = 516.36 \times 4 = 2065.44$

Answer:

The sum of squares among treatments is $\mathbf{2065.44}$.

Would you like more details or have any other questions?

Here are 5 related questions for further exploration:

1. How is the F-statistic used in hypothesis testing?
2. What assumptions must be met for ANOVA to be valid?
3. How do you interpret the significance of the F-statistic?
4. What is the total sum of squares in an ANOVA?
5. How does increasing the number of treatments affect the degrees of freedom?

Tip: In ANOVA, the degrees of freedom (DF) are crucial for calculating mean squares (MS), which are used to determine the F-statistic for significance testing. Always double-check DF for treatments and error terms.