To test the hypothesis that the treatment means are equal, we can use Analysis of Variance (ANOVA). In order to compute ANOVA, we need to calculate the following sums of squares:

1. SST (Sum of Squares for Treatment) - This represents the variability between the treatment groups.
2. SSE (Sum of Squares for Error) - This represents the variability within each treatment group (i.e., within-group variability).
3. SS Total - This represents the total variability in the data, combining both SST and SSE.

Steps to compute these values:

1. Grand Mean (Overall Mean)

We first calculate the grand mean, which is the mean of all the observations across all treatments.

2. Treatment Means

Next, we compute the mean for each treatment group.

3. SST (Sum of Squares for Treatment)

This is calculated as: $SST = \sum_{i=1}^{k} n_i (\bar{X}_i - \bar{X})^2$ Where:

• $n_i$ = number of observations in group $i$
• $\bar{X}_i$ = mean of group $i$
• $\bar{X}$ = grand mean
• $k$ = number of treatments

4. SSE (Sum of Squares for Error)

This is calculated as: $SSE = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (X_{ij} - \bar{X}_i)^2$ Where:

• $X_{ij}$ = individual observation in group $i$

5. SS Total (Total Sum of Squares)

This is the total variability and is calculated as: $SS\text{ Total} = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (X_{ij} - \bar{X})^2$ Alternatively: $SS\text{ Total} = SST + SSE$

Let me now compute these values for the data provided.

Data:

• Treatment 1: 3, 2, 5, 1, 3, 1
• Treatment 2: 9, 6, 5, 6, 8, 5, 4, 7, 6, 4
• Treatment 3: 6, 3, 5, 5, 5, 4, 1, 5

I'll calculate the required sums of squares.The computed sums of squares are:

• SST (Sum of Squares for Treatment): 46.96
• SSE (Sum of Squares for Error): 53.00
• SS Total (Total Sum of Squares): 99.96

Would you like more details on this process or how to proceed with the hypothesis test?

Here are 5 questions that extend this information:

1. How do we calculate the degrees of freedom for SST and SSE in ANOVA?
2. How do we compute the F-statistic in ANOVA from SST and SSE?
3. What assumptions must be met for an ANOVA test to be valid?
4. Can you explain the interpretation of the F-statistic in hypothesis testing?
5. What is the difference between one-way and two-way ANOVA?

Tip: In ANOVA, the sum of squares can be thought of as a measure of variance or dispersion in the data, divided into "explained" (SST) and "unexplained" (SSE) parts.