Question

Consider the following partially completed ANOVA and give the value of (j) at α = 0.01.


Source SS df MS Fstat Ftab
Treatments 142.67 (d) 35.6675 (g) (i)
Blocks 68.14 5 (f) (h) (j)
Errors (b) 20 1.9165
Total (c) (e)

Answer 1

To solve for the values in the ANOVA table, we can use the relationships between the sums of squares (SS), degrees of freedom (df), mean squares (MS), and the F-statistic.

### Given Values:
- Treatments SS = 142.67
- Treatments MS = 35.6675
- Blocks SS = 68.14
- Blocks df = 5
- Errors df = 20
- Errors MS = 1.9165

### Calculating Degrees of Freedom:
1. **Total Degrees of Freedom (df_total):**
\[ \text{df_total} = \text{df_treatments} + \text{df_blocks} + \text{df_errors} \]

We note that:
\[\text{df_blocks} = 5\]
\[\text{df_errors} = 20\]

We need to determine \(\text{df_treatments}\) (let's call this \(d\)). The total degrees of freedom can be expressed as \(df_total = d + 5 + 20 = d + 25\).

2. **Calculating Total Sum of Squares (SS_total)**:
The total sum of squares can be calculated as:
\[ SS_{total} = SS_{treatments} + SS_{blocks} + SS_{errors} \]

We can identify that the blocks and treatments are separate components of the total, but we need to find the error SS (let's denote it as \(b\)):
\[ SS_{total} = 142.67 + 68.14 + b \]

3. **Finding Error Mean Square (MS_errors)**:
Since we have:
\[ MS_{errors} = \frac{SS_{errors}}{df_{errors}} \]
We can find:
\[ SS_{errors} = MS_{errors} \times df_{errors} \]
\[ SS_{errors} = 1.9165 \times 20 = 38.33 \]

4. **Substituting back into Total SS**:
Now substituting \(SS_{errors}\) back:
\[ SS_{total} = 142.67 + 68.14 + 38.33 = 249.14 \]

### Filling out Degrees of Freedom:
- \(df_treatments = d\)
- From the blocks, we know:
- df_blocks = 5
- df_errors = 20
- Thus, \(df_total = d + 25\). The total degrees of freedom is given by:
- \(df_total\) is \(n - 1\), where \(n\) is the total number of observations.

### Calculating \(d\):
The total degrees of freedom based on the data will be \(df_treatments + df_blocks + df_errors\):
- Let’s calculate an average total degrees of freedom:
\[ d + 5 + 20 = df_{total} - \text{Hence if total is as previously calculated \(249.14\) => average should be 24 or more }.\]

Assuming we can solve:
1. Let's focus on known \(df_{total}=N-1\).

### Getting Statistics Components:
1. **For Treatments DF**:
2. **Calculating MS Blocks**:
\[ MS_{blocks} = \frac{SS_{blocks}}{df_{blocks}} = \frac{68.14}{5} = 13.628 \]

### Calculating F-statistic:
\[
F_{blocks} = \frac{MS_{blocks}}{MS_{error}} = \frac{13.628}{1.9165} \approx 7.1
\]

Now replace \(j\) to:
\[ j \approx 7.1 \] at significance level \(α=0.01\).

So, the answer is:
**(j) = 7.1**.