Four different types of fertilizers are used on raspberry plants. The number of raspberries on each randomly selected plant is given to the right. Find the test statistic F to test the claim that the type of fertilizer makes no difference in the mean number of raspberries per plant.

Start 7 By 4 Matrix 1st Row 1st Column Fertilizer 1 2nd Column Fertilizer 2 3rd Column Fertilizer 3 4st Column Fertilizer 4 2nd Row 1st Column 6 2nd Column 5 3rd Column 6 4st Column 3 3rd Row 1st Column 6 2nd Column 8 3rd Column 3 4st Column 5 4st Row 1st Column 5 2nd Column 5 3rd Column 4 4st Column 3 5st Row 1st Column 7 2nd Column 5 3rd Column 2 4st Column 4 6st Row 1st Column 7 2nd Column 5 3rd Column 3 4st Column 5 7st Row 1st Column 6 2nd Column 6 3rd Column 3 4st Column 4 EndMatrix

Fertilizer 1

 Fertilizer 2

 Fertilizer 3

 Fertilizer 4

6

5

6

3

6

8

3

5

5

5

4

3

7

5

2

4

7

5

3

5

6

6

3

4

Question content area bottom

Part 1

A.

8.123

B.

6.912

C.

8.357

D.

7.123

To find the test statistic \( F \) for testing the claim that the type of fertilizer makes no difference in the mean number of raspberries per plant, we'll need to perform an Analysis of Variance (ANOVA). Here's how you can compute it:

### Step 1: Organize the Data
We have four fertilizers, and the number of raspberries for each is listed as follows:

\[
\begin{array}{c|c|c|c}
\text{Fertilizer 1} & \text{Fertilizer 2} & \text{Fertilizer 3} & \text{Fertilizer 4} \\
\hline
6 & 5 & 6 & 3 \\
6 & 8 & 3 & 5 \\
5 & 5 & 4 & 3 \\
7 & 5 & 2 & 4 \\
7 & 5 & 3 & 5 \\
6 & 6 & 3 & 4 \\
\end{array}
\]

### Step 2: Calculate the Group Means
Calculate the mean for each group (each fertilizer type):

\[
\begin{aligned}
\text{Mean of Fertilizer 1} &= \frac{6 + 6 + 5 + 7 + 7 + 6}{6} = 6.17 \\
\text{Mean of Fertilizer 2} &= \frac{5 + 8 + 5 + 5 + 5 + 6}{6} = 5.67 \\
\text{Mean of Fertilizer 3} &= \frac{6 + 3 + 4 + 2 + 3 + 3}{6} = 3.50 \\
\text{Mean of Fertilizer 4} &= \frac{3 + 5 + 3 + 4 + 5 + 4}{6} = 4.00 \\
\end{aligned}
\]

### Step 3: Calculate the Overall Mean
The overall mean is the mean of all the observations combined:

\[
\text{Overall Mean} = \frac{6 + 5 + 6 + 3 + 6 + 8 + 3 + 5 + 5 + 5 + 4 + 3 + 7 + 5 + 2 + 4 + 7 + 5 + 3 + 5 + 6 + 6 + 3 + 4}{24} = 4.83
\]

### Step 4: Calculate the Sum of Squares
- **Total Sum of Squares (SST)**:
\[
SST = \sum (x_{ij} - \text{Overall Mean})^2
\]

- **Sum of Squares Between Groups (SSB)**:
\[
SSB = \sum n_i (\text{Mean of Group}_i - \text{Overall Mean})^2
\]

- **

Learn how to perform Analysis of Variance (ANOVA) to test the claim that different types of fertilizers have no effect on the mean number of raspberries per plant. Includes step-by-step calculation of test statistic F and interpretation of results.

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