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 critical value Upper F 0F0 to test the claim that the type of fertilizer makes no difference in the mean number of raspberries per plant. Use alpha equals 0.01 .Use α=0.01.
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 8 3rd Column 6 4st Column 3 3rd Row 1st Column 6 2nd Column 5 3rd Column 3 4st Column 5 4st Row 1st Column 7 2nd Column 5 3rd Column 3 4st Column 3 5st Row 1st Column 5 2nd Column 5 3rd Column 2 4st Column 4 6st Row 1st Column 7 2nd Column 5 3rd Column 4 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	8	6	3	6	5	3	5	7	5	3	3	5	5	2	4	7	5	4	5	6	6	3	4

We are asked to conduct an ANOVA (Analysis of Variance) test to determine if the type of fertilizer makes a difference in the mean number of raspberries per plant. We are given data for four different fertilizers and the number of raspberries on each plant. The significance level is \( \alpha = 0.01 \).

### Step 1: Set up the hypothesis
- Null Hypothesis (\( H_0 \)): The mean number of raspberries is the same for all fertilizers.
- Alternative Hypothesis (\( H_1 \)): At least one fertilizer has a different mean number of raspberries.

### Step 2: Structure the Data
The data is presented in matrix form with the number of raspberries per plant for each fertilizer:

| Fertilizer 1 | Fertilizer 2 | Fertilizer 3 | Fertilizer 4 |
|--------------|--------------|--------------|--------------|
| 6            | 8            | 6            | 3            |
| 6            | 5            | 3            | 5            |
| 7            | 5            | 3            | 3            |
| 5            | 5            | 2            | 4            |
| 7            | 5            | 4            | 5            |
| 6            | 6            | 3            | 4            |

### Step 3: Find the Critical Value (F-statistic)
To find the critical value \( F_{\alpha} \), we need to calculate the degrees of freedom for both the numerator (between groups) and the denominator (within groups).

- **Degrees of Freedom (between groups)**: \( df_1 = k - 1 \), where \( k \) is the number of groups (fertilizers). In this case, \( k = 4 \), so \( df_1 = 4 - 1 = 3 \).
  
- **Degrees of Freedom (within groups)**: \( df_2 = N - k \), where \( N \) is the total number of observations. There are 6 observations per group and 4 groups, so \( N = 24 \). Thus, \( df_2 = 24 - 4 = 20 \).

Using these degrees of freedom, the critical value \( F_0 \) can be found using an F-distribution table or calculator for \( \alpha = 0.01 \), \( df_1 = 3 \), and \( df_2 = 20 \).

I'll calculate the critical value next.import scipy.stats as stats

# Given values
alpha = 0.01
df1 = 3  # degrees of freedom for numerator (between groups)
df2 = 20  # degrees of freedom for denominator (within groups)

# Calculate the critical value F_0 for the given alpha and degrees of freedom
critical_value = stats.f.ppf(1 - alpha, df1, df2)
critical_valueThe critical value \( F_0 \) for the ANOVA test with \( \alpha = 0.01 \), degrees of freedom \( df_1 = 3 \), and \( df_2 = 20 \) is approximately **4.94**.

If the calculated F-statistic from the ANOVA test is greater than this critical value, we reject the null hypothesis and conclude that the type of fertilizer makes a difference in the mean number of raspberries per plant.

Would you like to proceed with the calculation of the ANOVA F-statistic, or do you have any other questions?

### Related Questions:
1. How do you calculate the F-statistic for ANOVA?
2. What is the interpretation of rejecting the null hypothesis in an ANOVA test?
3. What is the meaning of degrees of freedom in ANOVA?
4. Why is it important to choose the right significance level (α)?
5. What assumptions are made when performing ANOVA?
6. How do you interpret the p-value in an ANOVA test?
7. Can ANOVA be used if the sample sizes are different between groups?
8. What is the relationship between ANOVA and t-tests?

### Tip:
In ANOVA, always check the assumptions of normality and homogeneity of variances before interpreting the results.

Learn how to find the critical value for an ANOVA test to determine if different types of fertilizers affect the mean number of raspberries per plant. Given data and significance level α=0.01, understand the steps to calculate the critical F-statistic.

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