ANOVA

ANOVA typically uses the F-distribution. The F-statistic is a ratio of two scaled chi-square variables.

Let me provide you with an example of using ANOVA to test the difference between two or more samples:

**Example: ANOVA for Plant Growth**

Imagine you are testing the effect of three different types of fertilizers on plant growth. You have 15 plants, which are divided into three groups (5 plants per group), with each group receiving a different type of fertilizer. After a month, you measure the growth of each plant in centimeters:

- Fertilizer A: 8, 12, 9, 11, 10
- Fertilizer B: 13, 17, 12, 18, 15
- Fertilizer C: 10, 9, 11, 10, 13

To understand whether there is a significant difference in plant growth across the three fertilizers, you perform ANOVA. Here are the steps:

1. **Calculate Group Means and Overall Mean:**
- Mean of A: \( \mu_A = (8 + 12 + 9 + 11 + 10) / 5 = 10 \)
- Mean of B: \( \mu_B = (13 + 17 + 12 + 18 + 15) / 5 = 15 \)
- Mean of C: \( \mu_C = (10 + 9 + 11 + 10 + 13) / 5 = 10.6 \)
- Overall mean: \( \mu = (10 + 15 + 10.6) / 3 \)

2. **Calculate the Sum of Squares Between Groups (SSB):**
\[ SSB = n(\mu_A - \mu)^2 + n(\mu_B - \mu)^2 + n(\mu_C - \mu)^2 \]
where \( n \) is the number of observations per group.

3. **Calculate the Sum of Squares Within Groups (SSW):**
\[ SSW = \sum_{i=1}^{n} (x_{iA} - \mu_A)^2 + \sum_{i=1}^{n} (x_{iB} - \mu_B)^2 + \sum_{i=1}^{n} (x_{iC} - \mu_C)^2 \]
where \( x_{iA} \), \( x_{iB} \), and \( x_{iC} \) are the individual measurements in each group.

4. **Calculate the F-statistic:**
The F-statistic is calculated as follows:
\[ F = \frac{SSB / (k - 1)}{SSW / (N - k)} \]
where \( k \) is the number of groups and \( N \) is the total number of observations.

5. **Compare the F-statistic to the Critical Value:**
The critical value for F is determined by the degrees of freedom for the numerator (\( k - 1 \)) and denominator (\( N - k \)), and your chosen significance level (e.g., \( \alpha = 0.05 \)). If your calculated F is greater than the critical value from the F-distribution table, you reject the null hypothesis that there is no difference between the means.

In this example, if your calculated F-statistic is higher than the critical value from the F-distribution table, you can conclude that there is a statistically significant difference in plant growth due to fertilizer type. This example doesn't provide actual calculations for SSB and SSW because they would require more detailed computation. If you'd like to see those calculations, please let me know!