To determine whether the difference between the yields of the two varieties of wheat is significant, we will perform an Analysis of Variance (ANOVA). Here are the steps to solve the problem:
### Step 1: Calculate the Means
1. Calculate the mean yield for Variety A.
2. Calculate the mean yield for Variety B.
3. Calculate the overall mean yield across both varieties.
Given the yields:
- Variety A: [30, 32, 22]
- Variety B: [20, 18, 16]
Mean for Variety A:
[tex]\[ \text{mean}_A = \frac{30 + 32 + 22}{3} = 28.0 \][/tex]
Mean for Variety B:
[tex]\[ \text{mean}_B = \frac{20 + 18 + 16}{3} = 18.0 \][/tex]
Overall mean:
[tex]\[ \text{overall mean} = \frac{30 + 32 + 22 + 20 + 18 + 16}{6} = 23.0 \][/tex]
### Step 2: Calculate the Sum of Squares
1. Sum of Squares Between Groups (SSB):
[tex]\[ \text{SSB} = n_A(\text{mean}_A - \text{overall mean})^2 + n_B(\text{mean}_B - \text{overall mean})^2 \][/tex]
Where [tex]\( n_A = 3 \)[/tex] and [tex]\( n_B = 3 \)[/tex].
[tex]\[ \text{SSB} = 3(28.0 - 23.0)^2 + 3(18.0 - 23.0)^2 \][/tex]
[tex]\[ \text{SSB} = 3(5.0)^2 + 3(-5.0)^2 \][/tex]
[tex]\[ \text{SSB} = 3 \times 25 + 3 \times 25 \][/tex]
[tex]\[ \text{SSB} = 150.0 \][/tex]
2. Sum of Squares Within Groups (SSW):
[tex]\[ \text{SSW} = \sum (\text{Variety A individual yield} - \text{mean}_A)^2 + \sum (\text{Variety B individual yield} - \text{mean}_B)^2 \][/tex]
[tex]\[ \text{SSW} = (30 - 28.0)^2 + (32 - 28.0)^2 + (22 - 28.0)^2 + (20 - 18.0)^2 + (18 - 18.0)^2 + (16 - 18.0)^2 \][/tex]
[tex]\[ \text{SSW} = 4 + 16 + 36 + 4 + 0 + 4 \][/tex]
[tex]\[ \text{SSW} = 64.0 \][/tex]
3. Total Sum of Squares (SST):
[tex]\[ \text{SST} = \text{SSB} + \text{SSW} \][/tex]
[tex]\[ \text{SST} = 150.0 + 64.0 \][/tex]
[tex]\[ \text{SST} = 214.0 \][/tex]
### Step 3: Calculate the Degrees of Freedom
1. Degrees of freedom between groups, [tex]\(df_{between}\)[/tex]:
[tex]\[ df_{between} = 1 \][/tex]
2. Degrees of freedom within groups, [tex]\(df_{within}\)[/tex]:
[tex]\[ df_{within} = (n_A + n_B) - 2 = 6 - 2 = 4 \][/tex]
### Step 4: Calculate Mean Square Values
1. Mean Square Between Groups (MSB):
[tex]\[ \text{MSB} = \frac{\text{SSB}}{df_{between}} = \frac{150.0}{1} = 150.0 \][/tex]
2. Mean Square Within Groups (MSW):
[tex]\[ \text{MSW} = \frac{\text{SSW}}{df_{within}} = \frac{64.0}{4} = 16.0 \][/tex]
### Step 5: Calculate the F-statistic
[tex]\[ F = \frac{\text{MSB}}{\text{MSW}} = \frac{150.0}{16.0} = 9.375 \][/tex]
### Step 6: Determine Significance
Compare the calculated F-value with the table value at a 5% significance level:
- Calculated [tex]\( F = 9.375 \)[/tex]
- Table value of [tex]\( F \)[/tex] at [tex]\( v_1 = 1 \)[/tex] and [tex]\( v_2 = 4 \)[/tex] degrees of freedom = 7.71
Since [tex]\( 9.375 > 7.71 \)[/tex], the F-statistic is greater than the table value.
### Conclusion
The difference between the yields of the two varieties is significant at the 5% level. Therefore, we can conclude that the yields of the two varieties of wheat are significantly different.
