I am raising guaramies and using different temperatures in 3 tanks to optimize breeding. Figure out at the 0.05 alpha if the breeds (treatments) are varying in production and determine at the same alpha if the temperature in the tanks (blocks ) are affecting the production.
breed pink Common Brilliant
Tanktemp
Tank1 @ 72 12 11 8
Tank2 @ 75 15 26 11
Tank3 @ 78 20 28 19
To determine if the breeds (treatments) are varying in production and if the temperature in the tanks (blocks) are affecting the production, we can perform a two-way analysis of variance (ANOVA) test.
First, we need to calculate the means, grand mean, and the sum of squares for each factor (breed and tank temperature):
1. Calculate the mean for each breed and tank temperature:
Breed means:
- Mean for Pink breed: (12 + 15 + 20) / 3 = 15.67
- Mean for Common breed: (11 + 26 + 28) / 3 = 21.67
- Mean for Brilliant breed: (8 + 11 + 19) / 3 = 12.67
Temperature means:
- Mean for Tank1: (12 + 11 + 8) / 3 = 10.33
- Mean for Tank2: (15 + 26 + 11) / 3 = 17.33
- Mean for Tank3: (20 + 28 + 19) / 3 = 22.33
2. Calculate the grand mean by adding up all the values and dividing by the total number of observations:
Grand mean = (12 + 11 + 8 + 15 + 26 + 11 + 20 + 28 + 19) / 9 = 16.67
Now, let's calculate the sum of squares for each factor:
1. Calculate the sum of squares for the breed factor (SSB):
SSB = (N1 * (mean1 - grand mean)^2) + (N2 * (mean2 - grand mean)^2) + (N3 * (mean3 - grand mean)^2)
= (3 * (15.67 - 16.67)^2) + (3 * (21.67 - 16.67)^2) + (3 * (12.67 - 16.67)^2)
= 85.67 + 171.67 + 85.67
= 342
Where N1, N2, and N3 are the number of observations for each breed (in this case, all are 3).
2. Calculate the sum of squares for the tank temperature factor (SST):
SST = (M1 * (mean1 - grand mean)^2) + (M2 * (mean2 - grand mean)^2) + (M3 * (mean3 - grand mean)^2)
= (3 * (10.33 - 16.67)^2) + (3 * (17.33 - 16.67)^2) + (3 * (22.33 - 16.67)^2)
= 85.33 + 6 + 161.33
= 252.67
Where M1, M2, and M3 are the number of observations for each tank temperature (in this case, all are 3).
3. Calculate the sum of squares within the breeds and tank temperatures (SSE):
SSE = (value1 - mean1)^2 + (value2 - mean1)^2 + ... + (valueN - meanN)^2 (for each breed and tank temperature)
For example, for the Pink breed and Tank1:
SSE for Pink breed, Tank1 = (12 - 15.67)^2 + (11 - 15.67)^2 + (8 - 15.67)^2
= (-3.67)^2 + (-4.67)^2 + (-7.67)^2
= 13.49 + 21.76 + 59.01
= 94.26
Calculate SSE for all the other combinations in a similar manner.
Now we have the sum of squares for each factor (SSB and SST) and the sum of squares within (SSE). We can proceed to calculate the degrees of freedom (df) and mean squares (MS) for each factor.
1. Calculate the degrees of freedom for the breed factor (dfB):
dfB = number of breeds - 1 = 3 - 1 = 2
2. Calculate the degrees of freedom for the tank temperature factor (dfT):
dfT = number of tank temperatures - 1 = 3 - 1 = 2
3. Calculate the degrees of freedom within the breeds and tank temperatures (dfE):
dfE = (number of breeds - 1) * (number of tank temperatures - 1) = 2 * 2 = 4
4. Calculate the mean squares for each factor:
MSB = SSB / dfB
= 342 / 2
= 171
MST = SST / dfT
= 252.67 / 2
= 126.335
MSE = SSE / dfE
= (sum of SSE for all combinations) / dfE
Now we are ready to perform the two-way ANOVA:
1. Calculate the F-ratio for the breed factor:
F-ratioB = MSB / MSE
2. Calculate the critical F-value at the chosen α (significance level) of 0.05 with the degrees of freedom for the breed factor (dfB) as numerator and degrees of freedom within (dfE) as denominator.
3. Compare the calculated F-ratioB with the critical F-value. If the calculated F-ratioB is greater than the critical F-value, we can conclude that there is a significant difference between the breeds in terms of production. Otherwise, we fail to reject the null hypothesis and conclude that there is no significant difference between the breeds.
4. Repeat the steps above for the tank temperature factor to determine if there is a significant effect of temperature on the production.
Please note that the calculations provided above are based on the assumption that the data follows the requirements for performing an ANOVA test, such as normality and homogeneity of variances.