Answer :
Let's solve the problem step-by-step and detail the procedure for each part of the question.
### a. Calculate the Grand Mean
The grand mean is the average of all the observations across all treatments. To find the grand mean, we first list all the data points and then find their average.
1. Listing the Data Points:
- Treatment A: -12, -20, -11
- Treatment B: -20, -9, -13, -19
- Treatment C: -5, -18, -17, -7, -16
- Treatment D: -20, -20, -20
2. Combining All Data:
- All data points combined: -12, -20, -11, -20, -9, -13, -19, -5, -18, -17, -7, -16, -20, -20, -20
3. Calculating the Grand Mean:
- Sum of all data points: [tex]\(-12 + (-20) + (-11) + (-20) + (-9) + (-13) + (-19) + (-5) + (-18) + (-17) + (-7) + (-16) + (-20) + (-20) + (-20) = -227\)[/tex]
- Total number of data points: 15
- Grand Mean [tex]\( \bar{X} \)[/tex] = [tex]\(\frac{-227}{15} = -15.13\)[/tex]
So, the grand mean is [tex]\(-15.13\)[/tex].
### b. Calculate SSTR and MSTR
To calculate the SSTR (Sum of Squares for Treatments) and MSTR (Mean Square for Treatments), follow these steps:
1. Calculate the Mean of Each Treatment:
- Mean of Treatment A: [tex]\( \frac{-12 + (-20) + (-11)}{3} = \frac{-43}{3} = -14.33 \)[/tex]
- Mean of Treatment B: [tex]\( \frac{-20 + (-9) + (-13) + (-19)}{4} = \frac{-61}{4} = -15.25 \)[/tex]
- Mean of Treatment C: [tex]\( \frac{-5 + (-18) + (-17) + (-7) + (-16)}{5} = \frac{-63}{5} = -12.60 \)[/tex]
- Mean of Treatment D: [tex]\( \frac{-20 + (-20) + (-20)}{3} = \frac{-60}{3} = -20.00 \)[/tex]
2. Number of Observations in Each Treatment:
- [tex]\( n_A = 3 \)[/tex]
- [tex]\( n_B = 4 \)[/tex]
- [tex]\( n_C = 5 \)[/tex]
- [tex]\( n_D = 3 \)[/tex]
3. Number of Treatments:
- [tex]\( k = 4 \)[/tex]
4. Calculate SSTR:
[tex]\[ \begin{aligned} \text{SSTR} &= n_A (\overline{X}_A - \overline{X})^2 + n_B (\overline{X}_B - \overline{X})^2 + n_C (\overline{X}_C - \overline{X})^2 + n_D (\overline{X}_D - \overline{X})^2 \\ &= 3 (-14.33 + 15.13)^2 + 4 (-15.25 + 15.13)^2 + 5 (-12.60 + 15.13)^2 + 3 (-20.00 + 15.13)^2 \\ &= 3 (0.80)^2 + 4 (-0.12)^2 + 5 (2.53)^2 + 3 (-4.87)^2 \\ &= 3 (0.64) + 4 (0.0144) + 5 (6.4009) + 3 (23.7169) \\ &= 1.92 + 0.0576 + 32.0045 + 71.1507 \\ &= 105.1167 \end{aligned} \][/tex]
5. Calculate MSTR:
[tex]\[ \text{MSTR} = \frac{\text{SSTR}}{k - 1} = \frac{105.1167}{4 - 1} = \frac{105.1167}{3} = 35.0389 \][/tex]
The final results are:
- SSTR = 105.1167
- MSTR = 35.0389
### a. Calculate the Grand Mean
The grand mean is the average of all the observations across all treatments. To find the grand mean, we first list all the data points and then find their average.
1. Listing the Data Points:
- Treatment A: -12, -20, -11
- Treatment B: -20, -9, -13, -19
- Treatment C: -5, -18, -17, -7, -16
- Treatment D: -20, -20, -20
2. Combining All Data:
- All data points combined: -12, -20, -11, -20, -9, -13, -19, -5, -18, -17, -7, -16, -20, -20, -20
3. Calculating the Grand Mean:
- Sum of all data points: [tex]\(-12 + (-20) + (-11) + (-20) + (-9) + (-13) + (-19) + (-5) + (-18) + (-17) + (-7) + (-16) + (-20) + (-20) + (-20) = -227\)[/tex]
- Total number of data points: 15
- Grand Mean [tex]\( \bar{X} \)[/tex] = [tex]\(\frac{-227}{15} = -15.13\)[/tex]
So, the grand mean is [tex]\(-15.13\)[/tex].
### b. Calculate SSTR and MSTR
To calculate the SSTR (Sum of Squares for Treatments) and MSTR (Mean Square for Treatments), follow these steps:
1. Calculate the Mean of Each Treatment:
- Mean of Treatment A: [tex]\( \frac{-12 + (-20) + (-11)}{3} = \frac{-43}{3} = -14.33 \)[/tex]
- Mean of Treatment B: [tex]\( \frac{-20 + (-9) + (-13) + (-19)}{4} = \frac{-61}{4} = -15.25 \)[/tex]
- Mean of Treatment C: [tex]\( \frac{-5 + (-18) + (-17) + (-7) + (-16)}{5} = \frac{-63}{5} = -12.60 \)[/tex]
- Mean of Treatment D: [tex]\( \frac{-20 + (-20) + (-20)}{3} = \frac{-60}{3} = -20.00 \)[/tex]
2. Number of Observations in Each Treatment:
- [tex]\( n_A = 3 \)[/tex]
- [tex]\( n_B = 4 \)[/tex]
- [tex]\( n_C = 5 \)[/tex]
- [tex]\( n_D = 3 \)[/tex]
3. Number of Treatments:
- [tex]\( k = 4 \)[/tex]
4. Calculate SSTR:
[tex]\[ \begin{aligned} \text{SSTR} &= n_A (\overline{X}_A - \overline{X})^2 + n_B (\overline{X}_B - \overline{X})^2 + n_C (\overline{X}_C - \overline{X})^2 + n_D (\overline{X}_D - \overline{X})^2 \\ &= 3 (-14.33 + 15.13)^2 + 4 (-15.25 + 15.13)^2 + 5 (-12.60 + 15.13)^2 + 3 (-20.00 + 15.13)^2 \\ &= 3 (0.80)^2 + 4 (-0.12)^2 + 5 (2.53)^2 + 3 (-4.87)^2 \\ &= 3 (0.64) + 4 (0.0144) + 5 (6.4009) + 3 (23.7169) \\ &= 1.92 + 0.0576 + 32.0045 + 71.1507 \\ &= 105.1167 \end{aligned} \][/tex]
5. Calculate MSTR:
[tex]\[ \text{MSTR} = \frac{\text{SSTR}}{k - 1} = \frac{105.1167}{4 - 1} = \frac{105.1167}{3} = 35.0389 \][/tex]
The final results are:
- SSTR = 105.1167
- MSTR = 35.0389
