Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Discover in-depth answers to your questions from a wide network of experts on our user-friendly Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
