Answered

Random sampling from four normally distributed populations produced the following data:

\begin{tabular}{|cccc|}
\hline \multicolumn{4}{|c|}{Treatments} \\
\hline A & B & C & D \\
\hline -12 & -20 & -5 & -20 \\
-20 & -9 & -18 & -20 \\
-11 & -13 & -17 & -20 \\
& -19 & -7 & \\
& & -16 & \\
\hline
\end{tabular}

a. Calculate the grand mean. (Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.)

Grand mean: \_\_\_\_\_

b. Calculate SSTR and MSTR. (Round final answers to 4 decimal places.)

\begin{tabular}{|l|l|}
\hline & \\
\hline SSTR & \_\_\_\_\_ \\
\hline MSTR & \_\_\_\_\_ \\
\hline
\end{tabular}



Answer :

GinnyAnswer
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

Other Questions