Certainly! Let's go through the steps to calculate the pooled variance from the given data for Group 1 and Group 2.
### Step 1: Calculate the means of the groups
[tex]\[
\text{Mean of Group 1} (\overline{X_1}) = \frac{15 + 11 + 8 + 7 + 6 + 4 + 13}{7} = 9.142857142857142
\][/tex]
[tex]\[
\text{Mean of Group 2} (\overline{X_2}) = \frac{4 + 10 + 15 + 12 + 12 + 9 + 8}{7} = 10.0
\][/tex]
### Step 2: Calculate the variances of the groups
Variance is measured by the formula:
[tex]\[
s^2 = \frac{1}{N-1} \sum_{i=1}^{N} (X_i - \overline{X})^2
\][/tex]
For Group 1:
[tex]\[
s_1^2 = \frac{1}{7-1} \left[ (15 - 9.142857)^2 + (11 - 9.142857)^2 + (8 - 9.142857)^2 + (7 - 9.142857)^2 + (6 - 9.142857)^2 + (4 - 9.142857)^2 + (13 - 9.142857)^2 \right]
\][/tex]
[tex]\[
s_1^2 = 15.80952380952381
\][/tex]
For Group 2:
[tex]\[
s_2^2 = \frac{1}{7-1} \left[ (4 - 10.0)^2 + (10 - 10.0)^2 + (15 - 10.0)^2 + (12 - 10.0)^2 + (12 - 10.0)^2 + (9 - 10.0)^2 + (8 - 10.0)^2 \right]
\][/tex]
[tex]\[
s_2^2 = 12.333333333333334
\][/tex]
### Step 3: Calculate the sample sizes
[tex]\[
n_1 = 7
\][/tex]
[tex]\[
n_2 = 7
\][/tex]
### Step 4: Calculate the pooled variance
Pooled variance is calculated using the formula:
[tex]\[
s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{(n_1 + n_2 - 2)}
\][/tex]
Substituting the values obtained:
[tex]\[
s_p^2 = \frac{(7 - 1) \times 15.80952380952381 + (7 - 1) \times 12.333333333333334}{(7 + 7 - 2)}
\][/tex]
[tex]\[
s_p^2 = \frac{6 \times 15.80952380952381 + 6 \times 12.333333333333334}{12}
\][/tex]
[tex]\[
s_p^2 = \frac{94.85714285714286 + 74.0}{12}
\][/tex]
[tex]\[
s_p^2 = \frac{168.85714285714283}{12}
\][/tex]
[tex]\[
s_p^2 = 14.071428571428571
\][/tex]
Thus, the pooled variance is [tex]\( 14.071428571428571 \)[/tex].
