© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Identify the given information
• There are two data sets, each of size 5.
• Variance of the first data set, $ \sigma_1^2 = 4 $.
• Variance of the second data set, $ \sigma_2^2 = 5 $.
• Mean of the first data set, $ \overline{x} = 2 $.
• Mean of the second data set, $ \overline{y} = 4 $.
Step 2: Compute sums of values in each data set
Since each data set has 5 observations:
1. From $ \overline{x} = 2 $, we get
$ \displaystyle \sum x_i = 5 \times 2 = 10. $
2. From $ \overline{y} = 4 $, we get
$ \displaystyle \sum y_i = 5 \times 4 = 20. $
Step 3: Compute sums of squares of values in each data set
The formula for variance of each data set is
$ \displaystyle \sigma^2 = \frac{\sum (\text{data values})^2}{n} - (\text{mean})^2. $
For the first data set:
$ \displaystyle 4 = \frac{\sum x_i^2}{5} - (2)^2 \quad\Longrightarrow\quad 4 = \frac{\sum x_i^2}{5} - 4. $
Rearranging gives
$ \displaystyle \frac{\sum x_i^2}{5} = 8 \quad\Longrightarrow\quad \sum x_i^2 = 40. $
For the second data set:
$ \displaystyle 5 = \frac{\sum y_i^2}{5} - (4)^2 \quad\Longrightarrow\quad 5 = \frac{\sum y_i^2}{5} - 16. $
Rearranging gives
$ \displaystyle \frac{\sum y_i^2}{5} = 21 \quad\Longrightarrow\quad \sum y_i^2 = 105. $
Step 4: Determine the combined mean
The total number of observations in the combined data set is $ 5 + 5 = 10. $
The combined sum of all observations is:
$ \displaystyle \sum x_i + \sum y_i = 10 + 20 = 30. $
Hence, the combined mean $ \overline{z} $ is:
$ \displaystyle \overline{z} = \frac{30}{10} = 3. $
Step 5: Compute the combined variance
Use the formula:
$ \displaystyle \sigma^2 = \frac{\sum x_i^2 + \sum y_i^2}{n_1 + n_2} \;-\; (\overline{z})^2. $
Substitute the known values:
$ \displaystyle \sum x_i^2 + \sum y_i^2 = 40 + 105 = 145, \quad n_1 + n_2 = 10, \quad \overline{z} = 3. $
So,
$ \displaystyle \sigma^2 = \frac{145}{10} - (3)^2 = 14.5 - 9 = 5.5. $
Therefore,
$ \displaystyle \sigma^2 = \frac{11}{2}. $
Final Answer
$ \displaystyle \frac{11}{2}.
