For two data sets, each with size 5, the variances are given to be 3 and 4 and the corresponding means are given 2 and 4, respectively. The variance of the combined data set is

Options

A. `(11)/(2)`
B. `(9)/(2)`
C. `(13)/(2)`
D. `(5)/(2)`

Answer

To find the variance of the combined data set from two data sets, we can follow these steps:

### Step 1: Identify the given values
We have two data sets, each with:
- Size (n1 = n2) = 5
- Variance of the first data set (σ²₁) = 3
- Variance of the second data set (σ²₂) = 4
- Mean of the first data set (x̄₁) = 2
- Mean of the second data set (x̄₂) = 4

### Step 2: Calculate the sum of squares for each data set
The formula for variance is given by:
\[ \sigma^2 = \frac{\Sigma x_i^2}{n} - \bar{x}^2 \]
From this, we can rearrange to find the sum of squares:
\[ \Sigma x_i^2 = n \cdot \sigma^2 + \bar{x}^2 \]

For the first data set:
\[ \Sigma x_i^2 = 5 \cdot 3 + 2^2 = 15 + 4 = 19 \]

For the second data set:
\[ \Sigma y_i^2 = 5 \cdot 4 + 4^2 = 20 + 16 = 36 \]

### Step 3: Calculate the combined mean
The combined mean (M) of both data sets is given by:
\[ M = \frac{n_1 \cdot x̄_1 + n_2 \cdot x̄_2}{n_1 + n_2} \]
Substituting the values:
\[ M = \frac{5 \cdot 2 + 5 \cdot 4}{5 + 5} = \frac{10 + 20}{10} = \frac{30}{10} = 3 \]

### Step 4: Calculate the variance of the combined data set
The formula for the variance of the combined data set is:
\[ \sigma^2_{combined} = \frac{\Sigma x_i^2 + \Sigma y_i^2}{n_1 + n_2} - M^2 \]
Substituting the values we have:
\[ \sigma^2_{combined} = \frac{19 + 36}{10} - 3^2 = \frac{55}{10} - 9 = 5.5 - 9 = -3.5 \]
This is incorrect; we need to adjust our calculation.

### Step 5: Correct calculation for combined variance
The correct formula for combined variance is:
\[ \sigma^2_{combined} = \frac{(n_1 - 1) \sigma^2_1 + (n_2 - 1) \sigma^2_2 + n_1 (x̄_1 - M)^2 + n_2 (x̄_2 - M)^2}{n_1 + n_2 - 1} \]

Calculating each term:
- \( (n_1 - 1) \sigma^2_1 = 4 \cdot 3 = 12 \)
- \( (n_2 - 1) \sigma^2_2 = 4 \cdot 4 = 16 \)
- \( n_1 (x̄_1 - M)^2 = 5 \cdot (2 - 3)^2 = 5 \cdot 1 = 5 \)
- \( n_2 (x̄_2 - M)^2 = 5 \cdot (4 - 3)^2 = 5 \cdot 1 = 5 \)

Now substituting into the combined variance formula:
\[ \sigma^2_{combined} = \frac{12 + 16 + 5 + 5}{10 - 1} = \frac{38}{9} \approx 4.22 \]

### Final Answer
The combined variance of the two data sets is approximately \( \frac{38}{9} \).