If the first sample A of 100 items has the mean 15 and standard deviation 3 and second sample B has 150 items. If the combined mean and standard deviation of itmes of both the sample is 15.6 and `sqrt(13.44)` then then standard deviation of items of sample B is

Answer

To find the standard deviation of sample B, we can follow these steps:

### Step 1: Gather the given information
- Sample A:
- Number of items (nA) = 100
- Mean (x̄A) = 15
- Standard deviation (σA) = 3
- Sample B:
- Number of items (nB) = 150
- Mean (x̄B) = ?
- Standard deviation (σB) = ?
- Combined:
- Combined mean (x̄) = 15.6
- Combined standard deviation (σ) = √13.44

### Step 2: Calculate the mean of sample B
Using the formula for combined mean:
\[ \bar{x} = \frac{n_A \cdot \bar{x}_A + n_B \cdot \bar{x}_B}{n_A + n_B} \]
Substituting the known values:
\[ 15.6 = \frac{100 \cdot 15 + 150 \cdot \bar{x}_B}{100 + 150} \]
This simplifies to:
\[ 15.6 = \frac{1500 + 150 \cdot \bar{x}_B}{250} \]
Multiplying both sides by 250:
\[ 3900 = 1500 + 150 \cdot \bar{x}_B \]
Subtracting 1500 from both sides:
\[ 2400 = 150 \cdot \bar{x}_B \]
Dividing by 150:
\[ \bar{x}_B = 16 \]

### Step 3: Calculate the combined variance
The combined variance formula is:
\[ \sigma^2 = \frac{\Sigma x_i^2}{n} - \bar{x}^2 \]
Given that σ = √13.44, we find:
\[ \sigma^2 = 13.44 \]
Thus:
\[ 13.44 = \frac{\Sigma x_i^2}{250} - (15.6)^2 \]
Calculating (15.6)^2:
\[ 15.6^2 = 243.36 \]
Now substituting back:
\[ 13.44 + 243.36 = \frac{\Sigma x_i^2}{250} \]
\[ 256.8 = \frac{\Sigma x_i^2}{250} \]
Multiplying both sides by 250:
\[ \Sigma x_i^2 = 64200 \]

### Step 4: Calculate the variance of sample A
Using the variance formula for sample A:
\[ \sigma_A^2 = \frac{\Sigma x_i^2}{n_A} - \bar{x}_A^2 \]
Substituting the known values:
\[ 3^2 = \frac{\Sigma x_i^2}{100} - 15^2 \]
Calculating:
\[ 9 = \frac{\Sigma x_i^2}{100} - 225 \]
Adding 225 to both sides:
\[ 234 = \frac{\Sigma x_i^2}{100} \]
Multiplying by 100:
\[ \Sigma x_i^2 = 23400 \]

### Step 5: Calculate the variance of sample B
Using the variance formula for sample B:
\[ \sigma_B^2 = \frac{\Sigma x_i^2}{n_B} - \bar{x}_B^2 \]
Substituting the known values:
\[ \sigma_B^2 = \frac{64200 - 23400}{150} - 16^2 \]
Calculating:
\[ \sigma_B^2 = \frac{40800}{150} - 256 \]
\[ \sigma_B^2 = 272 - 256 = 16 \]

### Step 6: Calculate the standard deviation of sample B
Taking the square root of the variance:
\[ \sigma_B = \sqrt{16} = 4 \]

Thus, the standard deviation of sample B is **4**.