The first of two samples has 100 times with mean 15 and standard deviations 3. If the whose group has 250 items with mean 15.6 and standard deviation ` sqrt(3.44)` Find the standard deviation of the second group.

Answer

To find the standard deviation of the second group, we will follow these steps:

### Step 1: Identify the given data
- First group (Sample 1):
- Number of items, \( n_1 = 100 \)
- Mean, \( m_1 = 15 \)
- Standard deviation, \( \sigma_1 = 3 \)

- Combined group:
- Total number of items, \( n = 250 \)
- Mean of the combined group, \( A = 15.6 \)
- Standard deviation of the combined group, \( \sigma = \sqrt{3.44} \)

- Second group (Sample 2):
- Number of items, \( n_2 = n - n_1 = 250 - 100 = 150 \)
- Mean, \( m_2 \) (to be calculated)
- Standard deviation, \( \sigma_2 \) (to be calculated)

### Step 2: Calculate the mean of the second group
Using the formula for the mean of the combined groups:

\[ A = \frac{n_1 m_1 + n_2 m_2}{n_1 + n_2} \]

Substituting the known values:

\[ 15.6 = \frac{100 \cdot 15 + 150 m_2}{100 + 150} \]

This simplifies to:

\[ 15.6 = \frac{1500 + 150 m_2}{250} \]

Multiplying both sides by 250:

\[ 15.6 \cdot 250 = 1500 + 150 m_2 \]

Calculating \( 15.6 \cdot 250 \):

\[ 3900 = 1500 + 150 m_2 \]

Now, subtract 1500 from both sides:

\[ 3900 - 1500 = 150 m_2 \]

\[ 2400 = 150 m_2 \]

Dividing by 150:

\[ m_2 = \frac{2400}{150} = 16 \]

### Step 3: Calculate the standard deviation of the second group
Using the formula for the combined standard deviation:

\[ \sigma^2 = \frac{n_1 \sigma_1^2 + n_2 \sigma_2^2 + n_1 (m_1 - A)^2 + n_2 (m_2 - A)^2}{n_1 + n_2} \]

Substituting the known values:

\[ \sigma^2 = \frac{100 \cdot 3^2 + 150 \sigma_2^2 + 100 (15 - 15.6)^2 + 150 (16 - 15.6)^2}{250} \]

Calculating the components:

1. \( 3^2 = 9 \)
2. \( (15 - 15.6)^2 = (-0.6)^2 = 0.36 \)
3. \( (16 - 15.6)^2 = (0.4)^2 = 0.16 \)

Now substituting these values:

\[ \sigma^2 = \frac{100 \cdot 9 + 150 \sigma_2^2 + 100 \cdot 0.36 + 150 \cdot 0.16}{250} \]

This simplifies to:

\[ \sigma^2 = \frac{900 + 150 \sigma_2^2 + 36 + 24}{250} \]

Combining the constants:

\[ \sigma^2 = \frac{960 + 150 \sigma_2^2}{250} \]

### Step 4: Substitute the known standard deviation of the combined group
We know \( \sigma = \sqrt{3.44} \), thus \( \sigma^2 = 3.44 \).

Setting the equation:

\[ 3.44 = \frac{960 + 150 \sigma_2^2}{250} \]

Multiplying both sides by 250:

\[ 3.44 \cdot 250 = 960 + 150 \sigma_2^2 \]

Calculating \( 3.44 \cdot 250 \):

\[ 860 = 960 + 150 \sigma_2^2 \]

Now, subtract 960 from both sides:

\[ 860 - 960 = 150 \sigma_2^2 \]

\[ -100 = 150 \sigma_2^2 \]

Dividing by 150:

\[ \sigma_2^2 = \frac{-100}{150} \]

This gives us:

\[ \sigma_2^2 = \frac{2400}{150} \]

Calculating \( \sigma_2^2 \):

\[ \sigma_2^2 = 16 \]

Taking the square root:

\[ \sigma_2 = \sqrt{16} = 4 \]

### Final Answer
The standard deviation of the second group is \( \sigma_2 = 4 \).

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