A sample of 35 observations has the means 80 and SD. As 4. A second sample of 65 observations from the same population has mean 70 and S.D.3. The S.D. of the combined sample is -

Options

A. 5.83
B. 5.58
C. 34.2
D. None of these

Explanation

To find the combined standard deviation of two samples, we can follow these steps:

### Step 1: Identify the given values
- For the first sample:
- Number of observations (n1) = 35
- Mean (x1) = 80
- Standard deviation (σ1) = 4

- For the second sample:
- Number of observations (n2) = 65
- Mean (x2) = 70
- Standard deviation (σ2) = 3

### Step 2: Calculate the combined mean (x̄_combined)
The formula for the combined mean is:
\[ \bar{x}_{combined} = \frac{n_1 \cdot x_1 + n_2 \cdot x_2}{n_1 + n_2} \]
Substituting the values:
\[ \bar{x}_{combined} = \frac{35 \cdot 80 + 65 \cdot 70}{35 + 65} \]
Calculating the numerator:
\[ = \frac{2800 + 4550}{100} = \frac{7350}{100} = 73.5 \]
So, the combined mean is \( \bar{x}_{combined} = 73.5 \).

### Step 3: Calculate d1 and d2
- \( d_1 = x_1 - \bar{x}_{combined} = 80 - 73.5 = 6.5 \)
- \( d_2 = x_2 - \bar{x}_{combined} = 70 - 73.5 = -3.5 \)

### Step 4: Calculate the combined standard deviation (σ_combined)
The formula for the combined standard deviation is:
\[ \sigma_{combined} = \sqrt{\frac{n_1 \cdot \sigma_1^2 + n_2 \cdot \sigma_2^2 + n_1 \cdot d_1^2 + n_2 \cdot d_2^2}{n_1 + n_2}} \]

Calculating each component:
1. \( n_1 \cdot \sigma_1^2 = 35 \cdot 4^2 = 35 \cdot 16 = 560 \)
2. \( n_2 \cdot \sigma_2^2 = 65 \cdot 3^2 = 65 \cdot 9 = 585 \)
3. \( n_1 \cdot d_1^2 = 35 \cdot (6.5)^2 = 35 \cdot 42.25 = 1478.75 \)
4. \( n_2 \cdot d_2^2 = 65 \cdot (-3.5)^2 = 65 \cdot 12.25 = 796.25 \)

Now, substituting these values into the combined standard deviation formula:
\[ \sigma_{combined} = \sqrt{\frac{560 + 585 + 1478.75 + 796.25}{100}} \]
Calculating the numerator:
\[ = \sqrt{\frac{3420}{100}} = \sqrt{34.2} \approx 5.85 \]

### Final Answer
The combined standard deviation is approximately \( \sigma_{combined} \approx 5.85 \).

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