A sample of 35 observations has mean 80 men and standard deviation 4. A second sample of 65 observations has mean 70 and standard deviation =3. Find the combined mean and standard deviation.

Answer

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

### Step 1: Identify the given values
- For Sample 1:
- \( n_1 = 35 \)
- \( \bar{x}_1 = 80 \)
- \( \sigma_1 = 4 \)

- For Sample 2:
- \( n_2 = 65 \)
- \( \bar{x}_2 = 70 \)
- \( \sigma_2 = 3 \)

### Step 2: Calculate the combined mean
The formula for the combined mean \( \bar{x} \) is given by:

\[ \bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2} \]

Substituting the values:

\[ \bar{x} = \frac{35 \times 80 + 65 \times 70}{35 + 65} \]

Calculating the numerator:

\[ 35 \times 80 = 2800 \]
\[ 65 \times 70 = 4550 \]
\[ 2800 + 4550 = 7350 \]

Calculating the denominator:

\[ 35 + 65 = 100 \]

Now, substituting back into the mean formula:

\[ \bar{x} = \frac{7350}{100} = 73.5 \]

### Step 3: Calculate the combined standard deviation
The formula for the combined standard deviation \( \sigma \) is given by:

\[ \sigma = \sqrt{\frac{n_1 \sigma_1^2 + n_2 \sigma_2^2 + n_1 (\bar{x}_1 - \bar{x})^2 + n_2 (\bar{x}_2 - \bar{x})^2}{n_1 + n_2}} \]

First, we need to calculate \( \sigma_1^2 \) and \( \sigma_2^2 \):

\[ \sigma_1^2 = 4^2 = 16 \]
\[ \sigma_2^2 = 3^2 = 9 \]

Now substituting the values into the formula:

\[ \sigma = \sqrt{\frac{35 \times 16 + 65 \times 9 + 35 \times (80 - 73.5)^2 + 65 \times (70 - 73.5)^2}{100}} \]

Calculating each term:

1. \( 35 \times 16 = 560 \)
2. \( 65 \times 9 = 585 \)
3. \( 80 - 73.5 = 6.5 \) so \( (80 - 73.5)^2 = 42.25 \)
4. \( 35 \times 42.25 = 1478.75 \)
5. \( 70 - 73.5 = -3.5 \) so \( (70 - 73.5)^2 = 12.25 \)
6. \( 65 \times 12.25 = 796.25 \)

Now substituting back into the equation:

\[ \sigma = \sqrt{\frac{560 + 585 + 1478.75 + 796.25}{100}} \]

Calculating the numerator:

\[ 560 + 585 + 1478.75 + 796.25 = 3420 \]

Now substituting this into the standard deviation formula:

\[ \sigma = \sqrt{\frac{3420}{100}} = \sqrt{34.2} \approx 5.85 \]

### Final Results
- Combined Mean \( \bar{x} = 73.5 \)
- Combined Standard Deviation \( \sigma \approx 5.85 \)

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