Out of total sample survey of 100 nuts, in a first sample survey of 35 items has mean 80 and standard deviation 4. A second sample survey of 65 items has mean 70 and standard deviation 3. Find the mean and variance of combined 100 sample nuts.

Answer

To find the mean and variance of the combined sample of nuts, we will follow these steps:

### Step 1: Identify the given data
- First sample (n1): 35 items
- Mean of first sample (x̄1): 80
- Standard deviation of first sample (σ1): 4

- Second sample (n2): 65 items
- Mean of second sample (x̄2): 70
- Standard deviation of second sample (σ2): 3

### Step 2: Calculate the combined mean
The formula for the combined mean (x̄) of two samples 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:

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

Now, calculate the denominator:

\[ 35 + 65 = 100 \]

Now, substitute back into the formula for the mean:

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

### Step 3: Calculate the combined variance
To find the combined variance, we first need to calculate the individual variances. The variance (σ²) is the square of the standard deviation (σ).

- Variance of the first sample (σ1²):
\[ \sigma_1^2 = 4^2 = 16 \]

- Variance of the second sample (σ2²):
\[ \sigma_2^2 = 3^2 = 9 \]

Now, we will use the formula for the combined variance (σ²):

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

Where:
- \(d_1 = \bar{x}_1 - \bar{x} = 80 - 73.5 = 6.5\)
- \(d_2 = \bar{x}_2 - \bar{x} = 70 - 73.5 = -3.5\)

Now substituting the values into the variance formula:

\[ \sigma^2 = \frac{35 \times 16 + 65 \times 9 + 35 \times (6.5)^2 + 65 \times (-3.5)^2}{100} \]

Calculating each term:

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

Now, summing these:

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

Now, divide by the total number of samples:

\[ \sigma^2 = \frac{3420}{100} = 34.2 \]

### Final Results
- Combined Mean (x̄): 73.5
- Combined Variance (σ²): 34.2

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