A shopkeeper mixes a batch of 200 apples of mean mass 150 g and standard deviation 30 g with another batch of 300 apples of mean 100 g and standard deviation 20 g. Find the standard deviation for the combined batch of 500 apples.

Answer

To find the standard deviation of the combined batch of apples, we can follow these steps:

### Step 1: Gather the given data
- For the first batch:
- Number of apples, \( n_1 = 200 \)
- Mean mass, \( \bar{x}_1 = 150 \, \text{g} \)
- Standard deviation, \( \sigma_1 = 30 \, \text{g} \)

- For the second batch:
- Number of apples, \( n_2 = 300 \)
- Mean mass, \( \bar{x}_2 = 100 \, \text{g} \)
- Standard deviation, \( \sigma_2 = 20 \, \text{g} \)

### Step 2: Calculate the combined mean
The combined mean \( \bar{x}_{12} \) can be calculated using the formula:
\[ \bar{x}_{12} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2} \]

Substituting the values:
\[ \bar{x}_{12} = \frac{200 \times 150 + 300 \times 100}{200 + 300} = \frac{30000 + 30000}{500} = \frac{60000}{500} = 120 \, \text{g} \]

### Step 3: Calculate deviations from the combined mean
- Deviation for the first batch:
\[ d_1 = \bar{x}_{12} - \bar{x}_1 = 120 - 150 = -30 \]

- Deviation for the second batch:
\[ d_2 = \bar{x}_{12} - \bar{x}_2 = 120 - 100 = 20 \]

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

Substituting the values:
\[ \sigma = \sqrt{\frac{200 \times 30^2 + 200 \times (-30)^2 + 300 \times 20^2 + 300 \times 20^2}{500}} \]

Calculating each component:
- \( 30^2 = 900 \)
- \( (-30)^2 = 900 \)
- \( 20^2 = 400 \)

Now substituting these values:
\[ \sigma = \sqrt{\frac{200 \times 900 + 200 \times 900 + 300 \times 400 + 300 \times 400}{500}} \]
\[ = \sqrt{\frac{180000 + 180000 + 120000 + 120000}{500}} \]
\[ = \sqrt{\frac{600000}{500}} = \sqrt{1200} \]

### Step 5: Calculate the final standard deviation
\[ \sigma = \sqrt{1200} \approx 34.64 \, \text{g} \]

### Final Answer
The standard deviation for the combined batch of 500 apples is approximately \( 34.64 \, \text{g} \).

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