Given that the arithmetic mean and standard deviation of a sample of 15 observations are 24 and 0 respectively. Then which one of the following is the arithmetic mean of the smallesa five observations in the data?

Options

A. 0
B. 8
C. 16
D. 24

Answer

To solve the problem, we need to analyze the given information about the sample of 15 observations.

### Step-by-step Solution:

1. **Understanding the Given Data**:
- We have a sample of 15 observations.
- The arithmetic mean (AM) of these observations is 24.
- The standard deviation (SD) of these observations is 0.

2. **Implication of Standard Deviation**:
- A standard deviation of 0 indicates that all observations in the sample are the same. This means every observation in the sample is equal to the arithmetic mean.
- Therefore, all 15 observations must be equal to 24.

3. **Finding the Smallest 5 Observations**:
- Since all observations are equal to 24, the smallest 5 observations will also be 24.
- Thus, the smallest 5 observations are: 24, 24, 24, 24, 24.

4. **Calculating the Arithmetic Mean of the Smallest 5 Observations**:
- The arithmetic mean of these 5 observations can be calculated as follows:
\[ \text{AM} = \frac{\text{Sum of observations}}{\text{Number of observations}} = \frac{24 + 24 + 24 + 24 + 24}{5} = \frac{120}{5} = 24 \]

5. **Conclusion**:
- Therefore, the arithmetic mean of the smallest 5 observations is 24.

### Final Answer:
The arithmetic mean of the smallest 5 observations in the data is **24**.

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