Step 1: Understanding the Concept:
To provide a point estimate for the population standard deviation (\( \sigma \)) using sample data, we calculate the sample standard deviation (\( s \)), which uses the formula \( s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}} \).
Step 2: Detailed Explanation:
Data: 4, 5, 9, 10, 12.
1. Mean (\( \bar{x} \)) = \( \frac{4+5+9+10+12}{5} = \frac{40}{5} = 8 \).
2. Variance (\( s^2 \)) = \( \frac{(4-8)^2 + (5-8)^2 + (9-8)^2 + (10-8)^2 + (12-8)^2}{5-1} \)
\( s^2 = \frac{(-4)^2 + (-3)^2 + (1)^2 + (2)^2 + (4)^2}{4} = \frac{16 + 9 + 1 + 4 + 16}{4} = \frac{46}{4} = 11.5 \).
3. Standard Deviation (\( s \)) = \( \sqrt{11.5} \approx 3.391 \).
Step 3: Final Answer:
Rounding 3.391 to one decimal place gives 3.4.