The variance of 20 observations is 5. If each observation is multiplied by 3, then what is the new variance of the resulting observations?
(a)5
(b)10
(c)15
(d)45

Options

A. 5
B. 10
C. 15
D. 45

Answer

To find the new variance after multiplying each observation by 3, we can follow these steps:

1. **Understand the relationship between variance and scaling**: The variance of a set of observations measures how spread out the observations are from their mean. When we multiply each observation by a constant factor, the variance is multiplied by the square of that constant.

2. **Identify the given values**:
- The original variance (σ²) of the 20 observations is given as 5.
- The constant factor by which each observation is multiplied is 3.

3. **Apply the formula for new variance**:
- If each observation is multiplied by a constant \( k \), the new variance \( \sigma'^2 \) can be calculated using the formula:
\[ \sigma'^2 = k^2 \times \sigma^2 \]
- Here, \( k = 3 \) and \( \sigma^2 = 5 \).

4. **Calculate the new variance**:
\[ \sigma'^2 = 3^2 \times 5 = 9 \times 5 = 45 \]

5. **Conclusion**: The new variance of the resulting observations after multiplying each by 3 is 45.

Thus, the answer is (d) 45.