Question

The variance of 25 observations is 8. If each observation is multiplied by 3, then the new variance of the resulting observations is

• A. 8
• B. \( \frac{8}{9} \)
• C. 24
• D. 72

Hint:

If data is multiplied by \(k\), variance becomes \(k^2\) times. If a constant is added, variance remains unchanged.

Answer 1

The Correct Option is D

Step 1: Recall variance transformation rule.
If each observation is multiplied by a constant \(k\), then the variance is multiplied by \(k^2\).

Step 2: Identify given values.
Original variance:
\[ \sigma^2 = 8. \]
Each observation is multiplied by \(3\), so \(k = 3\).

Step 3: Apply transformation rule.
New variance is given by:
\[ \sigma^2_{\text{new}} = k^2 \cdot \sigma^2. \]

Step 4: Substitute values.
\[ \sigma^2_{\text{new}} = 3^2 \times 8. \]
\[ = 9 \times 8. \]

Step 5: Compute the value.
\[ = 72. \]

Step 6: Interpretation.
Multiplying data by a constant increases the spread of data, hence variance increases by the square of that constant.

Step 7: Final conclusion.
Thus, the new variance is 72.
Final Answer:
\[ \boxed{72}. \]