Question

Is the standard deviation of the set of measurements \( x_1, x_2, x_3, x_4, \dots, x_{20} \) less than 3?
(1) The variance for the set of measurements is 4.
(2) For each measurement, the difference between the mean and that measurement is 2.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are not sufficient

Hint:

When dealing with standard deviation, if the variance is known, you can directly calculate the standard deviation as the square root of the variance.

Answer 1

The Correct Option is A

Step 1: Analyze statement (1).
Statement (1) tells us that the variance for the set of measurements is 4. The variance \( \sigma^2 \) is the square of the standard deviation \( \sigma \). Since variance is 4, we have: \[ \sigma^2 = 4 \quad \implies \quad \sigma = 2 \] Since the standard deviation \( \sigma = 2 \), which is less than 3, statement (1) is sufficient to answer the question.
Step 2: Analyze statement (2).
Statement (2) tells us that the difference between the mean and each measurement is 2. This condition alone does not give us enough information to calculate the standard deviation because it does not describe the spread of the data, only the deviations from the mean. Hence, statement (2) alone is not sufficient.
\[ \boxed{A} \]