Question

Is the standard deviation of the set of measurements \(x_1, x_2, x_3, ..., x_{20}\) 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 to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

Recognize the direct relationship between standard deviation and variance. If one is given, the other is determined. Also, understand that standard deviation is a measure of the spread or dispersion of data points around the mean. Statement (2) gives a very specific description of this spread.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The question asks whether the standard deviation (SD) of a set of 20 measurements is less than 3. This is a ""Yes/No"" question. A statement is sufficient if it allows us to answer with a definitive ""Yes"" or a definitive ""No"". 
Step 2: Key Formula or Approach: 
The relationship between standard deviation and variance is fundamental: \[ \text{Standard Deviation} = \sqrt{\text{Variance}} \] The variance is the average of the squared differences from the mean (\(\mu\)): \[ \text{Variance} = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 \] Step 3: Detailed Explanation: 
The question is: Is SD \(< 3\)? 
Analyze Statement (1): The variance for the set of measurements is 4. 
Using the formula, we can calculate the standard deviation: \[ \text{SD} = \sqrt{\text{Variance}} = \sqrt{4} = 2 \] Now we can answer the question: Is \(2<3\)? Yes. Since we get a definitive ""Yes"", statement (1) is sufficient. 
Analyze Statement (2): For each measurement, the difference between the mean and that measurement is 2. 
This means that for every \(x_i\) in the set, the absolute difference \(|x_i - \mu|\) is 2. 
This implies that the squared difference \((x_i - \mu)^2\) is \(2^2 = 4\) for every measurement. 
Now we can calculate the variance for the 20 measurements (\(N=20\)): \[ \text{Variance} = \frac{1}{20} \sum_{i=1}^{20} (x_i - \mu)^2 = \frac{1}{20} \sum_{i=1}^{20} 4 \] \[ \text{Variance} = \frac{1}{20} (20 \times 4) = 4 \] With the variance being 4, we can find the standard deviation: \[ \text{SD} = \sqrt{4} = 2 \] Again, we can answer the question: Is \(2<3\)? Yes. Since we get a definitive ""Yes"", statement (2) is sufficient. 
Step 4: Final Answer: 
Both statements, independently, provide enough information to definitively answer the question.