Question

If \( x_1, x_2, x_3, \dots, x_{25} \) be 25 observations such that \( \sum_{i=1}^{25} (x_i + 5)^2 = 2500 \) and \( \sum_{i=1}^{25} (x_i - 5)^2 = 1000 \). Then the ratio of mean and standard deviation of the given observation, is:

• A. \( \frac{1}{3} \)
• B. \( \frac{1}{2} \)
• C. \( \frac{1}{4} \)
• D. \( \frac{1}{5} \)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We expand the given summations to find the values of \( \sum x_i \) and \( \sum x_i^2 \). Once these are known, we calculate the mean (\( \mu = \frac{\sum x_i}{n} \)) and the variance (\( \sigma^2 = \frac{\sum x_i^2}{n} - \mu^2 \)).
Step 2: Key Formula or Approach:
1. Subtract the two equations to find \( \sum x_i \). 2. Add the two equations to find \( \sum x_i^2 \). 3. \( \sigma = \sqrt{\text{Variance}} \).
Step 3: Detailed Explanation:
1. Subtracting the summations: \[ \sum (x_i^2 + 10x_i + 25) - \sum (x_i^2 - 10x_i + 25) = 2500 - 1000 \] \[ \sum 20x_i = 1500 \implies 20 \sum x_i = 1500 \implies \sum x_i = 75 \] 2. Mean \( \mu = \frac{75}{25} = 3 \). 3. Adding the summations: \[ \sum (x_i^2 + 10x_i + 25) + \sum (x_i^2 - 10x_i + 25) = 2500 + 1000 \] \[ 2 \sum x_i^2 + 2 \sum 25 = 3500 \implies 2 \sum x_i^2 + 2(25 \times 25) = 3500 \] \[ 2 \sum x_i^2 + 1250 = 3500 \implies 2 \sum x_i^2 = 2250 \implies \sum x_i^2 = 1125 \] 4. Variance \( \sigma^2 = \frac{1125}{25} - (3)^2 = 45 - 9 = 36 \). 5. Standard deviation \( \sigma = 6 \). 6. Ratio \( \frac{\mu}{\sigma} = \frac{3}{6} = \frac{1}{2} \).
Step 4: Final Answer:
The ratio of mean to standard deviation is 1/2.