Question

If \( x_1, i=2, 3, \ldots, n \) are \( n \) observations such that \( \sum_{i=1}^{n} x_i^2 = 550 \), mean \( \bar{x} = 5 \) and variance is zero, then the number of observations is equal to:

• A. 30
• B. 25
• C. 22
• D. 16
• E. 4

Hint:

When the variance is zero, all the observations must be equal to the mean.

Answer 1

The Correct Option is C

We are given the following information:
- The mean \( \bar{x} = 5 \)
- The variance is zero
- The sum of the squares of the observations \( \sum_{i=1}^{n} x_i^2 = 550 \)
Variance is calculated as: \[ {Variance} = \frac{1}{n} \sum_{i=1}^{n} x_i^2 - \bar{x}^2 \] Since the variance is zero: \[ \frac{1}{n} \sum_{i=1}^{n} x_i^2 - \bar{x}^2 = 0 \] Substitute \( \bar{x} = 5 \) into the equation: \[ \frac{1}{n} \sum_{i=1}^{n} x_i^2 - 25 = 0 \] Thus, \[ \frac{1}{n} \sum_{i=1}^{n} x_i^2 = 25 \] We are also given that \( \sum_{i=1}^{n} x_i^2 = 550 \), so: \[ \frac{550}{n} = 25 \] Solving for \( n \): \[ n = \frac{550}{25} = 22 \] Thus, the number of observations is \( n = 22 \).