First slide

Introduction to statistics

Question

Consider the first 10 positive integers. If we multiply each number by -1 and then add 1 to each number, the variance of the numbers so obtained is

Easy

A

8.25
B

6.5
C

3.87
D

2.87

Solution

The first 10 positive integers are 1 ,2,3,4,5,6,7,8,9 and 10.
On multiplying each number by -1, we get
-1, -2, -3,4, -5, -6, -7, -8, -9, -10
on adding 1 to each number , we get
$0, - 1, - 2, - 3, - 4, - 5, - 6, - 7, - 8, - 9$ .
$∴ Σ x_{i} = - \frac{9 \times 10}{2} = - 45$
$\begin{array}{l} and Σ x_{i}^{2} = 0^{2} + (- 1)^{2} + (- 2)^{2} + \dots + (- 9)^{2} = \frac{9 \times 10 \times 19}{6} = 285 \\ \begin{aligned} ∴ S.D, & = \sqrt{\frac{285}{10} - {(\frac{- 45}{10})}^{2}} = \sqrt{\frac{285}{10} - \frac{2025}{100}} \\ = \sqrt{\frac{2850 - 2025}{100}} = \sqrt{8.25} \\ Now, variance = (S.D.)^{2} = (\sqrt{8.25)^{2}} = 8.25 \end{aligned} \end{array}$

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