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

Options

•  8.25  

• 6.5       

• 3.87 

• 2.87 

Answer 1

The first 10 positive integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Multiplying each number by −1, we get

−1, −2, −3, −4, −5, −6, −7, −8, −9, −10

Adding 1 to each of these numbers, we get

0, −1, −2, −3, −4, −5, −6, −7, −8, −9

Now,

\[\sum_{} x_i = 0 + \left( - 1 \right) + \left( - 2 \right) + \left( - 3 \right) + \left( - 4 \right) + \left( - 5 \right) + \left( - 6 \right) + \left( - 7 \right) + \left( - 8 \right) + \left( - 9 \right) = - 45\]

\[\sum_{} x_i^2 = 0 + 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 = 285\]

∴ Variance of the obtained numbers

\[= \frac{\sum_{} x_i^2}{10} - \left( \frac{\sum_{} x_i}{10} \right)^2 \]

\[ = \frac{285}{10} - \left( \frac{- 45}{10} \right)^2 \]

\[ = 28 . 5 - 20 . 25\]

\[ = 8 . 25\]