Answer 1

Given:
Number of observations, n = 100
Mean,

\[\bar{x} \] = 40

Standard deviation,

\[\sigma\]   = 10

We know that \[x = \frac{\sum_{} x_i}{100}\]
\[ \Rightarrow \frac{\sum_{} x_i}{100} = 40\]
\[ \Rightarrow \sum_{} x_i = 4000\]

∴ Correct

\[ \sum_{} x_i = 4000 - \left( 30 + 70 \right) + \left( 3 + 27 \right) = 3930\]

Correct mean = \[\frac{\text{ Correct }  \sum_{} x_i}{100} = \frac{3930}{100} = 39 . 3\]

Now,
Incorrect variance,

\[\sigma^2 = \frac{\sum_{} x_i^2}{100} - \left( 40 \right)^2\]

\[\Rightarrow \frac{\sum_{} x_i^2}{100} = 100 + 1600 = 1700\]
\[ \Rightarrow \sum_{} x_i^2 = 170000\]

Correct \[\sum_{} x_i^2 = 170000 - {30}^2 - {70}^2 + 3^2 + {27}^2 = 164939\]

 

∴ Correct standard deviation \[= \sqrt{\frac{164939}{100} - \left( 39 . 3 \right)^2}\]
\[ = \sqrt{1649 . 39 - 1544 . 49}\]
\[ = \sqrt{104 . 9}\]
\[ = 10 . 24\]