Chapter 6.2, Problem 58E

To determine

Interpretation and calculation of the standard deviation of the sum S of two independent variable X and Y .

Answer to Problem 58E

Standard deviation of the sum S ,

  ${\sigma }_S\approx 1.5134\text{errors}$

Explanation of Solution

Given information:

X and Y are independent random variables.

For X :

Mean,

  ${\mu }_X=2.1\text{errors}$

Standard deviation,

  ${\sigma }_X=1.136\text{errors}$

For Y :

Mean,

  ${\mu }_Y=1.0\text{error}$

Standard deviation,

  ${\sigma }_Y=1.0\text{error}$

Calculations:

We know that

X: the number of non-word errors in a randomly selected essay

Y: the number of word errors in a randomly selected essay

Now,

Total mean of the mean number of errors for both X and Y :

  ${\mu }_{X+Y}=2.1+1.0=3.1\text{errors}$

When the random variables are independent, the variance of the sum is equal to the sum of their variances.

  ${\sigma }_{X+Y}^2={\sigma }_X^2+{\sigma }_Y^2={\left(1.136\right)}^2+{\left(1.0\right)}^2=2.290496\text{errors}$

We also know that

The standard deviation is the square root of the variance:

  ${\sigma }_{X+Y}=\sqrt{{\sigma }_{X+Y}^2}=\sqrt{2.290496}=1.5134$

On an average, the total number of errors (word and non-word both) varies by 1.5134 errors from the total mean number of errors i.e. 3.1 errors.