Question

Essay errors : Refer to Exercise 50.

Assume that the number of non-word errors $X$and word errors $Y$ in a randomly selected essay are independent random variables. Calculate and interpret the standard deviation of the sum$S=X+Y.$

Short answer

The overall number of errors (including word and non-word) ranges by $\mathbf{1}\mathbf{.}\mathbf{5134}$ errors on average from the entire mean number of errors, which is $3.1$ errors.

Step-by-step solution

Given Information 

Given:

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

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

For $X$:

Mean, ${\mu }_X:2.1$errors

Standard deviation , ${\sigma }_X:1.136$errors

For $Y$:

Mean, ${\mu }_Y:1.0$error

Standard deviation, ${\sigma }_Y:1.0$error

Calculating and interpreting the standard deviation of the sum S = X + Y.

For both $X\mathrm{and}Y,$the total mean of the mean number of errors is:

${\mu }_{x+y}=2.1+1.0=3.1$errors

The variance of the total is equal to the sum of the variances of the random variables when they are independent.

role="math" localid="1654176907587" $$\begin{gathered} {{\sigma }^2}_{X+Y}={{\sigma }^2}_X+{{\sigma }^2}_Y \\ ={(1.136)}^2+({1.0)}^2 \\ =2.290496\mathrm{errors} \end{gathered}$$

We also know that the standard deviation equals the variance squared:

$$\begin{gathered} {\sigma }_{X+Y}=\sqrt{{{\sigma }^2}_{X+Y}} \\ =\sqrt{2.290496} \\ =\mathbf{1}\mathbf{.}\mathbf{5134} \end{gathered}$$