Ms. Hall gave her class a $10$-question multiple-choice quiz. Let $X=$the number of questions that a randomly selected student in the class answered correctly. The computer output below gives information about the probability distribution of X. To determine each student’s grade on the quiz (out of localid="1649489099543" $100$), Ms. Hall will multiply his or her number of correct answers by 10. Let localid="1649489106434" $G=$the grade of a randomly chosen student in the class.

localid="1649489113566" $\begin{array}{llllllll}\mathrm{N}& \text{Mean}& \text{Median}& \text{StDev}& \text{Min}& \text{Max}& Q_1& Q_3\\ 30& 7.6& 8.5& 1.32& 4& 10& 8& 9\end{array}$

(a) Find the mean of localid="1649489121120" $G$. Show your method.

(b) Find the standard deviation of localid="1649489127059" $G$. Show your method.

(c) How do the variance of localid="1649489132289" $X$and the variance oflocalid="1649489138146" $G$compare? Justify your answer.

In the output, the mean for the variable $X$is given:

${\mu }_X=7.6$

Ms. Hall multiplies the number of correct answers $X$by $10$:

$G=10X$

Property mean:

${\mu }_{aX+b}=a{\mu }_X+b$

Then we can determine the mean for $G$:

localid="1649909794929" $$\begin{gathered} {\mu }_G={\mu }_{10X} \\ =10{\mu }_X \\ =10(7.6) \\ =76 \end{gathered}$$

In the output, the standard deviation "StDev" for the variable $X$is given:

${\sigma }_X=1.32$

Ms. Hall multiplies the number of correct answers $X$by $10$:

$G=10X$

Property standard deviation:

${\sigma }_{aX+b}=a{\sigma }_X$

Then we can determine the standard deviation for $G$:

localid="1649909831105" $$\begin{gathered} {\sigma }_G={\sigma }_{10X} \\ =10{\sigma }_X \\ =10(1.32) \\ =13.2 \end{gathered}$$

Given

$\begin{array}{llllllll}\mathrm{N}& \text{Mean}& \text{Median}& \text{StDev}& \text{Min}& \text{Max}& Q_1& Q_3\\ 30& 7.6& 8.5& 1.32& 4& 10& 8& 9\end{array}$

Ms. Hall multiplies the number of correct answers $X$by $10$:

$G=10X$

Property standard deviation:

${\sigma }_{aX+b}=a{\sigma }_X$

Then we can determine the standard deviation for $G$:

localid="1649909863743" $$\begin{gathered} {\sigma }_G={\sigma }_{10X} \\ =10{\sigma }_X \end{gathered}$$

The variance is the square of the standard deviation:

localid="1649909871172" $$\begin{gathered} {\sigma }_G^2={10}^2{\sigma }_X^2 \\ =100{\sigma }_X^2 \end{gathered}$$

Thus the variance of $G$is $100$times the variance of $X$.