Question

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 gives information about the probability distribution of $X$. To determine each student’s grade on the quiz (out of $100$), Ms. Hall will multiply his or her number of correct answers by $5$and then add $50.$Let $G=$the grade of a randomly chosen student in the class.

Easy quiz

a. Find the median of $G$.

b. Find the interquartile range (IQR) of $G$.

Short answer

a. The median is $\mathbf{92}\mathbf{.}\mathbf{5}$.

b. Interquartile range for G is$\mathbf{5}$.

Step-by-step solution

Part(a) Step 1 : Given Information    

Given :

$X=$the number of questions that a randomly selected student in the class answered correctly.

Ms. Hall multiplies his or her number of correct answers by $5$and then add $50$.

$G=$the grade of a randomly chosen student in the class.

Part(a) Step 2 : Simplification   

The grade is calculated by multiplying the number of right answers by $5$and increasing by $50$.

$G=5X+50$

This means that each data value in the $X$distribution is multiplied by the same constant $5$and then multiplied by the same constant $50$.

When the same constant is applied to each data value, the center of the distribution is also enlarged by the same constant.

Furthermore, if every data value is multiplied by the same constant, the distribution's center is multiplied by the same constant.

The median is the center's measurement, as we all know.

As a result, the median is :

$$\begin{gathered} Med{.}_G=5Med{.}_X+50 \\ =5(8.5)+50 \\ =\mathbf{}\mathbf{92}\mathbf{.}\mathbf{5} \end{gathered}$$

Part(b) Step 1 : Given Information    

Given :

$X=$ the number of questions that a randomly selected student in the class answered correctly.

Ms. Hall multiplies his or her number of correct answers by $5$and then add $50.$

$G=$the grade of a randomly chosen student in the class.

Part(b) Step 2 : Simplification   

$IQ{R}_X=Q_1-Q_1=9-8=1$is the interquartile range for $X$. The grade is calculated by multiplying the number of right answers by $5$and increasing by $50$.

$5X+50=G$

This means that each data value in the $X$distribution is multiplied by the same constant 5 and then multiplied by the same constant $50$. The spread of the distribution is unaltered if every data value is multiplied by the same constant.

Furthermore, if every data value is multiplied by the same constant, the distribution's spread is also multiplied by the same constant. The interquartile range (IQR) is a measure of the spread, as we all know. As a result, double the interquartile range (IQR) by $5$.

Thus,

$IQ{R}_G=5\left(IQ{R}_X\right)=5\left(1\right)=\mathbf{5}$interquartile range for G