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.

More easy quiz

a. Find the mean of $G$.

b. Find the range of $G$.

Short answer

a. The mean is $\mathbf{88}$

b. The range of$G$is$\mathbf{30}$

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 $50$.

$5X+50=G$

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

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

As a result, the mean is increased by $50$and multiplied by $5.$

$$\begin{gathered} {\mu }_G=5{\mu }_X+50 \\ =5(7.6)+50 \\ =\mathbf{88} \end{gathered}$$

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   

The range is defined as the difference between the maximum and least values.

As a result, for $X$, the range is

$$\begin{gathered} Rang{e}_X=Max-Min \\ =10-4 \\ =\mathbf{6} \end{gathered}$$

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

$5X+50=G$

This denotes Every data value in the $X$distribution is multiplied by $5$and then increased by$50.$

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.

We know that the spread is measured by the range. As a result, the range is multiplied by five.

Thus, Range is :

$$\begin{gathered} Rang{e}_G=5\left(Rang{e}_X\right) \\ =5\left(6\right) \\ =\mathbf{30} \end{gathered}$$