Chapter 6.2, Problem 45E

(a)

To determine

Median of G (the grade of randomly chosen student in the class).

Answer to Problem 45E

Median of G is 92.5.

Explanation of Solution

Given information:

X : the number of correct answers of a randomly chosen student in the class.

Such that

  $\text{Med}{\text{.}}_X=8.5$

G : the grade of a randomly chosen student in the class.

Such that

  $G=5X+50$

Calculations:

The grade is the number of correct answers multiplied by 5 and increased by 50.

  $G=5X+50$

This implies

Every data value in the distribution of X is multiplied by the same constant 5 and increased by the same constant 50.

If every data value is added by the same constant, the center of the distribution is also increased by the same constant.

Also,

If every data value is multiplied by the same constant, the center of the distribution is also multiplied by the same constant.

We know that

The median is the measure of the center.

Thus,

The median is multiplied by 5 and increased by 50.

  $\text{Med}{.}_G=5\text{Med}{.}_X+50=5\left(8.5\right)+50=92.5$

(b)

To determine

Interquartile range ( IQR ) ofG .

Answer to Problem 45E

Interquartile range ( IQR ) of G is 5.

Explanation of Solution

Given information:

First quartile,

  $Q_1=8$

Third quartile,

  $Q_3=9$

Calculations:

The difference between the third and first quartile gives the interquartile range.

Thus,

Interquartile range for X ,

  $IQ{R}_X=Q_3-Q_1=9-8=1$

The grade is the number of correct answers multiplied by 5 and increased by 50.

  $G=5X+50$

This implies

Every data value in the distribution of X is multiplied by the same constant 5 and increased by the same constant 50.

If every data value is added by the same constant, the spread of the distribution is unaffected.

Also,

If every data value is multiplied by the same constant, the spread of the distribution is also multiplied by the same constant.

We know that

The interquartile range ( IQR ) is the measure of the spread.

Thus,

The interquartile range ( IQR ) is multiplied by 5.

Thus,

Interquartile range for G ,

  $IQ{R}_G=5\left(IQ{R}_X\right)=5\left(1\right)=5$