Chapter 6.2, Problem 46E

(a)

To determine

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

Answer to Problem 46E

Mean of G is 88.

Explanation of Solution

Given information:

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

Such that

Mean,

  ${\mu }_X=7.6$

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 mean is the measure of the center.

Thus,

The mean is multiplied by 5 and increased by 50.

  ${\mu }_G=5{\mu }_X+50=5\left(7.6\right)+50=88$

(b)

To determine

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

Answer to Problem 46E

Range of G is 30.

Explanation of Solution

Given information:

Maximum number of correct answers,

  $\text{Max}=10$

Minimum number of correct answers,

  $\text{Min}=4$

Calculations:

The difference between the maximum value and minimum value gives the range.

Thus,

Range for X ,

  ${\text{Range}}_X=\text{Max}-\text{Min}=10-4=6$

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 range is the measure of the spread.

Thus,

The range is multiplied by 5.

Thus,

Range for G ,

  ${\text{Range}}_G=5\left({\text{Range}}_X\right)=5\left(6\right)=30$