Chapter 2.1, Problem 32E

To determine

Transformations that Ms. Martin should apply to each test score.

Answer to Problem 32E

Each score to be multiplied by 4, and then added with 27.

Explanation of Solution

Given information:

Before transformations:

Mean,

  $\mu =12$

Standard deviations,

  $\sigma =3$

After transformations:

New Mean,

  ${\mu }_{\text{N}}=75$

New Standard deviations,

  ${\sigma }_{\text{N}}=12$

In the first step,

The distance between the numbers has been taken into account by standard deviation.

The distance between the numbers gets increased by four times when each score is multiplied by four.

When the distance between each number gets increased by the factor of 4, their rate of deviation also gets multiplied by 4.

Thus,

The standard deviation gets transformed into 12 instead of 3.

In the next step,

The mean has been transformed into 48 instead of 12 according to the first step.

But we are required to achieve the mean of 75.

For that

We need to add 27 to each score and the average gets shifted to 75.

Whereas,

The standard deviation and the distance between numbers will not be affected.

Therefore,

For making the transformations,

Ms. Martin needs to multiply each score by 4, and then 27 to be added to each score.