Question

Quiz scores The scores on Ms. Martin’s statistics quiz had a mean of $12$ and a standard deviation of $3$Ms. Martin wants to transform the scores to have a mean of $75$ and a standard deviation of $12$What transformations should she apply to each test score? Explain your answer.

Short answer

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

Step-by-step solution

Given information

Before transformations:

Mean, $\mu =12$

Standard deviations, $\sigma =3$

After transformations: ${\mu }_N=75$

New Standard deviations,${\sigma }_N=12$

Calculation

The standard deviation was used to account for the distance between the numbers in the first stage.

When each score is multiplied by four, the distance between the numbers increases by four times.

When the distance between two numbers is multiplied by four, the rate of deviations is multiplied by four.

As a result, the standard deviation is converted to $12$ instead of $3$

In the next step,

According to the first phase, the mean has been changed to $48$ instead of $12$

However, we must reach a $75th$ percentile average.

To do this, we must add $27$ to each score, resulting in a $75-$point average.

Whereas,

The standard deviation and distance between numbers will remain unchanged.

Therefore,

To carry out the modifications,

Ms. Martin must multiply each score by four and then add $27$to each result.