Scores on a test have mean 100 and standard deviation 15. What is… Step by Step

Step by step

The question

Question: find the z-score of 115, when the mean is 100 and the SD is 15.

z-score of 115: mean 100, SD 15

A z-score tells how many SDs a number is from the mean.

z = how many SDs from the mean

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Write the formula

x minus the mean is how far x is from the mean.

Dividing by the SD counts how many SDs that is.

So z is x minus the mean, all over the SD.

z = x − meanSD

Careful!

Subtract first, then divide.

The whole top is divided by the SD.

Careful: not 115 − 10015 ✗

2 Put the numbers in

The top first: 115 minus 100 is 15.

Then 15 divided by 15 is 1.

z = 115 − 10015 = 1515 = 1

3 Read it

115 on a number line with the SD marks.

The z numbers count the SDs.

So 115 is 1 SD above the mean.

1 SD above the mean

4 Check

Let's check: start at the mean and go 1 SDs.

100 plus 1 times 15 is 115.

So it is right!

100 + 1 × 15 = 115

5 Pick the choice

Our answer is z = 1.

That is choice D.

D) 1

6 Why not the others

A is 15.

Wrong: that is the distance from the mean: divide it by the SD.

A) 15 ✗ that is the distance from the mean: divide it by the SD

B is 7.67.

Wrong: divided 115 by 15.

B) 7.67 ✗ divided 115 by 15

C is −1.

Wrong: 115 is above the mean.

C) −1 ✗ 115 is above the mean

Let's look back.

First: write the formula.

Then: put the numbers in.

Then: read it.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer