Scores on an exam have mean 200 and standard deviation 20. Which… Step by Step

Step by step

The question

Question: which number has a z-score of −1.5, when the mean is 200 and the SD is 20?

z = −1.5: mean 200, SD 20. x?

A z-score of −1.5 means 1.5 SDs below the mean.

z = −1.5: 1.5 SDs below the mean

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Write the formula

Start at the mean, then go z SDs: x is the mean plus z times the SD.

x = mean + z × SD

Careful!

A negative z-score means below the mean.

So x comes out smaller than the mean.

Careful: a negative z means below the mean: x is smaller

2 Put the numbers in

−1.5 times 20 is −30.

200 minus 30 is 170.

x = 200 + (−1.5) × 20 = 200 − 30 = 170

3 Read it

170 on the number line: 1.5 SDs below the mean.

4 Check

Let's check: the z-score of 170 is 170 minus 200, over 20.

That is −1.5.

So it is right!

z = 170 − 20020 = −3020 = −1.5

5 Pick the choice

Our answer is x = 170.

That is choice A.

A) 170

6 Why not the others

B is 230.

Wrong: −1.5 is below the mean: take 30 away.

B) 230 ✗ −1.5 is below the mean: take 30 away

C is 198.5.

Wrong: took away 1.5, not 1.5 standard deviations.

C) 198.5 ✗ took away 1.5, not 1.5 standard deviations

D is 180.

Wrong: used 1 standard deviation.

D) 180 ✗ used 1 standard deviation

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