CK-12 Probability and Statistics - Advanced

7.2. The z-Score and the Central Limit Theorem http://www.ck12.org

z=

x−μ

σ

z=

81 − 65

10

z=

16

10

z= 1. 6

Example:On a college entrance exam, the mean was 70 and the standard deviation was 8. If Helen’sz−score was
− 1 .5, what was her exam mark?

Solution:

z=

x−μ

σ

∴z·σ=x−μ

X=μ+z·σ

X= ( 70 )+(− 1. 5 )( 8 )

X= 58

Now you will see howz−scores are used to determine the probability of an event.

Suppose you were to toss 8 coins 2560 times. The following figure shows the histogram and the approximating
normal curve for the experiment. The random variable represents the number of tails obtained.

The blue section of the graph represents the probability that exactly 3 of the coins turned up tails. One way to
determine this is by the following

P(3 tails) =^8

C 3

28

P(3 tails) =

56

256

P(3 tails)∼= 0. 2186