Z Score
• The z value or z score tells the number of
standard deviations the original
measurement is from the mean.
• The z value is in standard units.
Formula for z score
x
z
Calculating z-scores
The amount of time it takes for a pizza delivery
is approximately normally distributed with a
mean of 25 minutes and a standard deviation
of 2 minutes. Convert 21 minutes to a z score.
x 21 25
z 2.00
2
Calculating z-scores
Mean delivery time = 25 minutes
Standard deviation = 2 minutes
Convert 29.7 minutes to a z score.
x 29 .7 25
z 2.35
2
Interpreting z-scores
Mean delivery time = 25 minutes
Standard deviation = 2 minutes
Interpret a z score of 1.6.
x z 1 .6( 2 ) 25 28 .2
The delivery time is 28.2 minutes.
Standard Normal Distribution:
=0
=1
-1 0 1
Values are converted to
x z =
z scores where
Importance of the Standard
Normal Distribution:
Standard
Normal
Distribution:
Any Normal 0 1
Distribution:
Areas will be equal.
1
Use of the Normal Probability
Table
(Table 5) - Appendix II
Entries give the probability that a
standard normally distributed
random variable will assume a
value to the left of a given negative
z-score.
Use of the Normal Probability
Table
(Table 5a) - Appendix II
Entries give the probability that a
standard normally distributed
random variable will assume a
value to the left of a given positive z
value.
To find the area to the left of
z = 1.34
_____________________________________
z … 0.03 0.04 0.05 ..…
_____________________________________
.
.
1.2 … .8907 .8925 .8944 ….
1.3 … .9082 .9099 .9115 ….
1.4 … .9236 .9251 .9265 ….
.
Patterns for Finding Areas
Under the Standard Normal
Curve
To find the area to the left of a given
negative z :
Use Table 5 (Appendix II) directly.
z 0
Patterns for Finding Areas
Under the Standard Normal
Curve
To find the area to the left of a given positive
z:
Use Table 5 a (Appendix II) directly.
0 z
Patterns for Finding Areas
Under the Standard Normal
Curve
To find the area between z values on either
side of zero:
Subtract area to left of z1 from area to left
of z2 .
z1 0 z2
Patterns for Finding Areas
Under the Standard Normal
Curve
To find the area between z values on the
same side of zero:
Subtract area to left of z1 from area to left
of z2 .
0 z1 z2
Patterns for Finding Areas
Under the Standard Normal
Curve
To find the area to the right of a positive z
value or to the right of a negative z value:
Subtract from 1.0000 the area to the left of the
given z.
Area under
entire curve
= 1.000.
0 z
Use of the Normal Probability
Table
a. .8925
P(z < 1.24) = ______
b. .4452
P(0 < z < 1.60) = _______
c. .4911
P( - 2.37 < z < 0) = ______
Normal Probability
d. .9974
P( - 3 < z < 3 ) = ________
e. .9322
P( - 2.34 < z < 1.57 ) = _____
f. .0774
P( 1.24 < z < 1.88 ) = _______
Normal Probability
g. .2254
P( - 2.44 < z < - 0.73 ) = _______
h. .9495
P( z < 1.64 ) = __________
i. .0084
P( z > 2.39 ) = _________
Normal Probability
j. .9236
P ( z > - 1.43 ) = __________
k. .0034
P( z < - 2.71 ) = __________
Application of the Normal Curve
The amount of time it takes for a pizza delivery is
approximately normally distributed with a mean of 25
minutes and a standard deviation of 2 minutes. If you order
a pizza, find the probability that the delivery time will be:
a. between 25 and 27 minutes. .3413
a. ___________
b. less than 30 minutes. .9938
b. __________
c. less than 22.7 minutes. .1251
c. __________
