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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 z

scores wherexz =

Importance of the Standard
Normal Distribution:
Standard
Normal
Distribution:
Any Normal
Distribution:
0
1
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. __________