Download Math 241 Notes 7.2

Section 7.2 The Standard Normal Distribution
Objective: Find the area under the standard normal curve; find Z-scores for a given
area.
Properties of the Standard Normal Distribution:
Standard Normal Distribution:

=0

=1
-1
0
1
Values are converted to z scores




The mean of the original distribution is always zero, in standard units.
The standard deviation is 1.
The area under the curve is 1.
An x value in the original distribution that is above the mean  has a positive z
score.
 An x value below the mean has a negative z value.
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Find the Area under the Standard Normal Curve
To find the area under the standard normal curve, first draw a standard normal curve
and shade the area that is to be found, then follow the calculator directions below.
Using the TI-83/84 for Finding Areas under the Standard Normal Curve:
Press 2nd DISTR
2:normalcdf( lowerbound, upperbound, 0, 1)
ENTER
* If there is no lowerbound, use E99
If there is no upperbound, use E99
 To find the area of the left z0: 2:normalcdf  E 99, UB , 0, 1
 To find the area of the right z0: 2:normalcdf  LB , E 99, 0, 1
 To find the area between z0 and z1: 2:normalcdf  LB , UB , 0, 1
1.
Determine the area under the standard normal curve in each of the following
(round to four decimal places):
a.
To the left of Z = 0.55
b. To the right of Z = 2.23
c.
Between Z = 3.03 and Z = 1.98
__________
__________
__________
7.2 - 2
Finding a Z-Score from a Specified Area to the Left
Using the TI-83/84 to Find Z-Scores Corresponding to an Area:
Press 2nd DISTR
3: invNorm (area left, 0, 1)
ENTER
 To find a Z-Score from a specified area to the right, you must first determine the
area left   1  area right 
area to the left then use invNorm on calculator:
 To find the Z-Score from an area in the middle, we use the fact that the standard
normal curve is symmetric about its mean:
area left    1  middle area  , then use invNorm to find  z
2


Work #1 - 4
The notation z is the Z-score such that the area under the standard normal curve to the
right of z is .
Work #5
Remember, the area under a normal curve can be interpreted as a probability. The
following inequalities are equivalent for the area under the standard normal curve:
 and  ,  and 
Work #6