Download Ch 5 Elementary Probability Theory

Section 7.2 The Standard Normal Distribution
Objective:
Find the area under the standard normal curve; find Z-scores for
a given area.
In order to compare data values with different centers and/or spreads, we
convert the data values to standard scores, called z-scores.
A z-score represents how many standard deviations a measurement lies from
the mean.
Population z-Score
z 
x  

Sample z-Score
z 
x  x
s
The z-score is unitless. It has mean 0 and standard deviation 1.
Z-scores measure the number of standard deviations an observation is above
or below the mean.
 If a data value is larger than the mean, the z-score will be positive.
 If a data value is smaller than the mean, the z-score will be negative.
1. 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.
2. The average 20- to 29 year old man is 69.6 inches tall, with a standard
deviation of 2.7 inches, while the average 20- to 29 year-old-women is 64.1
inches tall, with a standard deviation of 2.6 inches. Who is relatively taller,
a 75-inch man or a 70-inch woman?
7.2 - 2
Properties of the Standard Normal Distribution:
Standard Normal Distribution:

=0

=1
-1
0
1
Values are converted to z scores




Copyright (C) 2004 Houghton Mifflin Company. All rights reserved.
28
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.
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 1E99
If there is no upperbound, use 1E99
b
b
g
g
 To find the area of the left z0: 2:normalcdf 1E99, UB, 0, 1
 To find the area of the right z0: 2:normalcdf LB, 1E99, 0, 1
 To find the area between z0 and z1:
2:normalcdf LB, UB, 0, 1
b
g
7.2 - 3
3. 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
__________
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 to the left then use invNorm on calculator:
area left   1  area right 
 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


4. Find the indicated Z-scores. Round to two decimal places.
a. Area under the standard normal curve to the left is 0.2.
__________
b. Area under the standard normal curve to the right is 0.35 __________
c. Find the Z-scores that separate the middle 70% of the
distribution from the area in the tails of the standard
normal distribution.
__________
7.2 - 4
The notation z is the Z-score such that the area under the standard normal
curve to the right of z is .
5. Find the value of z .
a. z 0.35
__________
b. z 0.02
__________
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 
6. Find the indicated probability of the standard normal random variable Z.
Round to four decimal places.
a. P (z < 0.61) =
__________
b. P (z  0.92) =
__________
c. P (1.23 < z  1.56) =
__________
d. P (z  0.38 or z > 1.93) = __________