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Chapter 6, continued...
C. Standard Normal Probability
Distribution
The standard normal is a normal distribution where
all values of x have been converted to z-scores.
Thus =0 and =1 and the RV is referred to as z,
which represents how many standard deviations a
particular x falls from the mean.
How do we find the
probability that z falls
within 1 standard
deviation of the mean?
Need to use tables.
-1
0
+1
z
Using the standard normal
probability table
Look at a table like that found in the inside cover
of your textbook. Go down to z=1.0, then across
to .00. The result is P(0z1.00)=.3413
Z
.9
.00
.01
.02
.3159 .3186 .3212
1.0 .3413 .3438 .3461
1.1 .3643 .3665 .3686
This is an excerpt from
the larger table.
Example
Suppose the starting salaries of individuals with an
MBA degree is normally distributed with a mean
of $40,000 and a standard deviation of $5000.
What is the probability that a randomly selected
individual with an MBA degree will get a starting
salary of between $40,000 and $48,000?
Steps for solving
1. Diagram the situation
This is the area we
need to calculate.
=40K
48K
x
Steps, continued
2. Convert to z-scores.
This is the area we
need to calculate.
Z=(48,000-40,000)/5000= 1.60
0
1.6
z
Steps, continued...
We’ve converted the original normal distribution to
the standard normal.
3. Go to the table and find z=1.60.
The table gives P(0z1.60)=.4452. Since this is
exactly what we’re looking for, we’re done.
Thus the probability of a salary between $40,000
and $48,000 is .4452.
Another Example
What is the probability that a randomly selected
individual with an MBA degree will get a starting
salary of at least $47,500?
Another way of asking the same question is to ask
the probability of a person making $47,500 or
more.
Solving...
1. Diagram the situation.
This is the area we
need to calculate.
40K
47.5K
x
Solving...
2. Convert to z-scores.
This is the area we
need to calculate.
Z=(47,500-40,000)/5000=1.50
Look up z=1.50 in the table.
0
This is the area that
the table will
provide.
1.5
z
Solving...
3. From the table we find P(0z1.50)=.4332, but
we actually need the P(z>1.50).
Since we know that the area under the entire right
side of the curve is .50, and we know the area
between 0 and 1.50 is .4332, the area beyond 1.50
must be:
P(z>1.50) = .5 - P(0z1.50)=.0668
A diagram of what we did.
.5-.4332=.0668
.4332
0
1.50
Entire area = .5
z