Transcript The Normal Distribution Prepared by E.G. Gascon Properties of Normal Distribution Peak Image text page 487 • • • • It’s peak occurs directly above the mean  The curve.

The Normal Distribution
1
Prepared by E.G. Gascon
Properties of
Normal Distribution
2
Peak
Image text page 487
•
•
•
•
It’s peak occurs directly above the mean 
The curve is symmetric about the vertical line through the mean.
The curve never touches the x-axis
The area under the curve is always = 1. (This agrees with the fact
that the sum of the probabilities in any distribution is 1.)
Variations in Normal Curves
3
One standard
deviation is
smaller than
normal
One standard
deviation is
equal to the
normal
One standard
deviation is
larger than
normal
The Area Under the Standard
Normal Curve
4
1 standard
deviation
A
B
Image from text p 487
• The area of the shaded region under a normal curve form a point A
to B is the probability that an observed data value will be between A
and B
• Between -1 and +1 standard deviations there is 68% of the region,
therefore the probability of an observed data value being within 1
standard deviation is 68%, etc.
Problem solved using the Standard Normal Curve
5
The area under a normal curve to the left of x (the data) is the
same as the area under the standard normal curve to the left of
the z-score for x.
What does that mean?
The z-score is the formula that converts the raw data (x) from
a normal distribution into the lookup values of a STANDARD
NORMAL CURVE. [See table in appendix of text or use Excel
function =NORMSDIST(Z)]
Example: sales force drives
an average of 1200 miles,
with a standard deviations of
150 miles. 1600 miles is the
mileage in question.
First find the z-score
z
x1  


1600  1200
150
 2.67
What is the probability that a salesperson drives
less than 1600 miles?
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Ans: It is the area to the left of the standard normal curve.
Look up 2.67 in the Table of Normal Distributions.
There is a 99.62% probability that the salesperson drives
less than 1600 miles.
2.67
Using Table of the Normal Distribution
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Z = 2.67
Look up 2.6
in the row,
and .07 in
the column.
The
intersection
is the area to
the left, or
probability
Table
found
in text
page
A-1
back
of
book
Or Use Excel function
8
Enter:
Results:
What is the probability that a salesperson drives
more than 1600 miles?
9
2.67
Ans: It is the area to the right of the standard normal curve.
Since you know the are to the left of 2.67, the area to the
right must be 1 - .9962 = .0038, or .38% probability that
a salesperson drives more than 1600 miles.
What is the probability that a salesperson drives
between 1200 and 1600 miles?
It is the difference between driving less than 1600
10
and less than 1200.
2.67
Ans: The area to the left of 2.67 is already known, it is .9962.
Find the z value
for 1200, , then
look it up in the
table.
z
x1  


1200  1200
150
0
Between = .9962 - .5
= .4962
The probability that a salesperson drives between 1200 and 1600 miles is 49.62%
Questions / Comments /
Suggestions
11
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the main forum regarding this presentation.