Download Normal Distribution

Section 1.3: The Normal Distribution
Learning goals for this chapter:
Know when and how to use the empirical (68-95-99.7% rule).
Understand what the standard Normal distribution is and how it is related to other
Normal distributions.
Calculate both forwards and backwards Normal distribution problems.
Draw a Normal distribution curve appropriate for a story.
Normal curves
• N( , σ)
• Bell-shaped, unimodal, symmetric
• Mean, , always in the center of the curve (the peak)
• Standard deviation, σ, controls the spread of the graph (wide or narrow)
• Probabilities are just the area under the curve (integral) between the points of
interest
• Total area under the Normal curve = 1 (or 100%)
• Curve stretches from - to + , but the area under the curve gets very small the
farther you go from the mean.
In the last set of Chapter 1 notes, we discussed bell-shaped distributions, standardization,
and the 68-95-99.7% rule.
Standard Normal Distribution
What if you need different probabilities for X N( , )? Do we have to use Calculus?
No. We have a great shortcut—the Normal table, Table A, in the front cover of your
book.
You must convert X N( , ) to Z N(0, 1), where Z has the standard Normal
distribution. Convert using the formula:
x
Z
Z-scores are what you need in order to use Table A in the front cover of your book.
Z-scores also let you compare 2 values from different Normal distributions to see their
probabilities on the same scale.
P (Z < z-score) is what you will find on the Normal table. What if you want to know
something else?
P(Z > z-score) = 1 – P(Z z-score)
P(Z = z-score) = 0. Therefore
1
P(Z
z score)
P(Z
z score) and P(Z
z score)
P(Z
z score)
P(a < Z < b) = P(Z < b) – P(Z < a).
Z-scores tell you how far (measured in standard deviations) the original observations fall
from the mean.
To find a probability if you have X
N( , ) and a sample score to work with:
x
1.
Convert X to Z. Z
2.
Rearrange (if necessary) the inequality so that it uses < or
Remember that P(Z > z-score) = 1 – P(Z z-score).
3.
Look up the probability for your z-score on Table A.
4.
If z-score is between 2 table values, either pick the closer one or average the two
closest values.
.
Example: Checking account balances are ~ N(1325, 25). Bill has a balance of $1270.
a)
What is the probability an account will have less money than Bill’s?
b)
What is the probability an account balance will be more than $1380?
c)
What is the probability an account balance will be exactly $1380?
d)
What is the probability an account will have less than $1325 (the mean)?
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e)
What is the probability that an account will have between $1310 and
$1390?
f)
What is the probability an account will have less than $10?
“Backwards” Normal Problems
If you are given the probability and know X N( , ), but you don’t know the sample’s
score (backwards from the previous problems):
1.
Treat it as P(Z < z-score) = the probability. Work backward from the probability
in Table A to a corresponding z-score.
2.
Adjust to < if necessary by doing the ―1 –― trick.
3.
If you have a 2-sided probability, use P(-z0 < Z < z0) = 2 P(Z < z-score) – 1.
4.
Convert the z-score to x by converting with x =
+z .
Example: In the checking account example where the balances are ~ N (1325, 25),
a)
What is the account balance, x0, such that the percentage of balances less
than it is 23%?
b)
What is the account balance, x0, such that the probability of a balance
being more than it is 0.15?
c)
Between what 2 central values do 40% of the balances fall?
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Mixed forwards and backwards problems:
IQ Scores are Normally distributed with a mean of 100 and a standard deviation of 15.
1. What is the IQ range for the top 10% of people?
2. What percent of the population scores between 100 and 120?
3. Between what two scores does the central 20% of the population have?
4. If a person is selected at random, what is the chance he scores below an 85?
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