Statistics for business decision making and analysis robert stine and foster chapter 12

Chapter 12

The Normal
Probability Model

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12.1 Normal Random Variable
Black Monday (October, 1987) prompted
investors to consider insurance against
another “accident” in the stock market.
How much should an investor expect to pay
for this insurance?


Insurance costs call for a random variable
that can represent a continuum of values
(not counts)
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12.1 Normal Random Variable
Percentage Change in Stock Market Data

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12.1 Normal Random Variable
Prices for One-Carat Diamonds

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12.1 Normal Random Variable


With the exception of Black Monday, the
histogram of market changes is bellshaped



The histogram of diamond prices is also
bell-shaped



Both involve a continuous range of values
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12.1 Normal Random Variable
Definition
A continuous random variable whose
probability distribution defines a standard
bell-shaped curve.

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12.1 Normal Random Variable
Central Limit Theorem
The probability distribution of a sum of
independent random variables of
comparable variance tends to a normal
distribution as the number of summed
random variables increases.

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12.1 Normal Random Variable
Central Limit Theorem Illustrated

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12.1 Normal Random Variable
Central Limit Theorem


Explains why bell-shaped distributions are
so common



Observed data are often the accumulation
of many small factors (e.g., the value of the
stock market depends on many investors)
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12.1 Normal Random Variable
The Normal Probability Distribution


Defined by the parameters µ and σ2



The mean µ locates the center



The variance σ2 controls the spread

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12.1 Normal Random Variable
Normal Distributions with Different µ’s

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12.1 Normal Random Variable
Normal Distributions with Different σ’s

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12.1 Normal Random Variable
Standard Normal Distribution (µ = 0; σ2 = 1)

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12.1 Normal Random Variable
Normal Probability Distribution


A normal random variable is continuous
and can assume any value in an interval



Probability of an interval is area under the
distribution over that interval (note: total

area under the probability distribution is 1)
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12.1 Normal Random Variable
Probabilities are Areas Under the Curve

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12.2 The Normal Model
Definition
A model in which a normal random variable
is used to describe an observable random
process with µ set to the mean of the data
and σ set to s.

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12.2 The Normal Model
Normal Model for Stock Market Changes

Set µ = 0.972% and σ = 4.49%.
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12.2 The Normal Model
Normal Model for Diamond Prices

Set µ = $4,066 and σ = $738.
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12.2 The Normal Model
Standardizing to Find Normal Probabilities
Start by converting x into a z-score

z=

x−µ
σ

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12.2 The Normal Model
Standardizing Example: Diamond Prices
Normal with µ = $4,066 and σ = $738
Want P(X > $5,000)
5,000 − 4,066
 X − µ 5,000 − µ 



P( X > $5,000) = P
>
≈ 1.27 
 = P Z >
σ
738
 σ




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12.2 The Normal Model
The Empirical Rule, Revisited

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4M Example 12.1:
SATS AND NORMALITY
Motivation
Math SAT scores are normally distributed with
a mean of 500 and standard deviation of
100. What is the probability of a company
hiring someone with a math SAT score of
600?

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4M Example 12.1:
SATS AND NORMALITY
Method – Use the Normal Model

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4M Example 12.1:
SATS AND NORMALITY
Mechanics
A math SAT score of 600 is equivalent to
z = 1. Using the empirical rule, we find that
15.85% of test takers score 600 or better.

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