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

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The Normal Probability Model

Chapter 12

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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?

that can represent a continuum of values

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Percentage Change in Stock Market Data

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Prices for One-Carat Diamonds

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 With the exception of Black Monday, the

histogram of market changes is

bell-shaped

 The histogram of diamond prices is also

bell-shaped

 Both involve a continuous range of values

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Definition

A continuous random variable whose

probability distribution defines a standard

bell-shaped curve.

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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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Central Limit Theorem Illustrated

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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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The Normal Probability Distribution

 Defined by the parameters µ and σ 2

 The mean µ locates the center

 The variance σ 2 controls the spread

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Normal Distributions with Different µ’s

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Normal Distributions with Different σ’s

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Standard Normal Distribution (µ = 0; σ 2 = 1)

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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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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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Normal Model for Stock Market Changes

Set µ = 0.972% and σ = 4.49%.

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Normal Model for Diamond Prices

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Standardizing to Find Normal Probabilities

Start by converting x into a z-score

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σ−µ

z

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Standardizing Example: Diamond Prices

Normal with µ = $4,066 and σ = $738

5 000

,

5 000

, 5

µ

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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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SATS AND NORMALITY

Method – Use the Normal Model

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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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SATS AND NORMALITY

Message

About one-sixth of those who take the math

SAT score 600 or above Although not that common, a company can expect to find

candidates who meet this requirement.

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Using Normal Tables

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Example: What is P(-0.5 ≤ Z ≤ 1)?

0.8413 – 0.3085 = 0.5328

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12.3 Percentiles

Example:

Suppose a packaging system fills boxes such that the weights are normally distributed with a µ = 16.3

oz and σ = 0.2 oz The package label states the weight as 16 oz To what weight should the mean

of the process be adjusted so that the chance of an underweight box is only 0.005?

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12.3 Percentiles

Quantile of the Standard Normal

The p th quantile of the standard normal probability distribution is that value of z such that P(Z ≤ z ) = p.

Example: Find z such that P(Z ≤ z ) = 0.005.

z = -2.578

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12.3 Percentiles

Quantile of the Standard Normal

Find new mean weight (µ) for process

(2 5758) 16 52 2

0 16

5758

2 2

0

16

≈ +

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4M Example 12.2: VALUE AT RISK

Motivation

Suppose the $1 million portfolio of an investor

is expected to average 10% growth over the next year with a standard deviation of 30% What is the VaR (value at risk) using the

worst 5%?

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Method

The random variable is percentage change

next year in the portfolio Model it using

the normal, specifically N(10, 30 2 ).

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Mechanics

This works out to a change of -39.3%

µ - 1.645σ = 10 – 1.645(30) = -39.3%

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12.4 Departures from Normality

 Multimodality More than one mode

suggesting data come from distinct groups.

 Skewness Lack of symmetry.

 Outliers Unusual extreme values.

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Normal Quantile Plot

 Diagnostic scatterplot used to determine

the appropriateness of a normal model

 If data track the diagonal line, the data are normally distributed

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Normal Quantile Plot (Diamond Prices)

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Normal Quantile Plot

Points outside the dashed curves, normality not indicated.

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z

3

3 2

3 1 3

+ +

=

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4 1

4 = + + + −

n

z z

z

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Best Practices

 Recognize that models approximate what will happen.

 Inspect the histogram and normal quantile

plot before using a normal model.

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Best Practices (Continued)

 Estimate normal probabilities using a

sketch and the Empirical Rule.

 Be careful not to confuse the notation for

the standard deviation and variance.

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the distribution of data

prove that the data are normally distributed.

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