Você está na página 1de 6

# Chapter 6 Normal Distribution

6-1

## Chapter 6 Learning Objectives (LOs)

LO 6.1 Describe a continuous random variable.
LO 6.2 Calculate and interpret probabilities for a random
variable that follows the continuous uniform
distribution.
LO 6.3 Explain the characteristics of the normal distribution.
LO 6.4 Use the standard normal table (z table).
LO 6.5 Calculate and interpret probabilities for a random
variable that follows the normal distribution.
LO 6.6 Calculate and interpret probabilities for a random
variable that follows the exponential distribution.
LO 6.7 Calculate and interpret probabilities for a random
variable that follows the lognormal distribution.

Continuous Probability
Distributions
Communicating with Numbers, 2e
By Sanjiv Jaggia and Alison Kelly

11/13/2016

6-2

## 6.1 Continuous Random Variables and

the Uniform Probability Distribution

## 6.2 The Standard Normal Distribution

LO6.3Explainthecharacteristicsofthenormaldistribution.

LO6.1Describeacontinuousrandomvariable.

classified as
Discrete

## The random variable assumes a countable

number of distinct values.

Continuous

## The random variable is characterized by

(infinitely) uncountable values within any
interval.

LO6.1

6-3

LO6.3

## Mean = Median = Mode

Asymptoticthat is, the
tails get closer and
closer to the
P(X < ) = 0.5
horizontal axis,
but never touch it.

6-4

LO6.3

## 6.2 The Standard Normal Distribution

Symmetric
Bell-shaped
Closely approximates the probability distribution of a
wide range of random variables, such as the

Scores on SAT

## Cumulative debt of college graduates

Serves as the cornerstone of statistical inference.

## The normal distribution is completely described

by two parameters: and 2.

## is the population mean which describes the

central location of the distribution.

## is the population variance which describes

the dispersion of the distribution.

6-5

LO6.3

6-6

SOM@USM

6-2

## 6.2 The Standard Normal Distribution

Example:SupposetheagesofemployeesinIndustriesA,
B,andCarenormallydistributed.

Herearetherelevantparameters:

LetscompareindustriesusingtheNormalcurves.

## is the same, is different.

LO6.3

LO6.4Usethestandardnormaltable(ztable).

Aspecialcaseofthenormaldistribution:
Mean()isequaltozero(E(Z)=0).
Standarddeviation()isequaltoone
(SD(Z)=1).

6-7

LO6.4

## 6.2 The Standard Normal Distribution

TheStandardNormal(Z)Distribution.

6-8

## 6.2 The Standard Normal Distribution

StandardNormalTable(Z Table).

## GivesthecumulativeprobabilitiesP(Z < z)forpositive

andnegativevaluesofz.

meanof0,

## P(Z <0)=P(Z >0) =0.5.

thenacrossthetop.

LO6.4

6-9

LO6.4

## Transform normally distributed random variables into

standard normal random variables and use the z
table to compute the relevant probabilities.
The z table provides cumulative probabilities
P(Z < z) for a given z.

## Portion of right-hand page of z table.

If z = 1.52, then look up

LO6.4

6-10

6-11

## Remember that the z table provides cumulative

probabilities P(Z < z) for a given z.
Since z is negative, we can look up this
probability from the left-hand page of the z table.
Portion of left-hand page of Z Table.
If z = -1.96, then look up

LO6.4

6-12

SOM@USM

6-3

## Find P(-1.52 < Z < 1.96) =

P(Z < 1.96) P(Z < -1.52 ) =
P(Z < 1.96) = 0.9750

LO6.4

## Note the 1s are for

cumulative! A 0 would
be nonsensical for a
continuous distribution.

6-13

LO6.4

6-15

LO6.4

6-16

## 6.2 The Standard Normal Distribution

Example: The Empirical Rule

LO6.4

## For a standard normal variable Z, find the z

values that satisfy P(Z < z) = 0.6808.

## Round NORM.S.INVs output

to 2 decimal places in your
HW BEFORE doing other
calculations!

6-14

## Example: Alternately, use Excel to find a z

value for a given probability.

## For a standard normal variable Z, find the z

values that satisfy P(Z < z) = 0.6808.
Go to the standard normal table and find 0.6808
in the body of the table.
Find the corresponding
z value from the
row/column of z.
z = 0.47.

LO6.4

## Example: Finding a z value for a given

probability.

Find P(-1.52 < Z < 1.96) = P(Z < 1.96) P(Z < -1.52 )

Round NORM.S.INVs
output to 4 decimal places in
doing other
calculations!

## Example: Alternately, use Excel to look up

probabilities for standard normal variables.

## Example: Finding Probabilities for a

Standard Normal Random Variable Z.

6-17

LO6.4

## An investment strategy has an expected return of

4% and a standard deviation of 6%. Assume that
investment returns are normally distributed.
What is the probability of earning a return greater
than 10%?

## A return of 10% is one standard deviation

above the mean, or 10 = + 1 = 4 + 6.

## Since about 68% of observations fall within

one standard deviation of the mean, 32%
(100% 68%) are outside the range.
6-18

SOM@USM

6-4

## 6.3 Solving Problems with

the Normal Distribution

## 6.2 The Standard Normal Distribution

Example: The Empirical Rule

## An investment strategy has an expected return of

4% and a standard deviation of 6%. Assume that
investment returns are normally distributed.
What is the probability of earning a return greater
than 10%?

Using symmetry, we
conclude that 16%
(half of 32%) of the
observations are
greater than 10%.

## The Normal Transformation

Any normally distributed random variable X with
mean and standard deviation can be
transformed into the standard normal random
variable Z as:

16%

16%
68%

## with corresponding values

2
( )

LO6.4

LO6.5Calculateandinterpretprobabilitiesforarandomvariable
thatfollowsthenormaldistribution.

( + )

6-19

LO6.5

6-20

## 6.3 Solving Problems with

the Normal Distribution

## 6.3 Solving Problems with

the Normal Distribution

## A z value specifies by how many standard

deviations the corresponding x value falls
above (z > 0) or below (z < 0) the mean.

## Use the Inverse Transformation to compute

probabilities for given x values.

LO6.5

6-21

LO6.5

## 6.3 Solving Problems with

the Normal Distribution

## A standard normal variable Z can be transformed

to the normally distributed random variable X
with mean and standard deviation as

## A positive z indicates by how many standard

deviations the corresponding x lies above .
A zero z indicates that the corresponding x
equals
A negative z indicates by how many standard
deviations the corresponding x lies below

## First transform the random variable X to Z using the

transformation formula:

## Using the standard normal table, find

P(Z > -1.5) = 1 P(Z < -1.5) = 1 0.0668 = 0.9332

LO6.5

6-22

## 6.3 Solving Problems with

the Normal Distribution - Excel

## Example: Scores on a management aptitude exam

are normally distributed with a mean of 72 () and
a standard deviation of 8 ().
What is the probability that a randomly selected
manager will score above 60?

## Example: Scores on a management aptitude exam

are normally distributed with a mean of 72 () and
a standard deviation of 8 ().
What is the probability that a randomly selected
manager will score above 60?

6-23

LO6.5

## Alternately, Excel will return the probability directly if

you use NORM.DIST() and give it the mean and
standard deviation.

6-24

SOM@USM

6-5

## Introductory Case: Demand for Salmon

Based on this information, Akiko would like to:
Calculate the proportion of days that demand for
salmon at Little Ginza was above her earlier
purchase of 20 pounds.
Calculate the proportion of days that demand for
salmon at Little Ginza was below 15 pounds.
Determine the amount of salmon that should be
bought daily so that it meets demand on 90% of
the days.

## Akiko Hamaguchi, manager of a small sushi

restaurant, Little Ginza, in Phoenix, Arizona, has to
estimate the daily amount of salmon needed.
Akiko has estimated the daily consumption of
salmon to be normally distributed with a mean of
12 pounds and a standard deviation of 3.2 pounds.
Buying 20 lbs of salmon every day has resulted in
too much wastage.
Therefore, Akiko will buy salmon that meets the
daily demand of customers on 90% of the days.

6-25

6-26

## 6.3 Solving Problems with

the Normal Distribution
amount of salmon for Little Ginza. The daily consumption is
normally distributed with a mean of 12 pounds and a
standard deviation of 3.2 pounds.

## c. How much salmon should she buy so she meets customer

demand on 90% of the days?
In order to answer this, solve for the z-score of a 90%
probability and then plug that z into the equation:
or just use NORM.INV() in Excel:

## App A Table 1. Standard Normal Curve

LO6.5

amount of salmon for Little Ginza. The daily consumption is
normally distributed with a mean of 12 pounds and a
standard deviation of 3.2 pounds.

6-27

LO6.5

6-28

## 6.3 Solving Problems with

the Normal Distribution

SOM@USM

6-6

End of Chapter 6
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}