Statistical techniques in business ecohomics chap007

The Uniform distributionwhere a and b are the minimum and maximum values Has a standard deviation... Has its mean, , to determine its location and its standard deviation, , to det

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Chapter Seven

Continuous Probability Distributions

GOALS

When you have completed this chapter, you will

be able to: ONE

Understand the difference between discrete and continuous

Chapter Seven continued

GOALS

When you have completed this chapter, you

will be able to:

variables which can

assume only clearly

The Uniform distribution

where a and b are

the minimum and maximum values

Has a standard deviation

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Calculates its height as

(b-a)

Calculates its area as

The uniform distribution

Suppose the time

that you wait on the

telephone for a live

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What is the probability of waiting more than ten minutes?

The area from 10

is bell-shaped and has a single peak at the

center of the distribution

Is symmetrical about the mean.

is asymptotic That is the curve gets closer and

closer to the X-axis but never actually touches it.

Has its mean, , to determine its location and its standard deviation, , to determine its

dispersion.

The Normal probability distribution

The Standard Normal Probability Distribution

A z- value is the distance between a selected

value, designated X, and the population mean ,

divided by the population standard deviation, 

The formula is:

It is also called the

graduates follows the normal

distribution with a mean of $2,000 and

a standard deviation

of $200 What is the z- value for a salary of $2,200?

M B A

EXAMPLE 2 continued

50

1 200

$

000 ,

2

$ 700

, 1

What is the

z-value for

$1,700?

A z-v alue of 1 indicates that the value of

$2,200 is one standard deviation above the

mean of $2,000 A z-v alue of –1.50

indicates that $1,700 is 1.5 standard deviation

below the mean of $2000

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Areas Under the Normal

Curve

Practically all is within three standard

deviations of the mean

 + 3

About 68 percent of

the area under the

normal curve is within

one standard deviation

Example 3

The daily water usage per

person in New Providence,

New Jersey is normally

distributed with a mean of

20 gallons and a standard

deviation of 5 gallons

About 68 percent of those

living in New Providence

will use how many gallons

of water?

About 68% of the daily

water usage will lie between

15 and 25 gallons (+ 1  ).

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EXAMPLE 4

00

05

80

05

What is the probability that

a person from New Providence selected at random will use between 20 and 24 gallons per day?

Example 4 continued

The area under a normal

curve between a z-value of

0 and a z-value of 0.80 is

0.2881

We conclude that 28.81

percent of the residents use

between 20 and 24 gallons

of water per day.

See the following diagram

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EXAMPLE 4 continued

40

0 5

20

1 5

the population use

between 18 and 26

gallons per day?

z -value of 1.20 is

3849.

Adding these areas, the result is

5403.

EXAMPLE 5

Professor Mann has

determined that the scores

in his statistics course are approximately normally

distributed with a mean of

72 and a standard deviation

of 5 He announces to the class that the top 15 percent

of the scores will earn an A What is the lowest score a student can earn and still receive an A?

If 15 percent of the students score

more than X, then 35 percent must

score between the mean of 72 and X.

EXAMPLE 5 continued

2 77 2

5 72

) 5 ( 04

1 72

5

72 04

more earn an A.

We let z equal 1.04 and

solve the standard normal

equation for X The result

is the score that separates

students that earned an A

from those that earned a B.

distribution when n and

The Normal Approximation

continued

Recall for the binomial experiment:

oThere are only two mutually exclusive

outcomes (success or failure) on each trial.

oA binomial distribution results from counting

the number of successes.

oEach trial is independent

oThe probability is fixed from trial to trial, and

the number of trials n is also fixed.

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Continuity Correction Factor

The value 5 subtracted or added, depending on

the problem, to a selected value when a

binomial probability distribution (a discrete

probability distribution) is being approximated

by a continuous probability distribution (the

normal distribution).

Continuity Correction Factor

Continuity Correction Factor

For the probability at

least X occur, use the

area above (X-.5)

For the probability

that more than X

occur, use the area

above (X+.5)

For the probability that

X or fewer

occur, use the area below (X+.5)

many of the homes

would you expect

to have video

cameras?

This is the mean of a binomial distribution

EXAMPLE 6 continued

0498

5 5

.





What is the standard deviation?

What is the variance?

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EXAMPLE 6 continued

88

1 0498

5

0 30 5

From Appendix D the area between 0 and 1.88

on the z scale is 4699.

= 9699.

homes have a video camera is about 97%

EXAMPLE 6 continued

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