Download Chapter 7b 7b-1

Chapter 7b
7b-1
Business Statistics:
A First Course
5th Edition
Learning Objectives
In this part of the chapter, you learn:
To compute probabilities from the normal
approximation of the binomial distribution
! 
To determine probabilities from the logarithmic
distribution
! 
Chapter 7
Part b
Continuous
Probability Distributions
BUS210: Business Statistics
Continuous Probability Distributions - 2
Normal Approximation
Probability Distributions
Probability
Distributions
Discrete
Probability
Distributions
Binomial
Normal
Poisson
Exponential
! 
Continuous Probability Distributions - 3
! 
In this case, the normal probability distribution provides
an easy-to-use approximation of binomial probabilities
BUS210: Business Statistics
Continuous Probability Distributions - 4
Normal Approximation
of the Binomial
of the Binomial
Example: Coffee and a Donut
Recall from the binomial that!
and
! = n" (1! " )
! 
A large donut company states that 60% of their customers order
coffee along with their donut. What is the probability that 11 of
next 15 customers will order coffee?
We can use that information in the formula
z=
! 
A dataset is large whenever the following are all true:
!  n > 20, and
!  n! > 5, and
Note: "
Some textbooks use "
n! > 10 and n(1-!) > 10"
!  n(1 - !) > 5.
Normal Approximation
µ = n!
! 
! 
Ch. 7
BUS210: Business Statistics
When the number of trials, n, becomes large,
evaluating the binomial probability function by hand or
even with a calculator becomes difficult.
! 
Continuous
Probability
Distributions
Ch. 6
of the Binomial
x!µ
!
! 
First, calculate mean and standard deviation:
µ = n! = 15(0.6) = 9
However, we must add or subtract a 0.5 continuity
correction factor to x:
• 
• 
! = n" (1# " ) = 9(0.4) = 1.897
A discrete distribution is approximated by a continuous.
Example: P(x = 10) is approximated by P(9.5 < x < 10.5).
Note: The correction factor should always EXPAND the area of interest.
BUS210: Business Statistics
BUS210 Business Statistics
Continuous Probability Distributions - 5
BUS210: Business Statistics
Continuous Probability Distributions - 6
NSCC
Chapter 7b
7b-2
Normal Approximation
of the Binomial
Probability Distributions
Probability
Distributions
Example: Coffee and a Donut (Continued)
! 
Next, restate the problem in terms of the correction factor:
Discrete
Probability
Distributions
P(10.5 ! x ! 11.5)
! 
Next, solve for area under the normal curve:
zL =
x L ! µ 10.5 ! 9.0
=
= 0.79
"
1.897
zU =
xU ! µ 11.5 ! 9.0
=
= 1.32
"
1.897
P(10.5 ! x ! 11.5) = P(0.79 ! z ! 1.32) =
P(z ! 1.32) " P(z ! 0.79) = 0.4066 " 0.2852 = 0.1214
Continuous
Probability
Distributions
Binomial
Normal
Poisson
Exponential
Ch. 6
BUS210: Business Statistics
! 
! 
Ch. 7
BUS210: Business Statistics
Continuous Probability Distributions - 8
Exponential
Exponential
Probability Distribution
Probability Distribution
Used when you are interested in!
the interval in between occurrences.
! 
! 
Continuous Probability Distributions - 7
Distribution Function:
f (x) = ! e" ! x
Usually focused on time (or a function of time)
The reciprocal of the Poisson distribution
Exponential random variables can be used to
describe such things as:
!  Time between computer crashes
!  Time between vehicle arrivals at a toll booth
!  Distance between potholes on the highway
Where:
Continuous Probability Distributions - 9
1 ! x/µ
e
µ
(same as Poisson µ)
Cumulative Probabilities:
P(x ! x0 ) = 1" e" # x
P(x > x0 ) = e! " x
BUS210: Business Statistics
Continuous Probability Distributions - 10
Exponential
Exponential
Probability Distribution
Example:
Al would like to know the probability that the time
between two successive arrivals will be 2 minutes or less.
f (x) =
" = mean arrival rate
Probability Distribution
The time between arrivals of cars at Al s full-service gas
pump follows an exponential probability distribution with a
mean arrival of 1 car every 3 minutes.
Note: Sometimes
shown as
e = 2.71828
Note: In this case, distance is a function of time (distance = speed x time)
BUS210: Business Statistics
for x # 0
f(x)
P(x ! x0 ) = 1" e" # x = 1" 2.718 "2/3
= 1" .5134 = .4866
.4
.3
.2
.1
x
1
2
3
4
5
6
7
8
9 10
Time Between Successive Arrivals (mins.)
BUS210: Business Statistics
BUS210 Business Statistics
Continuous Probability Distributions - 11
BUS210: Business Statistics
Continuous Probability Distributions - 12
NSCC
Chapter 7b
7b-3
Exponential
Summary
Probability Distribution
Example: Customer Waiting Time
! 
Between 2 p.m. and 4 p.m. on Wednesday, patient insurance
inquiries arrive at Blue Choice insurance at a mean rate of 2.2
calls per minute.
What is the probability of waiting more than 30 seconds
(i.e., 0.50 minutes) for the next call?
Presented normal approximation of the binomial
distribution
! 
Presented logarithmic probability distribution
! 
Applied normal approximation and the
logarithmic distribution to problems
Set " = 2.2 events/min and x = 0.50 min
P(x > 0.50) = e–"x = e–(2.2)(0.5) = .3329
There is approximately a 33.3% chance of waiting more
than 30 seconds for the next call.
BUS210: Business Statistics
BUS210 Business Statistics
Continuous Probability Distributions - 13
BUS210: Business Statistics
Continuous Probability Distributions - 14
NSCC