Transcript Chapter 6, part C

Chapter 6, part C
III. Normal Approximation of
Binomial Probabilities
When n is very large, computing a binomial gets
difficult, especially with smaller pocket
calculators.
The Situation
If a binomial problem has the following
characteristics, you can use the normal probability
distribution to approximate the binomial
probability.
• n>20
• np  5, and
• n(1-p)  5 (recall that p is the probability of
“success”)
An Example
A firm has found that 10% of their sales invoices
contain errors. If the firm takes a sample of 100
invoices, what is the probability that 12 have
errors?
Steps to approximate with the
normal
1. Calculate a mean and standard deviation:
 = np = 100(.10) = 10
  np(1  p)  100(.10)(.90)  3
2. Create an interval around x=12 by adding and
subtracting .5 from 12.
=10 11.5 12 12.5 x
Steps continued...
3. Find P(11.5  x  12.5)
4. Convert the range to z-scores.
zL = (11.5-10)/3 = .5
zH = (12.5-10)/3 = .83
5. Use the standard normal probability table to find:
P(.5  z  .83)
Steps continued...
6. Find P(0  z  .83) - P(0  z  .5)
= .2967 - .1915 = .1052
The binomial solution to this same problem is .0988,
so our normal approximation is fairly accurate.
Check out this simulation (you browser needs to be
Java compatible) and choose p and sample size n.
IV. Exponential Probability
Distribution
The exponential is used to describe the time (and
probability) that it takes to do something.
For example, it can be used to calculate the
probability that a delivery truck will be loaded in
15 to 30 minutes time.
A. Exponential Probability Density
function
f ( x) 
1

e
x
( )

For x>0 and >0.
As an example, let’s suppose that a
delivery truck is loaded with a mean time
of  =10 minutes.
B. Computing Probabilities with the
Exponential
The function f(x)=(1/10)e(-x/10) draws the curve
below, but probabilities are still calculated as the
area under the curve.
For any x0, if you want the probability that the truck
is loaded in less than that time, use the following
formula:
P( x  x0 )  1  e
(
x0

)
A diagram of the exponential
f(x)
.10
f(x)=(1/10)e(-x/10)
0
5
10
20
30
x (time)
Example probabilities
Find the probability that the loading will take less
than 5 minutes:
P(x5) = 1-e(-5/10)=.3935
What about a loading time of less than 30 minutes?
P(x30) = 1-e(-30/10)=.9502