Chapter 6, part C
Transcript Chapter 6, part C
Chapter 6, part C
III. Normal Approximation of
When n is very large, computing a binomial gets
difficult, especially with smaller pocket
If a binomial problem has the following
characteristics, you can use the normal probability
distribution to approximate the binomial
• np 5, and
• n(1-p) 5 (recall that p is the probability of
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
Steps to approximate with the
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
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)
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
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
f ( 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
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
P( x x0 ) 1 e
A diagram of the exponential
Find the probability that the loading will take less
than 5 minutes:
P(x5) = 1-e(-5/10)=.3935
What about a loading time of less than 30 minutes?
P(x30) = 1-e(-30/10)=.9502