Transcript 8.1(II) mean,stddev,approx

Are You Psychic?
Mean and Standard Deviation
Remember, to find the mean of a probability distribution, you take
the sum of the probabilities times their value **sum of xP(x)
The binomial distribution has special formulas for the mean and
standard deviation
Mean(µ) = np
Standard Deviation(σ) =
Where: n = # of observations
np(1  p)
p = probability of success
Beware!!!
These formulas are BINOMIAL specific
and don’t work for other distributions.
Mean and Standard Deviation
A university claims that 80% of its basketball players get degrees.
An investigation examines the fate of all 20 players who entered
the program over a period of several years that ended six years
ago. Of these players, 11 graduated and the 9 remaining are no
longer in school.
Answer
How many athletes can the university expect
to graduate out of their next crop of 30
30(.55) = 16.5
athletes?
What is the standard deviation for the group of 30?
30(.55)(1  .55)  2.7249
If the same group had a probability of success of .8, what
would the standard deviation be?
30(.8)(1  .8)  2.1909
… p = .9?
30(.9)(1  .9)  1.6432
Notice that our σ gets smaller
as our p value gets closer to 1.
Normal Approximation
As the number of trials n, gets larger, a binomial
probability distribution gets close to becoming a
Normal distribution
What does that have to do with us?
Statisticians often use a “normal” approximation to find binomial
probabilities for high sample spaces.
How do use a normal curve
to find probabilities?
Change your X to Z and find
the appropriate area!!
Why would we use this?
This method was used to shorten large cumulative binomial
functions before we had cdf’s in the calculator. While the calculator
is better, we still need to know how and when to use the normal
approximation.
The “When” of the Normal
Approximation
In order to use the normal approximation, the following 2
conditions must be met:
 np ≥ 10
 n(1-p) ≥ 10
**Be sure to check these 2 conditions before you use a normal
approximation
Many local polls of public opinion use samples of size 400 to 800.
Consider a poll of 400 adults in Richmond that asks the question, “Do
you approve of President George W. Bush’s response to the World
Trade Center terrorist attacks in September 2001?” Suppose we know
the President’s approval rating on this issue nationally is 92%. You are
asked to calculate the probability that at most 358 of the 400 adults in
the Richmond Poll answer “yes” to the question.
Check the conditions and
decide if a normal
approximation would
appropriate.
400(.91) = 368 ≥ 10
400(.09) = 32 ≥ 10
Normal
Approximation
is appropriate.
The “How” of the Normal
Approximation
Here are the steps to a normal
approximation:



Find the µ and σ of the distribution.
Change the X to a Z score
Use the z chart or your calculator to find the
appropriate area under the curve.
Let’s Do It!!
Many local polls of public opinion use samples of size 400 to 800.
Consider a poll of 400 adults in Richmond that asks the question, “Do
you approve of President George W. Bush’s response to the World
Trade Center terrorist attacks in September 2001?” Suppose we know
the President’s approval rating on this issue nationally is 92%. Use the
Normal Approximation to calculate the probability of at most 358 people
in the sample approving.
µ = 400(.92) = 368
X -> Z
σ = sqrt(400*.92*.08) = 5.4259
(358 – 368) / (5.4259) = -1.84
Z-area for -1.84 = .0329
Try getting the answer using your calculator and the Binomial CDF
function.
binomcdf(400,.92,358 = ..0441
Note about the Normal
Approximation
Notice the difference between our approximate
answer and the exact answer of the pdf.

(.0441) – (.0329) = .0112 difference
Not large, but not really good either!!!
Accuracy of the Normal Approximation
The Normal Approximation is MORE accurate when p is closer to ½.
The Normal Approximation is LESS accurate when p is closer to
0 or 1.
Homework
Read Pages 457,58 on
Simulating Binomial
Experiments
Do Problem #’s
25,27-36
Bring today’s homework and what was Due today next class
to be checked for a homework grade. We will check answers
then as well.