Download AP Statistics: Section 8.1B Normal Approx. to a Binomial Dist.

AP Statistics: Section 6.3B
Normal Approx. to a Binomial Dist.
In section 6.2, we learned how to find the
find the mean, variance and standard
deviation of a probability distribution for a
discrete random variable X. Let’s use this
knowledge to find formulas for the mean
and variance of a random variable with a
binomial distribution.
Let’s consider a binomial random variable
B with a probability of success p. We could
describe B with the following probability
distribution where 1 represents a success
and 0 a failure.
Value of Bi
1 0
Probability pi p 1-p
0(1  p )  1( p )
p
(0  p) 2 (1  p)  (1  p) 2 p
p 2 (1  p)  (1  2 p  p 2 ) p
p 2  p3  p  2 p 2  p3
p  p 2  p(1  p)
B represents a single trial of some binomial
chance process. Let X = B1+B2+…..+Bn represent
the number of successes in n independent trials
of this chance process. In other words, X is a
binomial random variable with parameters n
and p. By the rules of section 7.2
p  p      p  np
p (1  p )  p (1  p )      p (1  p )  np (1  p )
If X has the distribution B(n, p),
then
x 
x 
n p
np (1  p )
Be careful : These short formulas are good only for binomial distributions.
They cannot be used for other discrete random variables.
Example 1: A Federal report finds that lie detector tests given to
truthful persons have a probability of 0.2 of suggesting that the
person is deceptive. A company asks 12 job applicants about
stealing from previous employers and used a lie detector test to
assess their truthfulness. Suppose that all 12 answered truthfully
and let X = the number of people who the lie detector test says
are being deceptive.
a) Find and interpret  x .
 x  (12)(.2)  2.4
If the lie detector test was given to many different groups of 12 job
applicants, the average number of times the test would indicate an
applicant was being deceptive is 2.4.
b) Find  x .
 x  12  .2  .8  1.386
The formula for binomial
probabilities gets quite
cumbersome for large values of n.
While we could use statistical
software or a statistical calculator,
here is another alternative.
The Normal Approximation to Binomial
Distributions:
Suppose that a count X has a binomial
distribution B(n, p). When n is large
(np _____
 10 and n(1 - p) _____),
 10
then the distribution of X is approximately
Normal, N(____,________)
np np(1 - p)
Example 2: Are attitudes towards shopping changing? Sample surveys show
that fewer people enjoy shopping than in the past. A survey asked a
nationwide random sample of 2500 adults if they agreed or disagreed that “I
like buying new clothes, but shopping is often frustrating and timeconsuming.” The population that the poll wants to draw conclusions about is
all U.S. residents aged 18 and over. Suppose that in fact 60% of all adult U.S.
residents would say “agree” if asked the same question. What is the
probability that 1520 or more of the sample would agree?
1  binomialcdf (2500,.6,1519)
1  .7869  .2131
Check conditions for Normal Approx.
(2500)(.6)  10
2500(.4)  10
1500  10
1000  10
 x  (2500(.6)  1500
 x  2500(.6)(.4)  24.4949
normalcdf (1520,100000,1500,24.4949)
.2071
The accuracy of the Normal approximation
improves as the sample size n increases.
It is most accurate for any fixed n when p
is close to ____
.5 and least accurate when p
is near ____
0 or ____
1 and the distribution is
________.
skewed