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

AP Statistics: Section 8.1B
Normal Approx. to a Binomial Dist.
In chapter 7, we learned how to find
the find the mean, variance and
standard deviation of a probability
distribution for a discrete random
variable X. This work is greatly
simplified for a random variable with a
binomial distribution.
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: The count X of bad switches is
binomial with n = 10 and p = 0.1.
Determine the mean and standard
deviation of the binomial distribution.
 x  (10)(.1)  1
 x  10(.1)(.9)  .9487
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
Binomial Distributions with the
Calculator
See pages 530-532 to determine how to
graph binomial distribution histograms on
your calculator.
See pages 533-534 to determine how to
simulate a binomial event on your
calculator.