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

pn

)1( pnp

. variablesrandom discreteother for used becannot They

ons.distributi binomialfor only good are formulasshort These :careful Be

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.

4.2)2)(.12( xa) Find and interpret x .

b) Find x .

2.4. is deceptive being wasapplicant

an indicate d test woul the timesofnumber average the,applicants

job 12 of groupsdifferent many given to est wasdetector t lie theIf

386.18.2.12 x

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 _____ and n(1 - p) _____), then the distribution of X is approximately

Normal, N(____,________)

10 10

p)-np(1 np

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 time-

consuming.” 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?

2131.7869.1

)1519,6,.2500(1

fbinomialcd

101000 101500

102500(.4) 10)6)(.2500(

Approx. Normalfor conditionsCheck

4949.24)4)(.6(.2500 1500)6(.2500( x x

.2071

)4949.24,1500,100000,1520(normalcdf

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 ____ and least accurate when p is near ____ or ____ and the distribution is

________.

5.0 1

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.