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 distributi ons. 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 t est was given to many different groups of 12 job applicants , the average number of times the test woul d 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? Check conditions 1 binomialcd f ( 2500 ,. 6 ,1519 ) 1 . 7869 . 2131 for Normal Approx. ( 2500 )(. 6 ) 10 2500(.4) 10 1500 10 1000 10 x ( 2500 (. 6 ) 1500 normalcdf x 2500 (. 6 )(. 4 ) 24 . 4949 (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.