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4.4 Normal Approximation to Binomial Distributions Binomial Probability Distribution The probability of k successes in n trials: n k P( X k ) ( p) (1 p) n k k This represents getting exactly k successes. E.g., Toss a coin 50 times. The probability of getting exactly 30 heads is P(X = 30) = C(50, 30)(0.5)30(0.5)20 = 0.042 Let’s complicate things… • What is the probability of getting between 20 and 30 heads if you flip a coin 50 times? • This is a more complex situation: = P(X = 20) + P(X = 21) + … + P(X=28) + P(X = 29) + P(X = 30) • Time-consuming calculation (even though numbers not very large) • Use a graphical representation of binomial distribution Go to the handout! Copy Jarvis C.I./Pick Up/Data Management/Unit 4/XL_NormalApproximation.xls into your home drive and open it. Follow directions on handout. We will discuss this in 15 minutes. Normal Approximation of Binomial Distribution • Binomial distributions can be approximated by normal distributions as long as the number of trials is relatively large. • Binomial distributions – Display discrete random variables (whole num.) • Normal distributions – Display continuous values Normal Approximation of Binomial Distribution • To use normal distribution (continuous variables) to approx. binomial distributions (discrete variables): • Use a range of values • If X = 5, consider all values from 4.5 to 5.5 • If X = 3, 4, 5, consider all values from 2.5 to 5.5 • This is called a continuity correction Why a range of values? • Recall that probability is area under the curve • P(X = x) = 0 • Create an area by extending x by 0.5 on either side ( x 0.5,xx 0.5) Recall • How do we find the probability that a given range of values will occur? • z-scores! xx z x np np(1 p) Why? Why? x E( X ) Just trust me. Example 1 What is the probability of getting between 20 and 30 heads if you toss a coin 50 times? P(20 X 30) P(19.5 X 30.5) x np 50(0.5) 25 np(1 p) 50(.5)(1 0.5) 3.5 z xx 19.5 25 3.5 1.57 z xx 30.5 25 3.5 1.57 Example 1 What is the probability of getting between 20 and 30 heads if you toss a coin 50 times? P(20 X 30) P(19.5 X 30.5) P( X 30.5) P( X 19.5) P( z 1.57) P( z 1.57) 0.9418 0.0582 0.8836 There is an 88% probability of getting between 20 and 30 heads – (binomial theorem gives 88.11%). WARNING! • Not all binomial distributions can be approximated with normal distribution • Left- or right-skewed distributions don’t fit • E.g. if n = 10, and p = 0.2 P(3 X 4) P( X 3) P( X 4) 10 10 3 7 (0.2) (0.8) (0.2) 4 (0.8) 6 3 4 Binomial Distribution 0.35 0.3 P(X=k) 0.25 0.2 0.15 0.1 0.29 0.05 0 1 2 3 4 5 6 7 8 9 10 Number of successes (Normal approx gives P(2.5 < X < 4.5) = 0.32 !) When can we approximate? How can we tell whether the data is symmetrical enough to approximate using normal distribution? If X is a binomial random variable of n independent trials, each with probability of success p, and if np > 5 and n(1 – p) > 5 then X can be approximated by normal distribution. Checking… • Example 1: np = 50(0.5) n(1 – p) = 50(0.5) = 25 = 25 >5 >5 Can be approximated by normal distribution • Example 2: np = 10(0.2) n(1 – p) = 10(0.8) =2 =8 <5 >5 Can’t be approximated by normal distribution