4.4 Normal Approximation to Binomial Distributions

Binomial Probability Distribution The probability of k successes in n trials: 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

Recall How do we find the probability that a given range of values will occur? z-scores! Why? Why? Just trust me.

Example 1 What is the probability of getting between 20 and 30 heads if you toss a coin 50 times?

Example 1 What is the probability of getting between 20 and 30 heads if you toss a coin 50 times? 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 (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: Example 2: 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