Normal Approximations to Binomial Distributions 5.5 SECTION 5.5 Normal Approximations to Binomial Distributions 275 Approximating a Binomial Distribution Continuity Correction Approximating Binomial Probabilities What You Should Learn How to determine when a normal distribution can approximate a binomial distribution How to find the continuity correction How to use a normal distribution to approximate binomial probabilities Approximating a Binomial Distribution In Section 4.2, you learned how to find binomial probabilities. For instance, consider a surgical procedure that has an 85% chance of success. When a doctor performs this surgery on 10 patients, you can use the binomial formula to find the probability of exactly two successful surgeries. But what if the doctor performs the surgical procedure on 150 patients and you want to find the probability of fewer than 100 successful surgeries? To do this using the techniques described in Section 4.2, you would have to use the binomial formula 100 times and find the sum of the resulting probabilities. This approach is not practical, of course. A better approach is to use a normal distribution to approximate the binomial distribution. If np Ú 5 and nq Ú 5, then the binomial random variable x is approximately normally distributed, with mean m = np and standard deviation s = 1npq where n is the number of independent trials, p is the probability of success in a single trial, and q is the probability of failure in a single trial. Normal Approximation to a Binomial Distribution To see why a normal approximation is valid, look at the binomial distributions for p = 0.25, q = 1 - 0.25 = 0.75, and n = 4, n = 10, n = 25, and n = 50 shown below. Notice that as n increases, the shape of the binomial distribution becomes more similar to a normal distribution. 1 0 2 3 4 x 0.10 0.20 0.30 0.40 0.05 0.15 0.25 0.35 0.45 n = 4 np = 1 nq = 3 P(x) 012345678910 x 0.05 0.10 0.15 0.20 0.25 0.30 n = 10 np = 2.5 nq = 7.5 P(x) 0 2 4 6 8 1012141618 x 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 P(x) n = 25 np = 6.25 nq = 18.75 0 2 4 6 8 1012141618202224 0.06 0.04 0.02 0.08 0.10 0.12 x P(x) n = 50 np = 12.5 nq = 37.5 Study Tip Here are some properties of binomial experiments (see Section 4.2). • n independent trials • Two possible outcomes: success or failure • Probability of success is p; probability of failure is q = 1 - p • p is the same for each trial
RkJQdWJsaXNoZXIy NjM5ODQ=