5.5 Normal Approximations to Binomial Distributions

• Key Concepts:– Binomial Distributions (Review)– Approximating a Binomial Probability– Correction for Continuity

5.5 Normal Approximations to Binomial Distributions

• Why do we need something new to find binomial probabilities?– Compare the following scenarios:

• Suppose a new medical procedure is successful 80% of the time. Seven patients are scheduled to have the procedure. What is the probability the procedure will be successful for at least five of the seven patients?

• Now suppose 2,000 patients are scheduled to have the procedure. What is the probability the procedure will be successful for at least 1,500 of the 2,000 patients?

5.5 Normal Approximations to Binomial Distributions

• How does the Normal Distribution fit in?– We need to go back to the shape of binomial

distributions for a moment.• A binomial distribution will be approximately bell-

shaped and symmetric if the success probability, p, is close to 0.5.

• The distribution becomes more and more skewed as p moves away from 0.5. However, if there are a large number of trials n, the skewness is mitigated (see p. 275).

5.5 Normal Approximations to Binomial Distributions

• As long as we have the right combination of trials and success probability, we can use the normal distribution to approximate binomial probabilities.

• Key Fact: If np ≥ 5 and nq ≥ 5, then the binomial random variable X is approximately normally distributed with mean and standard deviation .

np npq

5.5 Normal Approximations to Binomial Distributions

• We should always check the inequalities before we use the normal approximation.

#2 p. 281

#4

• A correction for continuity is necessary because we are using a continuous distribution to approximate something that is discrete (see p. 277).

#24 p. 282 (Medicare)

#27 p. 282 (Celebrities)