Statistics: Why np Min?

Objectives

• Students will understand that normal distributions can be used to approximate binomial distributions whenever both np and n(1 – p) are sufficiently large.
• Students will understand that when either np or n(1 – p) is small, the normal distribution probabilities for impossible numbers of successes (less than 0 or greater than n) are unreasonable.
• Students will formulate guidelines to determine what they mean by sufficiently large for good approximations.

Vocabulary

• binomial random variable
• mean
• normal random variable
• number of trials, n
• probability distribution function
• probability of success, p
• standard deviation

About the Lesson

This lesson involves examining the general shape of binomial distributions for a variety of values of n and p.
As a result, students will:

• Compare the shapes of binomial distributions to those of related normal distributions, recognizing the distinction between discrete and continuous random variables.
• Recognize that normal approximations of binomial probabilities become less and less accurate as either np or n(1–p) falls below 5 (or 10 or 15) by examining the probabilities calculated from the normal distribution for having the "number of successes" be less than 0 or greater than n.