Math 207—Normal Approximation for Certain Sampling Distributions

By the Central Limit Theorem (for “large” n), some statistics (not all) have an approximate normal distribution. Thus, the normal distribution (a distribution we have studied in detail) can be used to approximate certain probabilities.

Using the normal approximation:

Step 1:

Is it a binomial setting or not? Consider the variable being measured. Does it only take two values? Or does it take a range of values? If it takes a range of values, then it is not the binomial setting. If it only takes two values (and the other binomial characteristics are present), then it is the binomial setting (or approximate binomial setting—for example, we have approximate independence if the sampling is done without replacement, but the population size is 20 times the sample size).

Step 2:

Is it appropriate to use the normal approximation? That is, will the approximation be good? Note that it is always possible for the approximation to be used, but it will not always provide a good approximation—this is an important distinction for a statistician to make.

Setting	Check
Binomial
Not binomial

Recall if the original distribution is normal, then the sample average and sample total have exact (not approximate) normal distributions, regardless of sample size.

Step 3:

For what variable is the probability being calculated? Is it a total or an average?

Binomial Setting	Not Binomial Setting
Asking about a sample proportion or percentage? Then use and standardize with the appropriate mean and standard error:	Asking about a sample mean or average? Then use and standardize with the appropriate mean and standard error:
Asking about a sample count or number? Then use X and standardize with the appropriate mean and standard error:	Asking about a sample total or sum? Then use and standardize with the appropriate mean and standard error:

Step 4:

Draw the appropriate normal curve picture and standardize to obtain the probability of interest (or use reverse look-up to determine the value of interest).

Motivation for the Rule-of-Thumb Check when Using the Normal Approximation in the Binomial Setting

If we use a normal distribution to approximate a binomial distribution, the “tails” of the distribution must be cut off (the possible values for a binomial are 0 to n, yet for a normal distribution the possible values extend along the entire number line). This is not a big deal if the center of the distribution is far enough from 0 and n that the lost tails have negligible area.

One possible rule-of-thumb:

Ensure the mean is more than 3 standard deviations from both 0 and n:

This gives us the rule-of-thumb: . (This is probably the most commonly-used rule-of-thumb, but is not the one used by our textbook.)

Another possible rule-of-thumb:

Ensure the mean is more than 2 standard deviations from both 0 and n:

This gives us the rule-of-thumb: .

The textbook’s rule-of-thumb is in between these two (closer to the second):

The normal distribution should be a good approximation to binomial probabilities if

.