Wiki Page, Math 13 - Perez Sandbox : Module 17 - Distribution of Sample Proportions (5 of 6)

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2026-01-17T21:15:22-08:00

Sampling distributions are probability distributions.

If a normal curve is a good fit for a sampling distribution, then we can apply the Empirical Rule or use Z-scores and the standard normal distribution to determine probabilities, as we have done previously.

EXAMPLE

A 2007 report by the Pew Research Center stated that about 10% of the 3.1 million 18- to 24-year-olds in the United States were enrolled in a community college.

If we randomly selected 100 U.S. adults in this age group, what is the probability that less than 7% are enrolled in a community college.

Answer: 16%

How do we know?

First, we need to determine that a normal model is a good fit for the distribution of sample proportions.

Check normality conditions: Are and both equal to or greater than 10? YES. Therefore, we can use a normal curve to represent the probability distribution.

Here is our work: n is the sample size (100 students) and p is the population proportion. In this case, p=0.10 because the problem states that 10% of the population of 18- to 24-year-olds in the U.S. were enrolled in a community college.

and

Second, sketch the normal probability curve and mark the mean and several standard errors above and below the mean.

What is the standard error? We need to calculate it. (Calculator Keystrokes)

Mark the several standard errors above and below the mean. For example, above the mean: 0.10+0.03=0.13, 0.10+2(0.03)=0.16, etc. Similarly below the mean: 0.10-0.03 = 0.07, 0.10-2(0.03)=0.04, etc.

Third, use the Empirical Rule because the sample proportion of 0.07 is one standard error below the mean:

The central area for values within one standard error is 0.68.

The area in the two tails is 1 – 0.68 = 0.32.

The area in the left tail is half of 0.32 = 0.16

What if we cannot use the Empirical Rule? Then we find a z-score and use the Standard Normal Curve as we did before.

What is the z-score? As before, the z-score is a standardized unit; it is the number of standard errors the sample proportion is from the mean. We can say this another way: the z-score is the error in the statistic divided by the standard error.

For sample proportions, we have the following formulas:

Z =

where the standard error is

We can also write this as one formula:

Z =

As before, a positive z-score indicates that the sample proportion is larger than the parameter. A negative z-score indicates that the sample proportion is smaller than the parameter.

EXAMPLE Using the z-score and the Standard Normal Curve to find a probability

Let’s return to the example of community college enrollment: a 2007 report by the Pew Research Center stated that about 10% of the 3.1 million 18- to 24-year-olds in the United States were enrolled in a community college.

Let’s again suppose we randomly selected 100 young adults in this age group.

What is the probability that a random sample of this size has 15% or more enrolled in a community college?

To answer this question, we first determine if a normal model is a good fit for the sampling distribution.

Check normality conditions:
Yes, the conditions are met (duh!, we already did this above.) Therefore, we can use a normal model, which allows us to use a z-score to find the probability.
But before we move on, let's look at the normality conditions from a different point of view:

When 10% of the population is enrolled in a community college, we expect 10% of the sample of 100 students to be enrolled in a community college. This expected count is n p=100(0.10)=10.

Similarly, when 10% of the population is enrolled in a community college, 90% of the population is not enrolled; therefore, we expect 90% of the 100 is not enrolled, n(1−p)=100(0.90)=90.

In other words, another way to describe the normality conditions is to say: the count of expected successes (np) and failures (n(1-p)) are both at least 10.
Find the z-score:

We already found the standard error previously. It is 0.03.

Find the probability using the standard normal model using the Normal Distribution Calculator

Here are our results: We want the probability that the sample proportion is greater than 15%; therefore, we want the probability that the z-score is greater than 1.67. The probability is about 0.0475.

Conclusion: If it is true that 10% of the population of 18- to 24-year-olds are enrolled at a community college, then it is unusual to see a random sample of 100 with 15% or more enrolled. The probability is about 0.0475.

Comment

This probability is a conditional probability. Recall that we write a conditional probability P(A given B) as P(A | B).

We write the probability that a sample proportion is greater than or equal to 0.15 given the population proportion is 0.10. as

As we now know, we can use a normal model only when certain conditions are met. Whenever we want to use a normal model, we must check the conditions to make sure a normal model is a good fit.

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