Wiki Page, Math 13 - Perez Sandbox : Module 17 - Distribution of Sample Proportions (2a of 6)

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2026-01-17T21:15:20-08:00

Recall that our goal is to create a probability model that describes the long-run behavior of proportions from random samples. Previously, we used an applet to collect a few random samples to get acquainted with making a distribution of sample proportions. We randomly selected 25 students at a time from a population of part-time college students that is 60% female. In the next example, we predict what happens in the long run when we select many, many random samples of 25 students at a time from this population. Then we watch a simulation to see if our predictions are correct.

EXAMPLE Predicting the behavior of sample proportions

Based on our intuition and what we observed with the applet, we might expect the following about the distribution of sample proportions that come from a population where p = 0.60:

Center: Some sample proportions will be on the low side—such as 0.52 or 0.56—and others will be on the high side—such as 0.64 or 0.68. It is reasonable to expect all the sample proportions in repeated random samples to average out to the underlying population proportion, 0.6. In other words, the mean of the distribution of sample proportions should be p.

Spread: For samples of 25, we expect sample proportions of females not to stray too far from the population proportion 0.6. Sample proportions lower than 0.44 or higher than 0.72 will be unusual. Previously, we took smaller random samples of 8 and observed more variability in the sample proportions. We therefore think that sample size plays a role in the spread of the distribution of sample proportions. Smaller samples may be less accurate and more variable than larger samples.

Shape: Sample proportions closest to 0.6 will be most common, and sample proportions far from 0.6 in either direction will be progressively less likely. In other words, the shape of the distribution of sample proportions may be somewhat bell-shaped.

Now we use a simulation to collect numerous samples to see what happens in the long run. We use the simulation to check whether our intuition about the center, spread, and shape of the distribution of sample proportions is right.

This simulation collects random samples of 25 students from a population of part-time college students. The variable is gender. We examine how the proportion of females in the samples relate to the population proportion of 0.60 (60% of the part-time college student population is female.)

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