Distribution of Sample Proportions (2 of 6)

Recall that our goal is to create a probability model that describes the long-run behavior of proportions from random samples. Previously, we used a simulation 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.

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.

At this point, we have a good sense of what happens as we take random samples from a population. Our simulation suggests that our initial intuition about the shape and center of the distribution of sample proportions is correct.

Now we use another simulation to help us think more precisely about the variability we expect to see in the sample proportions. Our intuition tells us that larger samples will better approximate the population, so we might expect less variability in large samples. In the next walk-through, we use a simulation to investigate this idea. After that walk-through, we tie these ideas to more formal theory about the probability model for the long-run behavior of proportions from random samples.

We are now ready to use what we have observed to develop a formal probability model to describe the behavior of sample proportions. First, let’s return to the original question that prompted our investigation of sample proportions.

We based our investigation on the prediction that 60% of part-time college students will be female. In our investigation, we asked, How much could the sample proportion vary from a population proportion of 0.60 for us to feel confident in our prediction?

We don’t expect a sample proportion to be exactly equal to the population proportion. But how much error seems reasonable?

We now see that the answer to this question depends on the size of the sample.

• If we select a random sample of 25 students, the distribution of sample proportions has a standard deviation of about 0.1. We can see that most sample proportions fall within 2 standard deviations of 0.60. Therefore, we might decide that 2 × 0.10 = 0.20 is a reasonable margin of error, so a sample proportion between 0.40 and 0.80 is not surprising if 0.60 of all part-time college students are female.

• If we select a random sample of 100 students, the distribution of sample proportions has less variability. It has a standard deviation of about 0.05. Again we see that most sample proportions fall within 2 standard deviations of 0.60, so we might decide that 2 × 0.05 = 0.10 is a reasonable amount of error for these larger samples. For a sample of 100 students, then, a sample proportion between 0.50 and 0.70 is not surprising if 0.60 of all part-time college students are female.

We discuss this idea further in “Introduction to Statistical Inference” in this unit.