Learning OUTCOMES
- Describe the sampling distribution for sample proportions and use it to identify unusual (and more common) sample results.
Introduction
In this module, Linking Probability to Statistical Inference, we work with categorical variables, so the statistics and the parameters will be proportions. In the module Inference for Means, we work with quantitative variables, so the statistics and parameters will be means. In the Big Picture, we see that inference is based on probability. In this module, we begin the process of developing a probability model to describe the long-run behavior of proportions from random samples.
After we develop a probability model of how sample proportions behave, we can answer questions like the following:
- Do the majority of college students qualify for federal student loans?
- What proportion of all college students in the United States are enrolled at a community college?
The questions ask us to make an inference about a population. Our answers to these questions will be based on a sample. We will never be 100% sure of our answer, so we will make probability statements that describe the strength of the evidence and our certainty.
Brief Discussion of the Connection between These Questions and Probability
Do the majority of college students qualify for federal student loans?
- This question asks us to test a claim about college students. The claim is “the majority of college students qualify for federal student loans.” To test this claim, suppose we select a large random sample of college students and find that 40% of the sample qualify for these loans. A majority requires over 50%; 40% is definitely not a majority. Can we conclude from this sample that our claim is incorrect? Or could this sample have come from a population the majority of which qualify for loans? What is the probability that sample proportions will be 0.40 or less if the majority in the population qualify?
What proportion of all college students in the United States are enrolled at a community college?
- This question asks us to estimate a population proportion. Suppose we select a large random sample of college students and find that 46% are enrolled at a community college. What is the probability that an estimate from a sample is within 3% of the population proportion?
Note: Connected to each inference question about a population proportion, we see a probability question about the long-run behavior of sample proportions. We need to understand how proportions from random samples relate to the population proportion. We also need to understand how much variability we can expect in sample proportions. Therefore, in our early investigations, we will assume we know a population proportion and examine what happens when we select random samples from this population.
Now we begin an investigation of the long-run behavior of sample proportions.
Example
Gender in the Population of Part-time College Students
According to a 2010 report from the American Council on Education, females make up 57% of the U.S. college population. With the rising costs of education and a poor economy, many students are working more and attending college part time. We anticipate that if we look at the population of part-time college students, a larger percentage will be female. Let’s say we predict that 60% of part-time college students are female.
We don’t have information about the population of part-time college students, so we select a random sample of 25 part-time college students and calculate the proportion of the sample that is female. We don’t expect the sample proportion to be exactly 0.60. So, how much could the sample proportion vary from 0.60 for us to feel confident in our prediction?
To answer this question, we need to understand how much sample proportions will vary if the parameter is 0.60.
Try It
Refer to the previous example for the following questions. These questions focus on how the proportion of females will vary in random samples if we assume that 0.60 of the population of part-time college students is female.
Use the following simulation to select a random sample of 25 part-time college students. Repeat the selection many times to observe how the proportion of females in the samples vary. Then answer the following question.
Try It
