Wiki Page, Math 13 - Perez Sandbox : Module 17 - Parameters vs. Statistics (1 of 2)

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

The goal of inference is to use a sample to draw a conclusion about a population.

Probability is the basis of inference, so we have to use a random sample.

Obviously, random samples vary, so we need to understand how much they vary and how they relate to the population. Our ultimate goal is to create a probability model that describes the long-run behavior of sample measurements. We will use this model to make inferences about the population.

We begin our investigation with a simplified and artificial situation.

EXAMPLE Proportions from random samples vary

Imagine a small college with only 200 students, and suppose that 60% of these students are eligible for financial aid.

In this simplified situation, we can identify the population, the variable, and the population proportion.

Population: 200 students at the college.

Variable: Eligibility for financial aid is a categorical variable, so we use a proportion as a summary.

Population proportion: 0.60 of the population is eligible for financial aid.

Note: Populations are usually much larger than 200 people. Also, in real situations, we do not know the population proportion. We are using a simplified situation to investigate how random samples relate to the population. This is the first step in creating a probability model that will be useful in inference.

How accurate are random samples at predicting this population proportion of 0.60?

To answer this question, we randomly select 8 students and determine the proportion who are eligible for financial aid. We repeat this process several times. Here are the results for 3 random samples:

Random sample #1

Student # | 38 | 49 | 61 | 68 | 120 | 139 | 165 | 200
Aid? | Yes | Yes | Yes | No | No | Yes | Yes | Yes

6 Yes answers gives the Sample proportion = 6/8 = 0.750

Random sample #2

Student # | 1 | 45 | 48 | 82 | 86 | 100 | 148 | 183
Aid? | Yes | Yes | No | Yes | No | No | Yes | Yes

5 Yes answers gives the Sample proportion =5/8 = 0.625

Random sample #3

Student # | 25 | 56 | 63 | 75 | 130 | 131 | 193 | 195
Aid? | No | No | Yes | No | No | Yes | No | Yes

3 Yes answers gives the Sample proportion = 3/8 = 0.375

Notice the following about these random samples:

Each random sample came from a population in which the proportion eligible for financial aid is 0.60, but sample proportions vary. Each random sample has a different proportion who are eligible for financial aid.

Some sample proportions are larger than the population proportion of 0.60; some sample proportions are smaller than the population proportion.

Some samples give good estimates of the population proportion. Some do not. In this case, 0.625 is a much better estimate than 0.375.

A lot of variability occurs in these sample proportions. It is not surprising, therefore, that a sample of 8 students may give an inaccurate estimate for the proportion of those eligible for financial aid in the population. It makes sense that small samples of only 8 students may not represent the population accurately. Later we investigate the effect of increasing the size of the sample.

The variability we see in proportions from random samples is due to chance.

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