Wiki Page, Math 13 - Perez Sandbox : Module 18 - Introduction to Statistical Inference (1a of 3)

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2026-01-17T21:15:23-08:00

From the Big Picture of Statistics, we know that our goal in statistical inference is to infer from the sample data some conclusion about the wider population the sample represents.

In the first section, "Distribution of Sample Proportions," we investigated the obvious fact that random samples vary. Because different samples may lead to different conclusions, we cannot be certain that our conclusions are correct. Statistical inference uses the language of probability to say how trustworthy our conclusions are.

We will spend the rest of the course focused on two types of inference: confidence intervals and hypothesis tests.

We construct a confidence interval when our goal is to estimate a population parameter. We conduct a hypothesis test when our goal is to test a claim about a population parameter.

Both types of inference require random sampling (or random assignment.) With both types of inference we use the distribution of sample statistics that results from random sampling as a probability model and use probability to describe the amount of confidence we have in our results.

In this section, we build on the ideas in "Distribution of Sample Proportions" to develop the logic of statistical inference. Later we will add more formal details.

Confidence Intervals

When our goal is to estimate a population proportion, we select a random sample from the population and use the sample proportion as an estimate. Of course, random samples vary, so we want to include a statement about the amount of error that may be present. Because sample proportions vary in a predictable way, we can also make a probability statement about how confident we are in the process we used to estimate the population proportion.

We can find many examples of confidence intervals reported in the media. Here is an example.

EXAMPLE Do you have problems sleeping?

The National Sleep Foundation sponsors an annual poll. In 2011, the poll found that “43% of Americans between the ages of 13 and 64 say they rarely or never get a good night’s sleep on weeknights. More than half (60%) say that they experience a sleep problem every night or almost every night (i.e., snoring, waking in the night, waking up too early, or feeling unrefreshed when they get up in the morning” (as reported at the National Sleep Foundation).

Are these percentages sample statistics or population parameters? These percentages are statistics, not parameters. Why? Because they describe the responses of a sample of Americans.

Let’s focus on the 60% who say they experience a sleep problem every night or almost every night. Does this mean that 60% of all Americans have this same experience?

Well, no. This is a sample statistic from a poll. But from this sample, we want to infer what percentage of the population has sleep problems. Since the percentage with sleep problems will differ from one sample to the next, we need to make a statement about how much error we might expect between a sample percentage and the population percentage.

In the “Poll Methodology and Definitions” section of the article, we find more detailed information about the poll. According to the National Sleep Foundation website, “The 2011 Sleep in America® annual poll was conducted for the National Sleep Foundation by WB&A Market Research, using a random sample of 1,508 adults between the ages of 13 and 64. The margin of error is 2.5 percentage points at the 95% confidence level.”

There is a lot of important information here:

The sample is random.

The sample size is 1,508.

The margin of error is 2.5%.

The confidence level is 95%.

From this information, we can construct an interval that we are reasonably confident contains the population proportion.

Sample statistic ± margin of error

60% ± 2.5%

57.5% to 62.5%

This interval is an example of a confidence interval. We interpret the interval this way: We are 95% confident that between 57.5% and 62.5% of all Americans experience a sleep problem every night or almost every night.

How confident are we that this interval contains the population proportion? In this case, we are 95% confident. What does this mean? Here are several ways to say it:

95% of the time, a random sample of this size will have at most 2.5% error; 5% of the time the error will be larger than 2.5%.

95% of the confidence intervals with a 2.5% margin of error will contain the true population proportion.

This method accurately estimates the population proportion 95% of the time within 2.5%.

Note: Notice that the sample is a random sample. We can construct a confidence interval only with a random sample.

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