Wiki Page, Math 13 - Perez Sandbox : Module 23 - Estimating a Population Mean (1b of 3)

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2026-01-17T21:15:39-08:00

What Does 95% Confident Really Mean?

We have previously discussed the meaning of the term "95% confident;" but this probability statement is often misinterpreted, so we will revisit it again now.

We will use the applet below to randomly select a sample from a population with mean and standard deviation of . For each random sample, the applet computes the sample mean and shows the associated 95% confidence interval.

When the applet loads you will see a normal-shaped distribution, which represents the sampling distribution of the sample means . The green line marks the value of the population mean, .

To begin the simulation, click the "Sample" button on the left at the top of the applet (not the "Sample 50" button on the right). You will see a single horizontal line segment appear underneath the distribution with a tiny red dot in the middle. The red dot is the value. The line segment represents a confidence interval, so all of the values in the interval are estimates of .

Work with the applet to answer the following questions:

1. Click Sample. Did the resulting 95% confidence interval contain the population mean ?

If your confidence interval contains the population mean , the green line will go through your confidence interval and the applet will record 1 "hit" at the top. Otherwise the applet will record 0 "hits" at the top.

2. Now, click Sample again to select another sample. Is your second sample mean (i.e. the new red dot) the same as the first sample mean? Does the new interval contain the population mean ?

Your second sample mean probably differs from the first sample mean. After all, we expect sample means to vary. If your second confidence interval contains the population mean , the green line will go through your confidence interval and the applet will record another "hit" at the top.

3. Now click Sample 50 repeatedly until the applet tallies a "total" of around 1,000 samples. Note that the applet has computed the "percent hit" for all the 95% confidence intervals. A "hit" is a sample mean with a 95% confidence interval that contains the population mean .

In this large number of samples, what percentage of the 95% confidence intervals should contain the population mean ?

We anticipate that about 95% of the confidence intervals will contain the population mean; 5% will not. The 5% of intervals that do not contain appear as red line segments.

What is the main point?

The phrase "95% confident" means that in long run repeated random sampling, 95% of the confidence intervals will contain the population mean.

When we calculate a confidence interval, we do not know if that interval contains the population mean. But we can say that the method we are using to find confidence intervals will be accurate 95% of the time, i.e. will generate an interval that accurately estimates µ.

This does not mean that there is a 95% chance that is in the particular interval we calculated based on an observed sample mean. Why? Once we hit Sample and we get the interval, it either contains or it does not. However, before we hit Sample, we can say that there is a 95% chance that the next interval will contain . This is a subtle but important distinction that is hopefully clear after working with this applet.

Mark this page as done, then click the "Next" or > button to continue.

Content by the Open Learning Initiative and licensed under CC BY.