Learn by Doing – Connection between Confidence Intervals and Sampling Distributions

Published: February 16th, 2013

Category: Activity 1: Learn By Doing, Activity 3: Simulations and Tools

The purpose of this activity is to help give you a better understanding of the underlying reasoning behind the interpretation of confidence intervals. In particular, you will gain a deeper understanding of why we say that we are “95% confident that the population mean is covered by the interval.”

Video of Applet in Action (1:18)

The applet has a new look and different ways of entering the information but otherwises, works the same and still illustrates the same concepts.

Transcript – Video of Confidence Interval Applet in Action

When the applet loads, you see a normal-shaped distribution, which represents the sampling distribution of the mean (x-bar) for random samples of a particular fixed sample size, from a population with a fixed standard deviation of σ (sigma).

The green line marks the value of the population mean, μ (mu).

To begin the simulation, click the “sample” button. You will see a line segment appear underneath the distribution; you should see that the line segment has a dot in the middle.

You have used the simulation to select a single sample from the population; the applet has automatically computed the mean (x-bar) of your sample; your (x-bar) value is represented by the dot in the middle of the line segment. The line segment represents a confidence interval. Notice that, by default, the applet used a 95% confidence level.

If your confidence interval did cover the population mean μ (mu), then the applet will have recorded 1 “hit”.

Now, click to select another single sample.

Notice, under “total” on the right side of the applet, the number of total selected samples has been tallied.

Now, click “sample 25” repeatedly, until the applet tallies a “total” of around 1,000 samples. You will see that the applet computes the “percent hit” for all the intervals.

Based on what you’ve seen on the applet (with the level set at 95%), decide which of the following statements are true and which are false.

Note that in actual scientific practice, we only select one single sample and therefore we can only see the one corresponding interval, out of the potential thousands that are displayed using the applet. Give the best interpretation of a single 95% confidence interval as follows, based on what you’ve learned from the simulation:

Fill in the blanks in the following statement in next set of questions:

We are ___ confident that our interval is one that ____ cover _____

This document is linked from Population Means (Part 1).