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Statistics: Confidence Intervals

Confidence intervals are used to help you get a better feel for your estimated value. Confidence intervals are like nets. You don't know what the TRUE proportion value is so you throw a net (find a confidence interval based upon a survey). The confidence level indicates the percentage of times your net would "catch" the true population value. Alpha (which is 1 minus the confidence level), then indicates the percentage of times the net would NOT "catch" the true population value. When you are creating a 90% Confidence interval, what you are saying is that 90% of the time, the interval you find will contain the true parameter value. A 95% Confidence interval says that 95% of the time, the interval will contain the true parameter value.

There is another inherit relationship between the size or width of the interval and the confidence level chosen. The higher the confidence, the larger the interval will be because you want to be sure to include the true value more of the time.

At the following web address: Confidence Interval Applet, you will find another applet to play with. This applet assumes your population is the standard normal distribution. When you first load the page, it will have randomly created and plotted 50 confidence intervals for alpha = 0.05. Underneath the graph, a count is kept of the number of intervals that do NOT contain the true mean, 0. You will notice these intervals are colored red on the graph. When you click on "More intervals!", the applet will generate another 50 confidence intervals for the given alpha and the results at the bottom will also keep a running total. When you change the alpha, the counts will start over. Black lines are CIs that contain the true mean. Red lines are CIs that do NOT include the true mean.

Also an additional website applet to check out is: http://www.ruf.rice.edu/~lane/stat_sim/conf_interval/index.html. This applet also lets you visually see 95% and 99% Confidence intervals as they are simulated. It also keeps track of the number of intervals which did not contain the mean, so you can easily compare it to the confidence level. The orange lines are 95% CIs that include the true mean. The blue tips indicate the width of the 99% CIs that include the true mean. Red lines are 95% CIs that did NOT include the true mean and the white lines are 99% CIs that did NOT include the true mean.

With these applets, the TRUE population value is known and they are counting how many of the confidence intervals based on randomly generated samples "catch" the true value.

What you should observe is that the percentage that do not contain the mean will be VERY close to the alpha value.

Question

Using the first applet, select an alpha value and click on "More intervals!" until you have a total of 1000 intervals. Then take note of the number of intervals which did NOT contain the mean and compare to the alpha level. Try this for a number of different alphas. Then come back here and share your results and how this exercise helped your understanding of confidence intervals.

Additionally visit and explore the second website. Simulate at least 5000 intervals and note the proportion that contained the mean. Note how the proportions correspond to the confidence levels. Come back and report your findings.

Solution Preview

Here I will explain what I noticed by using the applets; you should also visit the websites in order to try these as well.

1. The first applet:
When I generated a total of 1000 intervals, here are the results:

alpha level Number of intervals that did NOT contain the mean
0.01 ...

Solution Summary

This solution discusses the two applets. Results are communicated.

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