# Category Archives: confidence intervals

## Before the semester starts … I’m playing with pictures!

I am sure I’m not alone in wanting to use the time between semesters to make adjustments to what I am teaching or how I am teaching it. By now, you probably recognize that I am a fan of learning about new pedagogical techniques. I am dedicated to helping students to truly understanding the concepts of statistics. Often, having visuals when you teach is useful for students.

I use the chalk and a board (OK, more like 8 boards that move). I draw a lot of pictures. However, a mathematics professor (who is both a great colleague and friend) has been bugging me about using Mathematic in addition to chalk (a delivery system she also loves).

With Mathematica, it is my hope that I will not only be able to present my students with a visual image of certain concepts during class time (like how a normal distribution changes when the size of the standard deviation gets larger or smaller) but by making these demonstrations available electronically to students for them to explore these concepts on their own, I am hoping students will gain a greater conceptual understanding of critical statistical concepts.

Mathematica is a software package, that among other things, provides demonstrations of statistical concepts. Each demonstration was designed by an instructor. For it to be published, it is my understanding that it goes through a rigorous peer-review process. As such, if it’s printed for use, you know it will work. The down side is that your university would have to pay for a subscription to Mathematica for the demonstrations to be useful. http://www.wolfram.com/solutions/education/higher-education/uses-for-education.html

As I stated last week, in my list of resolutions, my goal is to find five different demonstrations this semester. Why five? It seemed like a reasonable number … not too challenging.

This was really easier than I anticipated. I started by indentifying the concepts that would most benefit from being able to visualize and manipulate variables. Then I visited the Mathematica web site and searched the topics. Each search yielded anywhere from 5 to 25 demonstrations, some were appropriate, others weren’t. I looked through the demonstrations and selected the ones I liked.

Here are the concepts and the demonstrations I identified as being potentually useful this semester.

(1) The Normal Distribution, where students get to input mu and sigma, would make a nice visual demonstration.

http://demonstrations.wolfram.com/TheNormalDistribution/

This Normal Distribution also shows the area under the curve (i.e., you can manipulate the z-score)

http://demonstrations.wolfram.com/AreaOfANormalDistribution/

(2) Another good demonstration would be the Sampling Distribution of the Means, where students can see the impact of changing mu, sigma, or sample size on its shape.

http://demonstrations.wolfram.com/SamplingDistributionOfTheSampleMean/

I’m also going to throw in a demonstration on the Central Limit Theorem, as how can we talk about the Sampling Distribution of the Means without mentioning the Central Limit Theorem?

http://demonstrations.wolfram.com/TheCentralLimitTheorem/

(3) Of course, what changes in the Sampling Distribution of the mean is the standard error, thus showing how a standard error changes due to changes in the sample size and/or variability makes a great deal of sense. I was really hoping that a demonstration on the standard error would already be available, unfortunately, it doesn’t seem to be. A similar concept is the confidence interval, though even with this demonstration the writer of the Mathematica code for this demonstration did not include how variability (i.e., standard deviation) impacts the size of the “margin of error.” However, it still could be a useful demonstration.

http://demonstrations.wolfram.com/ConfidenceIntervalsConfidenceLevelSampleSizeAndMarginOfError/

Though not as clean looking at the one above, this demonstration also includes the size of the standard deviation. http://demonstrations.wolfram.com/ConfidenceIntervalExploration/

I would expect that the two demonstrations would be necessary for student to get a richer understanding of confidence intervals.

That having been said, I believe that two new Mathematica Demonstrations are in order … one dealing with the size of the standard error based on changes in sample size and variability and a possibily a new CI demonstration that merges the best of these two demonstrations.

(4) The effects of the sample size and population variance on hypothesis testing with the t-test seems like a great visual demonstration.

(5) How changes in the variables impact correlation’s (depending on how they are calculated) should be useful for my students.

http://demonstrations.wolfram.com/CorrelationAndRegressionExplorer/

(6) Those of you who know me, are probably not surprised that I can’t just stop at 5 examples for this first semester … so here is a great demonstration on Power. Though I can get students to define power, and identify threats to power, I am never fully certain that they truly get the beauty (and hassle) of power. This demonstration may help.

http://demonstrations.wolfram.com/ThePowerOfATestConcerningTheMeanOfANormalPopulation/

Of course, without proper instruction during class time and an accompanying explanation following class instruction, these demonstrations may end up being little more than pretty pictures to students.

In a few weeks, especially after I actually try these demonstrations with my students, I will provide for you the information I attached with the demonstrations as well as feedback as to what worked and what didn’t. After all … anyone who has taught long enough knows, even the best planned lessons and demonstrations some times flop.

Though not specifically having to do with teaching statistics … I found a nice article at Chronicle of Higher Education on Iphones, Blackberries, etc … and apps that could help professors. The attendance and learning students’ names apps look promising. http://chronicle.com/article/College-20-6-Top-Smartphone/125764/

I look forward to hearing from any of you who have used Mathematica Demonstrations (or others) during class and for homework.