## Thursday, February 1, 2018

### Teach 180: The Confidence Interval (Day 97)

In AP Statistics today, we looked at how changing confidence level and sample size impacts the width of a confidence interval.  We also saw how the confidence level is related to the long run capture rate of confidence intervals.  The website that we used to investigate this was http://digitalfirst.bfwpub.com/stats_applet/stats_applet_4_ci.html, but rather than having my students type this in, I told them to go to bit.ly/ConfidenceIntervalBasics. (Creating a shorter, custom link is easy to do at bit.ly or tinyurl.com and makes it easier for the students to correctly enter the web address in their browser.)

First, I asked students what they noticed about the two distributions that were shown on the screenshot above.  They commented on the fact that they had the same normal distribution shape and same mean, but that the sampling distribution was less variable than the population distribution.  Then, we used the applet to take several samples of size n = 20 and created 95% confidence intervals from each sample.  I asked them what they noticed about the intervals.  Students commented on the symmetry of the intervals, but weren't entirely sure if the dot in the center was the median or the mean.  They also noted that the one interval was red.  Why was that?  Noticing the sample under the interval gave them a sense that this sample had many values that were above the mean and that the entire interval was bigger than the population mean. Seeing this in action with several intervals helped the students to begin to understand that the probability that an individual interval captures the population parameter is 0 or 1, and not 95% (or whatever the confidence level is).  The 95% level refers to the long run capture rate of all possible intervals constructed by this method.

Next, I had half the room keep the confidence level at 95% and investigate the change of sample size on the width of the interval.  The other half of the room kept the sample size the same and investigated what happened as the confidence level changed.  If I had students change both variables at the same time, they may not have fully understood the relationship between confidence level and interval width and sample size and interval width.  Focusing on just one variable at a time was an important part of their discovery of the relationships.  Below are screenshots of the applet and what students discovered.

Changing Sample Size, Keeping Confidence Level Constant : Larger Sample, Narrower Intervals

Changing Confidence Level, Keeping Sample Size Constant: Larger Confidence Level, Wider Intervals

I believe that building an informal understanding without any formulas leads to a better understanding of how the formulas work to create wider or narrower intervals.  Tomorrow we toss Hershey Kisses to create our first intervals for the proportion of kisses that would land bottom side down when tossed.