### Paper airplane confidence intervals

As a demonstration of a confidence interval for a sample mean capturing a population mean I had the class throw paper airplanes from the second floor balcony. Three prior terms of data provided a population mean flight distance of 614 cm as measured perpendicular to the building.

Each section threw their airplanes and then I measured the perpendicular flight distances. Measuring and picking up tasks were assisted by the students.

Throwing an aircraft.

Measuring distances.

Recording data.

Fetching airplanes.

The data was put on the board and the 95% confidence interval was calculated for each section.

 Distances in cm m08 m09 m10 all time lower bound on the mean 469 275 330 536 upper bound on the mean 940 704 681 644 pop mean 614 614 614 590

Each section captured the population mean. The sections are all less than 30 students, so this exercise provides a vehicle for introducing the t-critical value and the TINV function in spreadsheets. Google Docs does not include the TINV function, those values were generated in LibreOffice.org Calc for this exercise. The data from this fall will lower the population mean to 590 cm in future terms.

The students watch the measurement process.

This term an impending off-island trip caused me to accelerate the syllabus into section 10.1, confidence interval hypothesis testing. As I was going to be gone the next week, and the students would be working on worksheets that included running sample against sample, I modified my 10.1 coverage to include sample on sample confidence interval hypothesis testing. This material is not in the text. The week went exceedingly well actually. The FiboBelly exercise begun the prior Friday led to confidence intervals for the sample mean FiboBelly ratio that did not include the Fibonacci ratio of 1.618. The paper airplane confidence intervals for the sample mean did include the pre-existing population mean distance. Those two examples provided examples of when the sample data supports the population mean or fails to support the population mean.

Sample on sample examples have, at a beginner's level, three possible outcomes. If the confidence intervals do not overlap at all, then the difference is significant. If each confidence interval overlaps the other mean, then the difference is not significant. If the confidence intervals overlap but not sufficiently to include the other mean, or only one overlaps the other mean, then the result is effectively indeterminate. Note that this really only makes sense for small samples - large samples can generate significant differences even when the effect size is small.