Are Albums Getting Longer?
Are Albums Getting Longer?
An Introduction to Confidence Intervals Using t
An Introduction to Confidence Intervals Using t
In addition to introducing confidence intervals using t, the goals of this lesson is to offer students the opportunity to express their interest, use technology in meaningful ways, and expand students perception of where math exists. Prior to this lesson students should be familiar with the concept of confidence intervals, calculating confidence intervals for means using normal distribution Z, and the conditions required to do so. By the end of the lesson, students have used technology to apply statistical methods in meaningful ways.
To start the lesson, I had students read through an article where the author argues the music streaming model (e.g. Spotify, Apple Music) has motivated music artist to release longer albums. Prior to reading the article, a couple questions as posed to students" "Have you noticed a similar trend", "Do you agree with the author?" and "What statistics were used in the article, if any?". Afterwards, these questions are discussed as a class.
Students are then tasked with making a confidence interval as a way to asses the authors claim. The class then selects a recently released album they enjoy and add its length, release date, and name to a Google Sheet. Below is template and the data provided by some students in the Google Sheet.
Students were then asked to use some of the previously learned statistical method to construct an argument either for or against the idea albums are getting longer. When I though this lesson, many began to construct confidence intervals but began asking questions when they realized the sample size they need not meet the necessary conditions (n being equal to or greater than 30). I took this opportunity to introduce t-statistics as an option they have when the sample is less than 30. I modeled how to implement this method for the class. They then had an opportunity to practice the method through a preprepared problem. This problem was constructed to meet all necessary conditions for t-statistics, mainly that the data provided is normally distributed. Students have learned to asses this through the use of technology. Most students elect to use Geogebra to do this as it offers a way to construct normal quantile plots relatively easily. At this point, students then were asked to return to the original data they co-created and use the newly introduce method to construct an argument.
I had designed the lesson so at this point students would be able to meet all conditions and calculate a confidence interval. However, like any teacher knows, the lesson did not go as I had planned. The data was not normal. I had not realized this until several students showed me the plot below.
Students were quick to point out to me the tails and outlier in the plot. They also noted it should be linear if normal. I was initially very proud of them for recognizing this. I then worried as to how to conclude the lesson. I was gonna have students come to a conclusion about album lengths based on their findings. However, I paused the class and had students explain to the others what they told me. I shifted the conversation towards highlighting the importance of checking the conditions and that although a powerful tool, statistics can be very fragile. Real data rarely behaves nicely. Ultimately, thanks to my students, the conclusion of the lesson was a lot more meaningful than I had planned for. Through the lesson, they engaged with many of the tools working statisticians might use and dealt with problems working statisticians most definitely face. In addition, throughout the lesson, all the math they did was taking place in a context that many don't consider when thinking math, music!