Appleton Arena is home to the St. Lawrence University men’s ice hockey team. Every home weekend it is packed to the brim, the cacophonous roar of fans making it impossible to hear anything but the sound of players hitting the boards and the referee’s whistle. Students arrive early in an attempt at getting a seat in the student section, also known as the Marsh Pit, and there is often not a spare inch of room in the stands. Appleton Arena is also home to the St. Lawrence University women’s ice hockey team. The team sees a decent number of local fans, but often the wood of the vintage bench seating is exposed between sparse groups of spectators.
In an effort to improve viewership of women’s hockey games, Laura, the social media manager for the team, wants to do some kind of promotion. Her idea is to give out free pub cookies at the next home game in hopes that word will spread about the delicious incentive and more people will turn up for the next game. But how many cookies will be needed? Too few and the people will not be happy that they didn’t get their decadent dessert, too many and there will be food waste. In order to figure out how many cookies she needs, she is going to find the typical attendance and go from there. To find these numbers, she consults Brennan, the hockey operations manager for the team. “Can we just use the attendance from the last game?” asks Laura. Brennan tells Laura that we can treat the attendance at a given game as a random variable. “The attendance at a given game will vary,” says Brennan to Laura, “and we have to account for this variability when making this decision on attendance, so we can’t just look at the numbers from last game because I’m sure it was different the game before that. We can try to find a good way to estimate the mean, which may give as a better representation of our typical attendance.” Laura has a good idea of where this is going now. If she takes random samples of games, she could find a way to use them to estimate the true mean number of fans that show up for the women’s games.
After collecting the raw data, Brennan presents a graph to Laura. The attendance appears normally distributed, its curve resembling that of the bell shape typical of a normal distribution, with a lot of games having values somewhere around 600. But there are also some with lower attendance, and some with attendance in the thousands! Laura thinks it would be great if every home game could get close to 1000. Looking at the graph, Laura tells Brennan that our true population mean is probably somewhere in the middle based on the graph. He tells Laura that it probably is, but in order to make sure we are going to use the maximum likelihood estimation technique to try and estimate this parameter of the normal distribution. “We’re going to try and find an equation that will be our estimator for mu and given some values this equation will maximize our likelihood function,” starts Brennan, “we also want to make sure that our estimator is not biased, meaning we aren’t going to be undershooting or overshooting our estimates. And we want to keep our estimates within a small range, hopefully close to our true mean, so ideally the variance is low too.” “Is there a way that we can measure that?” asks Laura. Brennan tells her that they can check if their estimator is consistent. “What does that even mean?” she asks. Brennan replies, “basically if our sample size was really huge, we want to know if the expected value of our estimator approaches the parameter and if our variance approaches zero.” Laura nods in understanding and they get to work on the math.
Together they find an estimator for the mean, input their values to get the estimate, and the pub cookies are requested. After the first game with cookies, word quickly spreads across campus about the delicious incentive. The promotion occurs occasionally over the next few seasons, and attendance only continues to rise. The year is 2030, and Brennan, now the head athletic director, decides to look at the attendance numbers for women’s hockey for every game from back in the 2025 season all the way up until now. He pulls up the data, the crooked smile on his face matching the left skewed distribution that appears on his screen.