A Good Enough Fit?

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In your previous studies of algebra, you explored linear, exponential and quadratic functions.  When given a set of relatively linear data, you calculated an equation for line of best fit.  But how did you know that the best fit for the data was a line?

In this lesson, you will review linear regression and analyze how well this model fits the data.  You will also be introduced to residuals and the correlation coefficient, and use these tools to determine whether or not a linear model is the best fit for a set of data.

Let’s begin with the following situation:

Andrew is the team captain of his school’s cross country team.  He has been tracking the winning times for the Maryland Cross Country 2A Finals at Hereford High for several years, as seen below:

Boys 2A Results
2012:  David Dorsey, Catoctin 16:25.2     
2011:  Austin McGinley, Rising Sun 16:21   
2010:  Robby Creese, Glenelg 16:46                
2009:  Robby Creese, Glenelg 17:30                  
2008:  Craig Morgan, River Hill 16:46                 
2007:  Kyle Andres, Oakland Mills 16:33            

Information Source: Maryland Public Secondary Schools Athletic Association

Andrew has been training hard to improve his time.  He wants to set a running time goal for himself based on the previous year’s results.

When presented with a set of data, it is important to find a model that best fits the data in order to make predictions about future trends.  For this problem, you will explore how to determine a model of best fit and then use the model to predict the winning time for future years.           

When presented with a data set or situation, follow these general steps:

1. Create and Analyze – Look at the data. Find various ways to model the mathematics, such as with a graph, table or equation. Make sense of the problem you are working with by reasoning through concepts you already know that may help you solve the problem.
2. Test and Confirm – Analyze the models and representations to develop strategies you think work best to solve the problem. You may need to try a few strategies and models until you get the one that is best. Persevere in solving the problem. Continually ask yourself if the steps you are following and what you are calculating makes sense in the context of the problem. Don’t be afraid to change your strategy if necessary.
3. Apply the Model – Employ your chosen models, representations and strategies. Input the necessary data, quantitative values, etc., and perform the necessary calculations with precision. Analyze the final results. Do they make sense in the context you have?

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