Regression

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Lines of best fit A line of best fit is drawn on a scatter plot to show the linear trend in a set of paired data. strong positive correlation weak positive correlation strong negative correlation weak negative correlation The stronger the correlation, the closer the points are to the line. We can find the equation of a line of best fit by calculating the slope and y-intercept.

Choosing a function type When we find the equation of the line of best fit for a scatter plot, we are effectively saying “This data follows a linear trend, so I can fit a linear function to the data.” Fitting a function to data in this way is called regression and the line we find is called the regression line. Not all data follow a linear trend though. Mathematical Practices 7) Look for and make use of structure. Students should be aware that sets of paired data may follow the trend of another function type other than linear. They should be able to recognize the exponential and quadratic trends in the graphs shown. It is important that students learn to choose which type of function will best fit a data set. exponential quadratic What type of function best fits these scatter plots?

What type of function? Teacher notes Encourage students to plot the data using the “STAT PLOT” feature of their graphing calculators. This will enable them to see the trend in the data. Mathematical Practices 5) Use appropriate tools strategically. Students should know that they can use their graphing calculators to quickly plot a given set of data. It will help them to choose the most appropriate type of regression if they see a graph of the data. 7) Look for and make use of structure. Students should be aware that sets of paired data may follow the trend of another function type other than linear. They should be able to recognize the exponential and quadratic trends in the graphs shown. It is important that students learn to choose which type of function will best fit a data set.

Regression using calculators Teacher notes Note that these instructions are based on using a TI-83/84. For some calculators, the first step is to ensure the correct mode is selected. Different calculators will have different buttons and modes. Point out to students that when the equation of a regression model is obtained, it can then be used to make an estimate of one variable given the value of the other variable. Mathematical Practices 5) Use appropriate tools strategically. Students should be aware of the regression features on their graphing calculators and know how to use them. They should also be able to interpret the results on the screen and understand the meanings of the values that they are given by the calculator.

The value of r When you use the linear regression feature, your calculator will display the value of r, the correlation coefficient. The value of r indicates the strength of association between two variables. It shows how close points lie to the regression line. r can be between 1 and –1, inclusive The closer the value of r is to 1 or –1, the better the regression line fits the data and the stronger the correlation. Teacher notes Only data sets that create a scatter plot with a perfectly straight line can have r = –1 or r = 1. If r is negative, the data has negative correlation, and if r is positive, the correlation will be positive. If a regression calculation returns a value close to zero, it might be a good idea to try a different type of model. As r approaches zero from either side, it indicates that the equation is not a good fit for the data.

Linear correlation

App downloads Teacher notes This question could be worked through in small groups. At the end, students have to discuss the accuracy of their prediction. Explain how and why we convert the years into smaller values. Let x = 1 represent 2008. We can subtract 2008 from each year to get the new value of x. Using smaller numbers in the exponent usually keeps the regression equation values from being extremely small decimals. In this example, the correlation coefficient is 0.957. Since this is very close to 1, it tells us that the exponential model provided is a very good fit for the data. Although the prediction is very accurate according to the exponential model, it is important that students realize that it can be risky to extrapolate too far using regression equations. These can only confidently be used to predict values of y that correspond to x-values that lie within the range of the data values available. This is because we cannot be sure that the relationship between the two variables will continue to be true. Mathematical Practices 4) Model with mathematics. This question shows how a function can be fitted to a set of real-life data. Students should be able to analyze a plot of the data and decide which function type would fit it best: exponential. They should be able to use the regression equation that they obtain to answer questions in context. 5) Use appropriate tools strategically. Students should know how to use their graphing calculators to fit a regression equation to a set of data. It will help them to choose the most appropriate type of regression if they see a graph of the data.

Estimating values The table below shows the average value of a particular model of car, depending on its age. age (years) 2 4 6 8 10 12 value ($) 24,340 18,290 12,530 8,760 5,510 2,780 Use your graphing calculator to find a regression line for this data. Use it to estimate the original value of the car. Plotting the data shows that it follows a linear pattern. Mathematical Practices 4) Model with mathematics. This question shows how a function can be fitted to a set of real-life data. Students should be able to analyze a plot of the data and decide which function type would fit it best: linear. They should be able to use the regression equation that they obtain to answer questions in context (in dollars). 5) Use appropriate tools strategically. Students should know how to use their graphing calculators to fit a regression equation to a set of data. It will help them to choose the most appropriate type of regression if they see a graph of the data. 7) Look for and make use of structure. Students should see that the original value of the car would have been at 0 years, i.e. at x = 0. It should occur to them that this is also the y-intercept of the graph and therefore see that they can quickly find this value from the equation of the regression line: $27,026. Photo credit: © Dim154, Shutterstock.com 2012 The regression equation is y = –2141.5714 x + 27026. The original value of the car will be at 0 years, which is the y-intercept of the regression line. The value of the car was around $27,026.