1.5 – Linear Models

1. 1.5 – Linear Models Essential Question: What do you do to find the least-squares regression line?

2. 1.5 – Linear Models • When given a set of data points, the first thing to do is determine whether a straight line would be a good model for the data. • A scatter plot can visually determine this.Approximately linearNot so much

3. 1.5 – Linear Models • Scatter plots can have a correlation • If the data appears to have a positive slope (up & to the right), it’s a positive correlation • If the data appears to have a negative slope (down & to the right), a negative correlation • If a straight line isn’t possible, we say the data has no correlation or very little correlation

4. 1.5 – Linear Models • You can also determine whether a line is a good fit by using finite differences. • Finite differences are determined by subtracting consecutive y-coordinate data points • Example 1: Linear Data • Estimated cash flows from a company over the five-year period 1988-1992 are shown in the table

5. 1.5 – Linear Models • 2.79 – 2.38 = 0.41 • 3.23 – 2.79 = 0.44 • 3.64 – 3.23 = 0.41 • 4.06 – 3.64 = 0.42 • Because the differences are approximately equal, a line is a good model for this data, and the data shows a positive correlation

6. 1.5 – Linear Models(Bonus Content) • We now enter the area of stuff that I’m not going to test you on, but is just really cool • A residual is the distance from a real data point to it’s point on a linear model (see board) • To find the line of best fit, we square the residuals. The closer the square of the residuals for all points is to 0, the better the model • For any scatter plot, there is only one true “line of best fit”, which is the line where the square of the residuals is as small as possible.

7. 1.5 – Linear Models(Bonus Content) • Your calculator has a function for linear regression (LinReg), which can calculate the line of best fit. • Refresher: Adding functions to the calculator • 2nd, Custom, F1, F3 • We’ll need to add three functions for all of this: • LinR, Plot1, Zdata

8. 1.5 – Linear Models(Bonus Content) • We’ll use the data from example 2 in the book • We need to store our x and y data in lists • Open the list menu (2nd, ‘−’) • Use the curly braces (F1, F2) to input the x data points, then use the STO button (above the On key) to store your value as a variable • {0,1,2,3,4,5,6}X (note, we’re using a capital X – the ‘+’ key – here) • {1,2,2,3,3,5,5} Y (the capital Y is the ‘0’ button)

9. 1.5 – Linear Models(Bonus Content) • We now use the linear regression function • LinR(xData,yData) • We stored our xData in ‘X’ and yData in ‘Y’, so the command for the calculator is: • LinR(X,Y,y1) • You should receive the following output • LinReg y=a+bx a=.964285714 b=.678571429corr=.959644917 n=7 • So what the heck does that all mean?

10. 1.5 – Linear Models(Bonus Content) • LinReg y=a+bx a=.964285714 b=.678571429 corr=.959644917 n=7 • If we substituted the a and b values into the top equation, we’d get • y = .964285714 + .678571429x [let’s reorder] • y =.678571429x + .964285714 • So a line with an intercept ≈ .964 and a slope ≈ .679 is our line of best fit • ‘corr’ stands for coefficient correlation. The closer corr is to 1 or -1, the better the fit. • ‘n’ is the number of points

11. 1.5 – Linear Models • So what’s the use? • We can have the calculator graph the line for us using the Plot1 function • Plot1(graph type, xData, yData) • We need graph type 1 (scatter plot), so our command is: • Plot1(1,X,Y) • The calculator will simply say ‘Done’ in response • To see the plot, we need ZData • Just press ZData and Enter • The graph will auto draw for you • You can change the view of the graph with ‘Wind’ (F2)

12. 1-5 Linear Models • HW: Pg. 53, 4-15 (All) • This is potential quiz/test material, covering describing correlations and finite differences • EC: 16-22 (All) • These are questions using linear regression. You will need a graphing calculator to accomplish these. • Completing 16-22 (all) will result in an extra credit homework assignment