Objectives: INTERPRET the slope and y intercept of a least-squares regression line USE the least-squares regression line to predict y for a given x CALCULATE and INTERPRET residuals and their standard deviation EXPLAIN the concept of least squares DETERMINE the equation of a least-squares regression line using a variety of methods CONSTRUCT and INTERPRET residual plots to assess whether a linear model is appropriate
DEFINITION: REGRESSION LINE: A line that describes how a response variable (y) changes as an explanatory variable (x) changes This is a MODEL of the data
EQUATION FOR A REGRESSION LINE: y is the OBSERVED VALUE at x ( From the point ( Ky ) ) (read y-hat ) is the PREDICTED VALUE a is the y-intercept Loading b is the slope of the regression line when we plug X into the ) equation ( The value we get Ryon Mutt Know the difference between y and Or observed, J and expected
- Example: This scatterplot shows the price and miles driven on 16 used Ford F-150's " ergadpreadsictedprice y ya - intercept = miles driven PROBLEM: Identify the slope and y intercept of the regression line Interpret each value in context SOLUTION: The slope b = -01629 tells us that the price of a used Ford F-150 is predicted to go down by 01629 dollars (1629 cents) for each additional mile that the truck has been driven The y intercept a = 38,257 is the predicted price of a Ford F-150 that has been driven 0 miles
- 0 1629400,000 PREDICTIONS: You can use a regression line to predict the value of a response variable for a specific value of the explanatory variable Use the regression equation from the previous example to predict the price of a Ford F-150 that has 100,000 miles on it ^ price = 38,257 = 21,967 price ) Loading The predicted price of a Ford F- 150 that has 100,000 miles on it is $21,967 * Note : you Mutt clarify that this is a predicted price, not the actual price
We can use a regression line to predict the response ŷ for a specific value of the explanatory variable x The accuracy of the prediction depends on how much the data scatter about the line While we can substitute any value of x into the equation of the regression line, we must exercise caution in making predictions outside the observed values of x Extrapolation is the use of a regression line for prediction far outside the interval of values of the explanatory variable x used to obtain the line Such predictions are often not accurate other words, you - extrapolate shouldn't
RESIDUALS: A residual is the difference between an actual observed value of the response variable (y), and the predicted value ( :% ) Gt Note : Order matters! To help you for ' observed ' comes before " p " remember, the " o " for 'predicted ' in the alphabet
In most cases, no line will pass exactly through all the points in a scatterplot A good regression line makes the vertical distances of the points from the line as small as possible ( Residual )
LEAST-SQUARES REGRESSION LINES (LSRL s) Different regression lines produce different residuals The regression line we want is the one that minimizes the sum of the squared residuals The least-squares regression line of y on x is the line that makes the sum of the squared residuals as small a possible
Let s take a look at this in motion Go to http://wwwmacmillanlearningcom/catalog/studentresources/tps5e# and use the Correlation & Regression Applet
CALCULATOR: When we have actual data, we will use the calculator to find our prediction line There are actually 2 linear regression options in your calculator, STAT/CALC/4:LinReg(ax+b), and STAT/CALC/8:LinReg(a+bx) Both will give you the same values for the slope and y-intercept, but you need to note that in one case a is the slope and b is the y-intercept, and in the other we have the opposite Loading For consistency, we will use the STAT/CALC/8:LinReg(a+bx) command Once you ve got the LinReg(a+bx) on your calculator screen, you will need to specify the lists in which you ve put your x and y values If you want to put the equation into your equation editor, you also do that at this time
Here is the actual data for our used Ford F-150 s Put the data into your calculator and find the LSRL equation Miles Price ($) Miles Price ($) ux ix y Put X 's in 4 and y 's in 4
No AP Stats: 3B ~ Least Squares Regression and Residuals Huh idea why this slide got blown up the scatter plot when use the Un Reg 8 Command q and name of the equation function name mum VARSAVARVI : Function /z:y, puts the regression equation into Y in the, equation editor [linregslaxtb ) 4,4,'D
How do we know if our linear model is appropriate? We can use our residuals and make a residual plot A RESIDUAL PLOT is a scatterplot of the residuals against the explanatory variable (so each point is at (x, residual)) The pattern (or lack thereof) tells us if the model is a good one
These are some scatterplots of a regression line s residual values plotted against the explanatory variable To show that our model is a good one, we want to have a scatterplot that is evenly scattered above and below the x-axis, and that has no particular pattern or clusters
To plot this on your calculator: Input x-values into List 1, y- values into List 2, and use the LinReg(a+bx) command This automatically populates a list called RESID (you can find this list by going into LIST (press 2nd then STAT) and scrolling down Set up your stat plot as follows, then graph with Zoom 9 (ZoomStat) Note : You must run Lin Reg to populate the list Resid before making a residual plot
his axis Example: Find and sketch the residual plot for our Ford F-150 data Is our model a good one? ' the model is pretty evenly good scattered above and & aim o bdowthex X Residual
Homework: p 193: #35-47, 49, 51 Read pp 177-192