Presentation on theme: "Least Squares Regression"— Presentation transcript:
1Least Squares Regression D3: 3.2aTarget Goal: I can make predictions using a least square regression line.Hw: pg 162: 27 – 32, 36, 38, 40, 42, 62
2LSRL: least squares regression line a model for the dataa line that summarizes the two variablesIt makes the sum of the squares of the vertical distances of the data points as small as possible
3The LSRL minimizes the total area of the squares. The LSRL makes the sum of the squares of these distances as small as possible.The LSRL minimizes the total area of the squares.
4Regression Line Straight line Describes how the response variable y changes as the explanatory variable x changes.Use regression line to predict value of y for given value of x.Regression (unlike correlation) requires both an explanatory and response variable.
5The dashed line shows how to use the regression line to predict. You can find the vertical distance of each point on the scatterplot from the regression line.
6Predictions and Error Error (residual) = observed y – predicted ŷ We are interested in the vertical distance of each point on the scatterplot from the regression line.If we predict 4.9, and the actual value turns out to be 5.1, our error is the vertical distance.Error (residual) = observed y – predicted ŷ
7Equation of the least squares regression line We have data on an explanatory variable x and a response variable y for n individuals.From the data, calculate the means x bar, y bar, sx, sy of the two variables, and their correlation r.
8The Least Squares Regression Line (LSRL): ŷ = a + bxwith slope,b =and intercept,a = y – b
9ŷ = a + bx y: the observed value ŷ: the predicted value every LSRL passes throughslope: rate of changeWe will usually not calculate by hand, we will use the calculator.
10Exercise: Gas Consumption The equation of the regression line of gas consumption y on the degree-days x is:ŷ = x
11Verifying ŷ = xUse your calculator to find the mean and standard deviation of both x and y and their correlation r from data in the following table.
13x bar == 22.31Sx == 17.74y bar == 5.306Sy == 3.368r =
14Using what we’ve found, find the slope b and intercept a of the regression line from these. This Verifies ŷ = x except for round off error.
15Least squares lines on the calculator Use the same data you entered into L1 and L2. (Turn off other plots & graphs.)Define the scatterplot using L1 and L2 and the use ZoomStat to plot.
16Press STAT:CALC:(8)LinReg(a+bx):L1,L2,Y1:enter To enter Y1, VARS:Y-VARS:(1)FUNCTION}If r2 and r do not appear on your screen, press 2nd:0 (catalog).Scroll down to “DiagnosticOn” and press enter.
17Press GRAPH to overlay the LSRL on the scatterplot. Note: verify LSRL equation at Y1 to beŷ = x
18Least-Squares Regression Interpreting a Regression LineConsider the regression line from the example“Does Fidgeting Keep You Slim?” Identify the slope and y-intercept and interpret each value in context.Least-Squares RegressionThe y-intercept a = kg is the fat gain estimated by this model if NEA does not change when a person overeats.The slope b = tells us that the amount of fat gained is predicted to go down by kg for each added calorie of NEA.
19Least-Squares Regression PredictionWe can use a regression line to predict the response ŷ for a specific value of the explanatory variable x.Use the NEA and fat gain regression line to predict the fat gain for a person whose NEA increases by 400 cal when she overeats.Least-Squares RegressionWe predict a fat gain of 2.13 kg when a person with NEA = 400 calories.
20Least-Squares Regression Interpreting Computer Regression OutputA number of statistical software packages produce similar regression output. Be sure you can locatethe slope b,the y intercept a,and the values of s and r2.Least-Squares Regression
21The slope b = tells us that the amount of Pct is predicted to go down by units for each additional pair.The y-intercept a = is the Pct estimated by this model when there are no pairs.