3.2 (part 1) 9.22.2017

Y-hat Why do we label the predicted value y-hat instead of just y?

Y-hat Why do we label the predicted value y-hat instead of just y? Because the “true” y values are just labeled as y So “true” or “observed” y values are just y “Predicted” y values are y-hat

GPA Example GPA Stats Grade Student 1 3.317 94.37 Student 2 2.500 76.06 Student 3 4.024 100.73 Student 4 4.238 97.11

GPA Example When I calculate a regression line, I get: 𝐺𝑟𝑎𝑑𝑒 =46.73+12.88(𝐺𝑃𝐴) GPA Stats Grade Student 1 3.317 94.37 Student 2 2.500 76.06 Student 3 4.024 100.73 Student 4 4.238 97.11

GPA Example When I calculate a regression line, I get: 𝐺𝑟𝑎𝑑𝑒 =46.73+12.88(𝐺𝑃𝐴) So if I want to estimate (or predict) the grade in AP Stats for someone with a 3.12 GPA, what would I do? GPA Stats Grade Student 1 3.317 94.37 Student 2 2.500 76.06 Student 3 4.024 100.73 Student 4 4.238 97.11

GPA Example When I calculate a regression line, I get: 𝐺𝑟𝑎𝑑𝑒 =46.73+12.88(𝐺𝑃𝐴) So if I want to estimate (or predict) the grade in AP Stats for someone with a 3.120 GPA, what would I do? 𝐺𝑟𝑎𝑑𝑒 =46.73+12.88 3.120 So our predicted grade is 86.916 GPA Stats Grade Student 1 3.317 94.37 Student 2 2.500 76.06 Student 3 4.024 100.73 Student 4 4.238 97.11

Practice (Corvettes) 𝑃 =371.6−27.9(𝐴𝑔𝑒) Use the regression equation to predict the value of a brand new Corvette Predict the value of a 3 year old Corvette Predict the value of a 20 year old Corvette

Practice (Corvettes) Note: Price in hundreds of dollars 𝑃 =371.6−27.9(𝐴𝑔𝑒) Use the regression equation to predict the value of a brand new Corvette $37,160 Predict the age of a 3 year old Corvette $28,790 Predict the age of a 20 year old Corvette -$18,640 Someone will PAY YOU to take their 20 year old Corvette!!!!!!

Extrapolation

Residuals

Positive and Negative Residuals So if a positive residual is above the line, that means that 𝑦> 𝑦 Or in other words, the model UNDERpredicted for that value of x If the residual is below the line, it is negative So 𝑦< 𝑦 The model OVERpredicted for that value of x

Residuals Practice So 𝐺𝑟𝑎𝑑𝑒 =46.73+12.88(𝐺𝑃𝐴) Find residuals for each student Y 𝒚 Residual U/O Student 1 94.37 89.45 4.92 Under-Predicted Student 2 76.06 78.93 -2.87 Over Student 3 100.73 98.56 2.17 Under Student 4 97.11 101.32 -4.21

Next Class We will learn how to actually calculate the regression line You’ve done this before on your calculaltors We will learn ways to assess whether linear regression is appropriate for the situation And we will learn how to assess how well the model is doing at predicting the outcome variable Can compare which predictor is more effective