1. Stat 217 – Day 26 Regression, cont.

2. Last Time – Two quantitative variables Graphical summary  Scatterplot: direction, form (linear?), strength Numerical summary  Correlation coefficient, r, measures the strength of the linear relationship  -1 < r < 1 Model to allow predictions Inference beyond sample data

3. Example 1: Price vs. Size r =.780 What do you learn from these numerical and graphical summaries?

4. Describing Scatterplots Activity 26-6 (p. 539) Positive, nonlinear, fairly strong Causation? Strength: How closely do the points follow the pattern? Direction Strength Form: Linear or not

5. 3) Model IF it is linear, what line best summarizes the relationship?  Demo Demo Moral: The “least squares regression line” minimizes the sum of the squared residuals

6. Interpreting the equation (p. 577)  a = intercept, b = slope  Slope = predicted change in response associated with a one-unit increase in the explanatory  Intercept = predicted value of response when explanatory variable = 0 Explanatory variable Response variable

7. 3) Model? Price-hat = 265222 + 169  size  Slope = each additional square foot in house size is associated with a $169 increase in predicted price (price per foot) Be a little careful here, don’t sound too “causal” I really do like the “predicted” in here  Intercept = a house of size zero (empty lot?) is predicted to cost $265,222 Be a little careful here, don’t have any houses in data set with size near 0…

8. Using the model Price-hat = 265222 + 169  size  Predicted price for a 1250 square foot house?  Predicted price for a 3000 square foot house? Extrapolation: Very risky to use regression equation to predict values far outside the range of x values used to derive the line!

9. 4) Is this relationship statistically significant? Is it possible there is no relationship between house price and size in the population of all homes for sale at that time, and we just happened to coincidently obtain this relationship in our random sample? Or is this relationship strong enough to convince us it didn’t happen just by chance but reflects a genuine relationship in the population?

10. p. 605 Let  represent the slope of the population regression line H 0 :  = 0; no relationship between price and size in population H a :  ≠ 0; is a relationship  positive Idea: Want to compare the observed sample slope to zero, does it differ more than we would expect by chance?

11. Assume  = 0 How many standard deviations away? Variation in sample slopes  Sample slopes  our slope? Standard error = SE(b) 169

12. Minitab The regression equation is Price = 265222 + 169 Size (sq ft) Predictor Coef SE Coef T P Constant 265222 42642 6.22 0.000 Size (sq ft) 168.59 31.88 5.29 0.000 Regression equation (add hat) b a SE(b) Two-sided t=(observed slope-hypothesized slope) standard error of slope = (b – 0)/SE(b) = (168.59-0)/31.88 = 5.29

13. Turn in, with partner  Price vs. pages: (a) Interpret slope (b)Identify and evaluate p-value For Tuesday  Activities 26-7, 28-5  Be working on Lab 9 and HW 7 The regression equation is Price = - 3.4 + 0.147 Pages Predictor Coef SE Coef T P Constant -3.42 10.46 -0.33 0.746 Pages 0.14733 0.01925 7.65 0.000