1. A student wonders if tall women tend to date taller men than do short women. She measures herself, her dormitory roommate, and the women in the adjoining rooms. Then she measures the next man each woman date. Draw & discuss the scatterplot and calculate the correlation coefficient. Women (x) Men (y) 6672 6468 6670 6568 7071 65

2. Find the correlation Coefficient: XYZxZyZx*Zy 46 815 22 1918 2227

3. Find the correlation Coefficient: XYZxZyZx*Zy 3227 4082 3034 1814 151 2522

4. Find the correlation Coefficient: XYZxZyZx*Zy 272 860 1064 1452 2843 3240 1832

5. Linear Regression

6. Guess the correlation coefficient http://istics.net/stat/Correlations/

7. Can we make a Line of Best Fit

8. Regression Line This is a line that describes how a response variable (y) changes as an explanatory variable (x) changes. It’s used to predict the value of (y) for a given value of (x). Unlike correlation, regression requires that we have an explanatory variable.

9. Let’s try some! http://illuminations.nctm.org/ActivityDetail.asp x?ID=146 http://illuminations.nctm.org/ActivityDetail.asp x?ID=146

10. Regression Line

11. The following data shows the number of miles driven and advertised price for 11 used Honda CR-Vs from the 2002-2006 model years (prices found at www.carmax.com). The scatterplot below shows a strong, negative linear association between number of miles and advertised cost. The correlation is -0.874. The line on the plot is the regression line for predicting advertised price based on number of miles. Thousand Miles Driven Cost (dollars) 2217998 2916450 3514998 3913998 4514599 4914988 5513599 5614599 6911998 7014450 8610998

12. The regression line is shown below…. Use it to answer the following. Slope: Y-intercept:

13. Predict the price for a Honda with 50,000 miles.

14. Extrapolation This refers to using a regression line for prediction far outside the interval of values of the explanatory variable x used to obtain the line. They are not usually very accurate predictions.

15. Slope: Y-int: Predict weight after 16 wk Predict weight at 2 years:

16. Residual

17. The equation of the least-squares regression line for the sprint time and long- jump distance data is predicted long-jump distance = 304.56 – 27.3 (sprint time). Find and interpret the residual for the student who had a sprint time of 8.09 seconds with a long jump of 84 inches.

18. Regression Let’s see how a regression line is calculated.

19. Fat vs Calories in Burgers Fat (g)Calories 19410 31580 34590 35570 39640 39680 43660

20. Let’s standardize the variables FatCalz - x'sz - y's 19410-1.959-2 31580-0.42-0.1 34590-0.0360 355700.09-0.2 396400.60.56 396800.61 436601.120.78 The line must contain the point and pass through the origin.

21. Let’s clarify a little. (Just watch & listen) The equation for a line that passes through the origin can be written with just a slope & no intercept: y = mx. But, we’re using z-scores so our equation should reflect this and thus it’s Many lines with different slope pass through the origin. Which one fits our data the best? That is which slope determines the line that minimizes the sum of the squared residuals.

22. Line of Best Fit –Least Squares Regression Line It’s the line for which the sum of the squared residuals is smallest. We want to find the mean squared residual. Focus on the vertical deviations from the line. Residual = Observed - Predicted

23. Let’s find it. (just watch & soak it in) St. Dev of z scores is 1 so variance is 1 also. This is r!

24. Continue…… Since this is a parabola – it reaches it’s minimum at Hence – the slope of the best fit line for z-scores is the correlation coefficient → r.

25. Slope – rise over run A slope of r for z-scores means that for every increase of 1 standard deviation in, there is an increase of r standard deviations in. “Over 1 and up r” Translate back to x & y values – “over one standard deviation in x, up r standard deviations in y. Slope of the regression line is:

26. Why is correlation “r” Because it was calculated from the regression of y on x after standardizing the variables – just like we have just done – thus he used r to stand for (standardized) regression.

27. The number of miles (in thousands) for the 11 used Hondas have a mean of 50.5 and a standard deviation of 19.3. The asking prices had a mean of $14,425 and a standard deviation of $1,899. The correlation for these variables is r = -0.874. Find the equation of the least-squares regression line and explain what change in price we would expect for each additional 19.3 thousand miles.

28. So let’s write the equation! Fat (g)Calories 19410 31580 34590 35570 39640 39680 43660 Slope: Explain the slope:

29. Now for the final part – the equation! Y-intercept: Remember – it has to pass through the point. Solve for y-intercept:

30. Now it can be used to predict. How many calories do I expect to find in a hamburger that has 25 grams of fat?

31. Try another problem Mean call - to-shock time Survival Rate 290 645 730 95 122

32. Interpret the slope: Interpret the y-int: Predict the survival rate for a 10 min. call to shock time Predict the survival rate for a 20 min. call to shock time

33. Try another problem SAT MathSAT Verbal 600650 720800 540600 450500 620

34. Interpret the slope: Interpret the y-int: Predict the verbal score for a math score of 400 Predict the verbal score for a math score of 500

35. That’s…all…..Folks!