1. Chapter 10 Correlation and Regression

2. SCATTER DIAGRAMS AND LINEAR CORRELATION

3. Note: Studies of correlation and regression of two variables usually begin with a graph of paired data values (x,y). We call this a scatter diagram

4. Scatter Plot/Scatter Diagram It is a graph in which data pairs (x,y) are plotted as individual points on a grid with horizontal axis x and vertical axis y. We call x the explanatory variable and y the response variable.

5. Why do we use scatter plot? We use scatter plot to observe whether there seems to be a linear relationship between x and y values.

6. Example: Phosphorous is a chemical used in many household and industrial cleaning compound. Unfortunately, phosphorous tends to find its way into surface water, where it can kill fish, plants, and other wetland creatures. Phosphorous reduction programs are required by law and are monitored by the EPA. A random sample of eight sites in a California wetlands study gave the following information about phosphorous reduction in drainage water. X is a random variable that represents phosphorous concentration and y is a random variable that represents total phosphorous concentration. Graph the scatter plot and then comment on the relationship between x and y x5.27.36.75.96.18.35.57.0 y3.35.94.84.54.07.13.66.1

7. Group Work Here are the safety report of different divisions: Make a scatter plot Does a line fit the data reasonably well? Draw a line that “fits best” DivisionX (hours in safety training) Y (accidents) 11080 219.565 33068 44555 55035 66510 78012

8. Scatter Diagrams with its correlation

9. Correlation coefficient r

10. How do you compute the sample correlation coefficient r?

11. Example: calculate r X70115105829312588 y345217166212

12. Group work: calculate r X101516146181214 y521978153

13. Note:

14. TI 83/TI 84 Enter the data into two columns. Use Stat Plot and choose the first type. Use option 9:ZoomStat under Zoom. CATALOG, find DiagnosticON, press enter twice. Then, press STAT, CALC, then option 8:LinReg(a+bx)

15. Homework Practice P503 #1-16 even

16. LINEAR REGRESSION AND THE COEFFICIENT OF DETERMINATION

17. Least-squares criterion The sum of the squares of the vertical distances from the data points (x,y) to the line is made as small as possible.

18. What does it mean? It means that for any given point, the distance from the point to the line has the least amount of error (the sum of the squares of the vertical distance from the points to the line be made as small as possible). We use the least-squares criterion to find the best linear equation.

20. Example: Find the least-squares line y=a+bx X30342725172320 y66797060485560

21. Sketch the scatter plot and the least- squares line

22. Remember

23. Using the Least-Squares Line for Prediction

24. Group Work a)Find the least-squares line. b)Predict when x = 51 X= Super soldier serum y=Captain America

25. Coefficient of determination

26. Homework Practice Pg 520 #1-18 eoo

27. INFERENCES FOR CORRELATION AND REGRESSION

28. Sample Statistic to Population Parameter Sample StatisticPopulation Parameter

29. Important Note:

31. Example: Learn more, earn more! We have probably all heard this platitude. The question is whether or not there is some truth in the statement. Do college graduates have an improved chance at a better income? Is there a trend in the general population? Consider the following variables: x=percentage of the population 25 or older with at least four years of college and y = percentage growth in per capita income over the past seven years. A random sample of six communities in Ohio gave the info. Use 1% level of significance

32. Answer

33. Group Work x9.210.19.012.58.89.19.5 y5.04.84.55.75.14.64.2

34. Standard Error of Estimate

35. TI 83/TI 84

36. How to Find a Confidence Interval for a predicted y from the Least-Squares Line

37. Confidence interval Continue.

38. Example: XY 1017 2021 3025 4028 5033 6040 7049

39. It is important because it measures the rate at which y changes per unit change in x

42. Example: X12083605035302017 Y301615.514.52218120

43. Group Work x8101420284045 Y7131723303435

44. Homework Practice Pg 543 #1-12 odd