1. Linear Regression Hypothesis testing and Estimation

2. Assume that we have collected data on two variables X and Y. Let ( x 1, y 1 ) ( x 2, y 2 ) ( x 3, y 3 ) … ( x n, y n ) denote the pairs of measurements on the on two variables X and Y for n cases in a sample (or population)

3. The Statistical Model

4. Each y i is assumed to be randomly generated from a normal distribution with mean  i =  +  x i and standard deviation . ( ,  and  are unknown) yiyi  +  x i  xixi Y =  +  X slope =  

5. The Data The Linear Regression Model The data falls roughly about a straight line. Y =  +  X unseen

6. The Least Squares Line Fitting the best straight line to “linear” data

7. Let Y = a + b X denote an arbitrary equation of a straight line. a and b are known values. This equation can be used to predict for each value of X, the value of Y. For example, if X = x i (as for the i th case) then the predicted value of Y is:

8. The residual can be computed for each case in the sample, The residual sum of squares (RSS) is a measure of the “goodness of fit of the line Y = a + bX to the data

9. The optimal choice of a and b will result in the residual sum of squares attaining a minimum. If this is the case than the line: Y = a + bX is called the Least Squares Line

10. The equation for the least squares line Let

11. Computing Formulae:

12. Then the slope of the least squares line can be shown to be: This is an estimator of the slope, , in the regression model

13. and the intercept of the least squares line can be shown to be: This is an estimator of the intercept, , in the regression model

14. The residual sum of Squares Computing formula

15. Estimating , the standard deviation in the regression model : This estimate of  is said to be based on n – 2 degrees of freedom Computing formula

16. Sampling distributions of the estimators

17. The sampling distribution slope of the least squares line : It can be shown that b has a normal distribution with mean and standard deviation

18. Thus has a standard normal distribution, and has a t distribution with df = n - 2

19. The sampling distribution intercept of the least squares line : It can be shown that a has a normal distribution with mean and standard deviation

20. Thus has a standard normal distribution and has a t distribution with df = n - 2

21. (1 –  )100% Confidence Limits for slope  : t  /2 critical value for the t-distribution with n – 2 degrees of freedom

22. Testing the slope The test statistic is: - has a t distribution with df = n – 2 if H 0 is true.

23. The Critical Region Reject df = n – 2 This is a two tailed tests. One tailed tests are also possible

24. (1 –  )100% Confidence Limits for intercept  : t  /2 critical value for the t-distribution with n – 2 degrees of freedom

25. Testing the intercept The test statistic is: - has a t distribution with df = n – 2 if H 0 is true.

26. The Critical Region Reject df = n – 2

27. Example

28. The following data showed the per capita consumption of cigarettes per month (X) in various countries in 1930, and the death rates from lung cancer for men in 1950. TABLE : Per capita consumption of cigarettes per month (X i ) in n = 11 countries in 1930, and the death rates, Y i (per 100,000), from lung cancer for men in 1950. Country (i)X i Y i Australia4818 Canada5015 Denmark3817 Finland11035 Great Britain11046 Holland4924 Iceland236 Norway259 Sweden3011 Switzerland5125 USA13020

30. Fitting the Least Squares Line

31. First compute the following three quantities:

32. Computing Estimate of Slope (  ), Intercept (  ) and standard deviation (  ),

33. 95% Confidence Limits for slope  : t.025 = 2.262 critical value for the t-distribution with 9 degrees of freedom 0.0706 to 0.3862

34. 95% Confidence Limits for intercept  : -4.34 to 17.85 t.025 = 2.262 critical value for the t-distribution with 9 degrees of freedom

35. Y = 6.756 + (0.228)X 95% confidence Limits for slope 0.0706 to 0.3862 95% confidence Limits for intercept -4.34 to 17.85

36. Testing the positive slope The test statistic is:

37. The Critical Region Reject df = 11 – 2 = 9 A one tailed test

38. and conclude we reject

39. Confidence Limits for Points on the Regression Line The intercept  is a specific point on the regression line. It is the y – coordinate of the point on the regression line when x = 0. It is the predicted value of y when x = 0. We may also be interested in other points on the regression line. e.g. when x = x 0 In this case the y – coordinate of the point on the regression line when x = x 0 is  +  x 0

40. x0x0  +  x 0 y =  +  x

41. (1-  )100% Confidence Limits for  +  x 0 : t  /2 is the  /2 critical value for the t-distribution with n - 2 degrees of freedom

42. Prediction Limits for new values of the Dependent variable y An important application of the regression line is prediction. Knowing the value of x (x 0 ) what is the value of y? The predicted value of y when x = x 0 is: This in turn can be estimated by:.

43. The predictor Gives only a single value for y. A more appropriate piece of information would be a range of values. A range of values that has a fixed probability of capturing the value for y. A (1-  )100% prediction interval for y.

44. (1-  )100% Prediction Limits for y when x = x 0 : t  /2 is the  /2 critical value for the t-distribution with n - 2 degrees of freedom

45. Example In this example we are studying building fires in a city and interested in the relationship between: 1. X = the distance of the closest fire hall and the building that puts out the alarm and 2. Y = cost of the damage (1000$) The data was collected on n = 15 fires.

46. The Data

47. Scatter Plot

48. Computations

49. Computations Continued

52. 95% Confidence Limits for slope  : t.025 = 2.160 critical value for the t-distribution with 13 degrees of freedom 4.07 to 5.77

53. 95% Confidence Limits for intercept  : 7.21 to 13.35 t.025 = 2.160 critical value for the t-distribution with 13 degrees of freedom

54. Least Squares Line y=4.92x+10.28

55. (1-  )100% Confidence Limits for  +  x 0 : t  /2 is the  /2 critical value for the t-distribution with n - 2 degrees of freedom

56. 95% Confidence Limits for  +  x 0 :

57. 95% Confidence Limits for  +  x 0 Confidence limits

58. (1-  )100% Prediction Limits for y when x = x 0 : t  /2 is the  /2 critical value for the t-distribution with n - 2 degrees of freedom

59. 95% Prediction Limits for y when x = x 0

60. 95% Prediction Limits for y when x =  x 0 Prediction limits