1 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Simple Linear Regression Estimates for single and mean responses

2 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Properties of the Sampling Distribution of a + bx for a Fixed x Value Let x* denote a particular value of the independent variable x. When the four basic assumptions of the simple linear regression model are satisfied, the sampling distribution of the statistic a + bx* has the following properties: 1.The mean value of a + bx* is  +  x*, so a + bx* is an unbiased statistic for estimating the average y value when x = x*

3 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Properties of the Sampling Distribution of a + bx for a Fixed x Value 3.The distribution of the statistic a + bx* is normal. 2.The standard deviation of the statistic a + bx* denoted by  a+bx*, is given by

4 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Addition Information about the Sampling Distribution of a + bx for a Fixed x Value The estimated standard deviation of the statistic a + bx*, denoted by s a+bx*, is given by When the four basic assumptions of the simple linear regression model are satisfied, the probability distribution of the standardized variable is the t distribution with df = n - 2.

5 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Confidence Interval for a Mean y Value When the four basic assumptions of the simple linear regression model are met, a confidence interval for a + bx*, the average y value when x has the value x*, is a + bx*  (t critical value)s a+bx* Where the t critical value is based on df = n -2. Many authors give the following equivalent form for the confidence interval.

6 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Confidence Interval for a Single y Value When the four basic assumptions of the simple linear regression model are met, a prediction interval for y*, a single y observation made when x has the value x*, has the form Where the t critical value is based on df = n -2. Many authors give the following equivalent form for the prediction interval.

7 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example - Mean Annual Temperature vs. Mortality Data was collected in certain regions of Great Britain, Norway and Sweden to study the relationship between the mean annual temperature and the mortality rate for a specific type of breast cancer in women. * Lea, A.J. (1965) New Observations on distribution of neoplasms of female breast in certain European countries. British Medical Journal, 1,

8 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example - Mean Annual Temperature vs. Mortality Regression Analysis: Mortality index versus Mean annual temperature The regression equation is Mortality index = Mean annual temperature Predictor Coef SE Coef T P Constant Mean ann S = R-Sq = 76.5% R-Sq(adj) = 74.9% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Unusual Observations Obs Mean ann Mortalit Fit SE Fit Residual St Resid RX R denotes an observation with a large standardized residual X denotes an observation whose X value gives it large influence.

9 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example - Mean Annual Temperature vs. Mortality The point has a large standardized residual and is influential because of the low Mean Annual Temperature.

10 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example - Mean Annual Temperature vs. Mortality Predicted Values for New Observations New Obs Fit SE Fit 95.0% CI 95.0% PI ( 42.79, 63.57) ( 33.95, 72.41) X ( 52.48, 68.96) ( 42.57, 78.88) ( 67.20, 77.82) ( 55.48, 89.54) ( 79.30, 87.39) ( 66.66, ) ( 90.37, ) ( 78.93, ) ( 92.71, ) ( 81.74, ) X denotes a row with X values away from the center Values of Predictors for New Observations New Obs Mean ann These are the x* values for which the above fits, standard errors of the fits, 95% confidence intervals for Mean y values and prediction intervals for y values given above.

11 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Example - Mean Annual Temperature vs. Mortality 95% prediction interval for single y value at x = 45. (67.62,100.98) 95% confidence interval for Mean y value at x = 40. (67.20, 77.82)