The Simple Linear Regression Model

Estimators in Simple Linear Regression and

Sampling distributions of the estimators

Recall that if y 1, y 2, y 3 …, y n are 1.Independent 2.Normally distributed with means  1,  2,  3 …,  n and standard deviations  1,  2,  3 …,  n Then L = c 1 y 1 + c 2 y 2 + c 3 y 3 + … + c n y n is normal with mean and standard deviation

Sampling distribution the slope

Note : Also

Thus Hence where and standard deviation is normal with mean

Thus since and

Also

and standard deviation Henceis normal with mean

Sampling distribution of the intercept

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

Proof: where Thus

Also now

Hence and

and standard deviation Summary is normal with mean is normal with mean and standard deviation 1. 2.

Sampling distribution of the estimate of variance

The sampling distribution of s 2 This estimate of  is said to be based on n – 2 degrees of freedom

The sampling distribution of s 2 Recall that y 1, y 2, …, y n are independent, normal with mean  +  x i and standard deviation  Let Then z 1, z 2, …, z n are independent, normal with mean 0 and standard deviation 1, and Has a  2 distribution with n degrees of freedom

If  and  are replaced by their estimators: then has a  2 distribution with n-2 degrees of freedom Note:

Thus This verifies the statement made earlier that s 2 is an unbiased estimator of  2. and

and standard deviation Summary is normal with mean is normal with mean and standard deviation 1. 2.

and standard deviation Recall is normal with mean Therefore has a standard normal distribution

has a t distribution with n – 2 degrees of freedom and

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

and standard deviation Also is normal with mean Therefore has a standard Normal distribution

and has a t distribution with n – 2 degrees of freedom

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

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 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 Country (i)X i Y i Australia4818 Canada5015 Denmark3817 Finland11035 Great Britain11046 Holland4924 Iceland236 Norway259 Sweden3011 Switzerland5125 USA13020

Fitting the Least Squares Line

First compute the following three quantities:

Computing Estimate of Slope and Intercept

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

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

(1 –  )100% Confidence Limits for a point on the regression line  +  x 0 : x y regression line  +  0 x 0 x0x0 y =  +  0 x

Let then and

Proof: where Note and Thus

Also now

Hence and

(1 –  )100% Confidence Limits for a point on the regression line intercept  +  x 0 : t  /2 critical value for the t-distribution with n - 2 degrees of freedom

Prediction In linear regression model

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