Lecture 10: THE LINEAR REGRESSION MODEL

The population regression line can be written as:

Y_i = A + B.X_i + e_i

where Y_i is the ith observed value of the dependent variable, X_i is the ith observed value of the independent variable, and e_i is a normally distributed random variable with mean zero and standard deviation s_e. Because of the presence of the error term e_i, the observed values of Y_i fall

around the population regression line, and not on it.

Figure 12.4 (page 457) explains the situation.

The sample regression line:

To carry out a regression analysis, we must obtain the mathematical equation for a line that dscribes the average relationship between the dependent variable and independent variable. This line is calculate from the sample of observations and is called the sample or estimated regression line.

The sample regression line is

Y_i^hat= a + b.X_i,

Where Y_i^hat is the value of the dependent variable predicted by the regression line, and a and b are estimated values of A and B.

Figure 12.5 gives the estimated regression based on data in table 12.1.

Y_i^hat= 1.266 + 0.752.X_i.

It is important to be able to interpret a (intercept) and b (slope).

The Method of Least Squares:

The method of least squares dictates that we choose the line where the sum of the squared deviations of the points from the line is minimum.

See Figure 2.6.

It can be shown that b and a can be calculated using the formulae:

b = { nSX_iY_i – (SX_i)( (SY_i)}/{nSX_i² – (SX_i)²}

a = (SY_i)/n – b. (SX_i)/n

See example in Table 12.2.

Do exercises 12.1, 12.2, 12.3, 12.4, 12.5, 12.6, and 12.7 on pages 464-467.