Statistical Analysis for Education and Psychology Researchers

before a regression analysis could be properly used to make inferences about the
dependence of one variable on another and these are discussed in section 8.2.

The Simple Linear Regression Model

Regression analysis may be used to investigate the straight line (linear) relationship in a
population between a random response variable, Y, and an independent explanatory
variable, X. This linear relationship can be expressed as a regression equation which takes
the general form
y=β 0 +β 1 x+ε
Simple
linear
regressio
n
equation
—8.1

This equation says that the observed value of the response variable, y, varies as a linear
function of the explanatory variable x and a random error term, ε, (the Greek letter
epsilon). The term linear function refers to the additive sum of the two parameters in the
model, β 0 (Greek letter beta 0 ), the intercept and β 1 (Greek letter beta 1 ) the population
regression weight for the value of the explanatory variable x. The model can be thought
of as consisting of two components, the deterministic part of the model, β 0 +β 1 x, which
describes the straight-line relationship, and a random component, ε. The observed
response variable, y, can be predicted from a weighted value of the explanatory variable,
x which is the explained straight-line part of the model and an unexplained error
component, ε, which allows for random variation of the y values about their mean. This
error component accounts for random fluctuations of the y variable values and possibly
other important variables not included in the statistical model.

The Linear Regression Line

A simple estimated linear regression line with Y as the predicted response variable and X
as the explanatory variable is described as the regression of Y on X. (The regression of X
on Y would give a different regression line.) We can think of the regression line in the
population being described by two parameters: β 0 , the intercept which is the point at
which the regression line cuts the Y axis, that is the value of the variable Y when the value
of variable X is zero, and β 1 the regression coefficient (weight) which represents the slope
of the regression line, that is the increase or decrease in the variable Y corresponding to a
unit change in the value of variable X. A third parameter, a, the standard deviation of the
response variable Y about the regression line is frequently estimated in regression
analysis as this provides an indication of extent of the linear relationship between the
response and explanatory variable.
As in previous statistical procedures, we use sample estimates of these population
parameters namely, b 0 is the sample statistic which estimates β 0 the unknown population
intercept and b 1 is the sample regression statistic which is used to estimate β 1 the

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