Page 35 - Industrial Process Plant Construction Estimating and Man Hour Analysis

Introduction to construction statistics using Excel Chapter 1 5

1.3 Linear regression and method of least squares
1.3.1 Linear regression: Fitting a straight line

The straight-line relationship can be valuable in summarizing the observed
dependence of one variable on another. The most common type of linear regres-
sion is called ordinary least-squares regression. Linear regression uses the
values from an existing data set consisting of measurements of the values of
two variables, x and y, to develop a model that is useful for predicting the value
of the dependent variable, y for given values of x. Thus, the statistical tech-
niques used in linear regression can be used to uncover the relationships that
exist in construction.

1.3.2 Elements of a regression equations (linear, first-order model)
Regression equation:
y ¼ aþbxþε
y is the value of the dependent variable (y), what is being predicted or
explained.
a, a constant, equals the value of y when the value of x¼0.
b is the coefficient of X, the slope of the regression line, how much Y
changes for each change in x.
x is the value of the independent variable (x), what is predicting or explain-
ing the value of y.
ε is the error term, the error in predicting the value of y, given the
value of x

1.3.3 Assumptions of linear regression
1. Both the independent (x) and the dependent (y) variables are measured at
interval or ration level.
2. The relationship between the independent (x) and the dependent (y) vari-
ables is linear.
3. Errors in prediction of the value of y are distributed in a way that approaches
the normal curve.
4. Errors in prediction of the value of y are all independent of one another.
5. The distribution of the errors in prediction of the value of y is constant
regardless of the value of x.

1.3.4 Method of least squares
Least square line
The least-squares line approximating the set of points (x1, y1), (x2, y2) … (xn,
yn) has the equation