Business Statistics Lecture Notes 08 - MN1025 Business...

MN1025 – Business Statistics 35
Lecture 8—Friday 29/2/2008
LINEAR REGRESSION
Reference: Lind et al. , Chapter 13.
8.1 Regression: introduction
In the last lecture we introduced the concept of the
best-fit line, which is an approximation to the data.
The closeness of this approximation is measured by
the correlation coefficient r . In this lecture we will
see how the best-fit line can be used for prediction.
Example: Suppose the College wishes to save money
and asks: can we predict exam results well from
weekly work? If the answer is yes, we dispense with
exams. So to test this we need a sample of students
already examined and see if for each student, their
average weekly mark predicts their exam result. To
estimate the predictive power, one uses Linear Re-
gression .
8.2 Back to sales and scores
Back to Example 7.7 (sales and scores). Here are
the data again:
Data Display
Row scores sales
1 4 5
2 7 12
3 3 4
4 6 8
5 10 11
We wish to analyse how good an approxima-
tion to the data the best-fit line is. We use
STAT → REGRESSION → REGRESSION. We are
asked to choose a RESPONSE column and a PRE-
DICTOR column. In this case the only reasonable
choice is “scores” as predictor (or cause) and “sales”
as response (or effect). We get the Regression Anal-
ysis table shown below.
Regression Analysis: sales versus scores
The regression equation is
sales = 1.20 + 1.13 scores
Predictor Coef SE Coef T P
Constant 1.200 2.313 0.52 0.640
scores 1.1333 0.3569 3.18 0.050
S = 1.955 R-Sq = 77.1% R-Sq(adj) = 69.4%
Analysis of Variance
Source DF SS MS F P
Regression 1 38.53 38.53 10.08 0.050
Residual Error 3 11.47 3.82
Total 4 50.00
The regression equation
sales = 1.20 + 1.13 scores in the printout is
the equation of the best-fit line. We can plot this on
the scatter plot or get Minitab to plot it for us: we
use STAT → REGRESSION → FITTED LINE PLOT
and enter again “sales” as response, “scores” as pre-
dictor.
In all these examples we assume that the underlying
populations have an approximately normal distribu-
tion, and that a relation of the form
sales = m × scores + c + random error
is reasonable. In general, there could be more than
one predictor. For instance, we could think that staff
experience was a relevant factor and get a relation
of the form
sales = m
1
× scores + m
2
× experience+ c +random error .
Here we have two predictors, “scores” and “experi-
ence”. Generally, by a suitable choice of additional
predictors we can reduce the random error. In this
course, we will always use a single predictor only.
8.3 Testing if the slope is nonzero
For the population of scores and sales there is an
underlying (population) regression line:
sales = m
population
× scores + c
population
.
In this equation, m
population
is the slope of the (pop-
ulation) regression line, and c
population
is its inter-
cept. The sample slope of m = 1 . 13 is our estimate
for m
population
, and the sample intercept of c = 1 . 20
is our estimate for c
population
.