h)
Each Sum of squares has a corresponding degrees of freedom, df
associated with it.
i)
Total df is n - 1, one less than the number of observations.
j)
Regression df is the number of independent variables in the model. For SLR, it is 1.
k)
Error df is the difference between total df and Regression df.
l)
Mean Squares (MS) are the sum of squares divided by the corresponding df.
m)
F statistic or F ratio is the ratio of SS divided by the corresponding df.
n)
In the context of regression, the
p value
reported in the table gives us an overall test for the
significance of our model.
o)
As p value (2 e-16) is less than 0.05, we reject the null hypothesis that there is no relationship
between predictor and response and conclude that there is a significant relationship between the
predictor and response variables.
VIII.
Tests of significance using ANOVA

Regression
5-Jan-20
Great Learning
26
6.
Simple Linear Regression (SLR) Model building
IX.
Hypothesis test for regression coefficient (t test) and p values
H
0
:
There is no relationship between X and Y or
β
1
= 0
H
A
:
There is a relationship between X and Y or
β
1
≠ 0
Test statistic follows a t distribution with n-2 df where n is the number of observations.
We reject the null hypothesis at 5% level of significance, when the p value
corresponding the t statistic < 0.05.

Regression
5-Jan-20
Great Learning
27
IX.
Hypothesis test for regression coefficient (t test) and p values
We observe that the overall mean square of
prediction error is 4.53
Model parameters estimated:
β0
= 2.2526;
β1
= 1.9689
Regression equation is given as
Revenue = 2.2526 + 1.9689 * Promotion Expenses
Interpretation of the equation:
For every one unit increase in Promotion expenses,
revenue will increase at the rate of 1.9689 on the
average.
R square is 0.996 tells us the model explains 99.6% of
the variation in the dependent variable, Revenue.
6.
Simple Linear Regression (SLR) Model building

Regression
5-Jan-20
Great Learning
28
6.
Simple Linear Regression (SLR) Model building
IX.
Hypothesis test for regression coefficient (t test) and p values
✓
Standard error of the residual is 2.04 on 48 degrees of freedom
✓
Standard error of the X is 0.0191 and t value is 103.16.
✓
For this two-tailed test with alpha = 0.05 & degrees of freedom =
48 (50 -2), t critical value is 2.01
✓
Our t calculated value of 103.16 > t critical value of 2.01.
Alternatively, p value of 2e-16 < 0.05
✓
So we have evidence to reject the null hypothesis and conclude
that there exists a relationship between X and Y and
β
1
≠ 0

Regression
5-Jan-20
Great Learning
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