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Therefore, the F test statistics is F
o
= s
M
2
/s
m
2
=0.36/0.25=1.44
The P-value is P-value = P(F>F
o
) = P(F>1.44) = UTPF(ν
N
, ν
D
,F
o
) =
UTPF(20,30,1.44) = 0.1788…
Since 0.1788… > 0.05, i.e., P-value > α, therefore, we cannot reject the null
hypothesis that H
o
: σ
1
2
= σ
2
2
.
Additional notes on linear regression
In this section we elaborate the ideas of linear regression presented earlier in
the chapter and present a procedure for hypothesis testing of regression
parameters.
The method of least squares
Let x = independent, non-random variable, and Y = dependent, random
variable. The regression curve
of Y on x is defined as the relationship between
x and the mean of the corresponding distribution of the Y’s.
Assume that the regression curve of Y on x is linear, i.e., mean distribution of
Y’s is given by Α + Βx. Y differs from the mean (Α + Β⋅x) by a value ε, thus
Y = Α + Β⋅x + ε, where ε is a random variable.
To visually check whether the data follows a linear trend, draw a scattergram or
scatter plot.
Suppose that we have n paired observations (x
i
, y
i
); we predict y by means of
∧
y = a + b⋅x, where a and b are constant.
Define the prediction error
as, e
i
= y
i
-
∧
y
i
= y
i
- (a + b⋅x
i
).
The method of least squares requires us to choose a, b so as to minimize the
sum of squared errors (SSE)
the conditions
2
11
2
)]([
i
n
i
i
n
i
i
bxayeSSE +−==
∑∑
==
0)( =SSE
a
∂
∂
0)( =SSE
b
∂
∂