Page 18-49
Example1 -- Consider two samples drawn from normal populations such that
n
1
= 21, n
2
= 31, s
1
2
= 0.36, and s
2
2
= 0.25. We test the null hypothesis, H
o
:
σ
1
2
= σ
2
2
, at a significance level α = 0.05, against the alternative hypothesis,
H
1
: σ
1
2
≠ σ
2
2
. For a two-sided hypothesis, we need to identify s
M
and s
m
, as
follows:
s
M
2
=max(s
1
2
,s
2
2
) = max(0.36,0.25) = 0.36 = s
1
2
s
m
2
=min(s
1
2
,s
2
2
) = min (0.36,0.25) = 0.25 = s
2
2
Also,
n
M
= n
1
= 21,
n
m
= n
2
= 31,
ν
N
= n
M
- 1= 21-1=20,
ν
D
= n
m
-1 = 31-1 =30.
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.
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