Confidence Intervals and Hypothesis Tests: Two Samples - Florida ...

STATSprofessor.com
Chapter 9
2
σ1
Confidence Interval for : We won’t form this interval in the course, but it is provided here for your
2
σ
reference.
L,
α / 2
U , α / 2
2
A (1 − α) × 100% Confidence Interval for σ / σ
2 2 2
⎛ s ⎞⎛ 1 1 ⎞ ⎛ σ ⎞ ⎛ 1 s ⎞⎛
1 1 ⎞
⎜ 2 ⎟⎜ < < 2 2
s ⎜
⎟
2 F ⎟ ⎜ ⎟ ⎜ ⎟⎜
⎟
U , α / 2 σ 2 s ⎜
2 F ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝
L,
α / 2 ⎠
2 2
1 2
where F leaves ( α/2)%
of the distribution in a lower tail
and F leaves ( α/2)%
of
the distribution in an upper tail.
Degrees of freedom are n −1
in the numerator and
n −1
in the denominator.
2
Another way to write the above is:
⎡ 2 2
S ⎛
1 1 ⎞ S ⎤
1
⎢ 2 ⎜ , F 2 U , / 2
S ⎜ ⎟
α
2 F ⎟
⎥
⎢⎣ ⎝ L,
α / 2 ⎠ S2
⎥⎦
1
Where FL, α / 2 places alpha/2 area in the upper tail of a
and FU , α / 2 places alpha/2 area in the upper tail of a
following relationship,
f
1
n1 −1, n2
−1,1 −α
/ 2
=
f
n2 −1, n1
−1,
α / 2
Fn1 −1, n2
−1 distribution,
Fn2 −1, n1
−1 distribution. Note: this comes from the
28