HP 50G, User Manual

Page: 601 / 887

Page 18-34
The confidence interval for the population variance
σ
2
is therefore,
[(n-1)
⋅
S
2
/
χ
2
n-1,
α
/2
; (n-1)
⋅
S
2
/
χ
2
n-1,1-
α
/2
].
where
χ
2
n-1,
α
/2
, and
χ
2
n-1,1-
α
/2
are the values that a
χ
2
variable, with
ν
= n-1
degrees of freedom, exceeds with probabilities
α
/2 and 1-
α
/2, respectively.
The one-sided upper confidence limit for
σ
2
is defined as (n-1)
⋅
S
2
/
χ
2
n-1,1-
α
.
Example 1 – Determine the 95% confidence interval for the population variance
σ
2
based on the results from a sample of size n = 25 that indicates that the
sample variance is s
2
= 12.5.
In Chapter 17 we use the numerical solver to solve the equation
α
= UTPC(
γ
,x).
In this program,
γ
represents the degrees of freedom (n-1), and
α
represents the
probability of exceeding a certain value of x (
χ
2
), i.e., Pr[
χ
2
>
χ
α
2
] =
α
.
For the present example,
α
= 0.05,
γ
= 24 and
α
= 0.025. Solving the
equation presented above results in
χ
2
n-1,
α
/2
=
χ
2
24,
0.025
= 39.3640770266.
On the other hand, the value
χ
2
n-1,
α
/2
=
χ
2
24,
0.975
is calculated by using the
values
γ
= 24 and
α
= 0.975. The result is
χ
2
n-1,1-
α
/2
=
χ
2
24,
0.975
=
12.4011502175.
The lower and upper limits of the interval will be (Use ALG mode for these
calculations):
(n-1)
⋅
S
2
/
χ
2
n-1,
α
/2
= (25-1)
⋅
12.5/39.3640770266 = 7.62116179676
(n-1)
⋅
S
2
/
χ
2
n-1,1-
α
/2
= (25-1)
⋅
12.5/12.4011502175 = 24.1913044144
Thus, the 95% confidence interval for this example is:
7.62116179676 <
σ
2
< 24.1913044144.