Page 18-34
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.
Hypothesis testing
A hypothesis is a declaration made about a population (for instance, with
respect to its mean). Acceptance of the hypothesis is based on a statistical
test on a sample taken from the population. The consequent action and
decision-making are called hypothesis testing.
The process of hypothesis testing consists on taking a random sample from the
population and making a statistical hypothesis about the population. If the
observations do not support the model or theory postulated, the hypothesis is
rejected. However, if the observations are in agreement, then hypothesis is
not rejected, but it is not necessarily accepted. Associated with the decision
is a level of significance
α
.
