In a given hypotheses testing problem, the maximum probability with which we would be willing to tolerate the occurrence of type I error is called level of significance of the test.

**LEVEL OF SIGNIFICANCE, CRITICAL** **REGION AND CRITICAL VALUE(S)**

In a given hypotheses testing problem, the *maximum probability*
with which we would be willing to tolerate the occurrence of type I error is
called **level
of significance** of the test. This probability is usually denoted by ‘*α*’.
Level of significance is specified before samples are drawn to test the
hypothesis.

The level of significance normally chosen in every hypotheses
testing problem is 0.05 (5%) or 0.01 (1%). If, for example, the level of
significance is chosen as 5%, then it means that among the 100 decisions of
rejecting the null hypothesis based on 100 random samples, maximum of 5 of
among them would be wrong. It is emphasized that the 100 random samples are
drawn under identical and independent conditions. That is, the null hypothesis *H*_{0}
is rejected wrongly based on 5% samples when *H*_{0} is actually
true. We are about 95% confident that we made the right decision of rejecting *H*_{0}.

**Critical region **in a hypotheses testing problem is a subset of
the sample space whose** **elements lead to rejection of *H*_{0}. Hence, its
elements have the dimension as that of the sample size, say, *n*(*n* >
1). That is,

Critical Region = { * x* = (

A subset of the sample space whose elements does not lead to
rejection of *H _{0}* may be termed as

**S** = {Critical Region} U {Acceptance Region}.

Test statistic, a function of statistic(s) and the known value(s)
of the underlying parameter(s),is used to make decision on H_{0}.
Consider a hypotheses testing problem, which uses a** test statistic **t ( __X__) and a constant c for deciding on H_{0}
. Suppose that H_{0} is rejected, when t(__x__) > c . It is to be
noted here that t ( __X__) is a scalar and is of dimension one. Its sampling
distribution is a univariate t (__X__) satisfying the condition t ( x) >
c will identify the probability distribution. The values of samples in the
sample space, which lead to rejection of H_{0}. It does not mean that {
t | t(* x*) > c } is the
corresponding critical region. The value ‘c’, distinguishing the elements of
the critical region and the acceptance region, is referred to as

**Example 1.4**

Suppose an electrical equipment manufacturing industry receives
screws in lots, as raw materials. The production engineer decides to reject a
lot when the number of defective screws is one or more in a randomly selected
sample of size 2.

Then, *X*_{1}
and *X*_{2} are *iid* random variables and they have the *Bernoulli* (*P*) distribution.

Let *H*_{0} : *P* = 1/3 and *H*_{1}
: *P* = 2/3

The sample space is S
= {(0,0),(0,1),(1,0),(1,1)}

If *T*(*X*_{1}, *X*_{2}) represents
the number of defective screws, in each random sample, then the statistic *T*(*X*_{1},
*X*_{2}) = *X*_{1} + *X*_{2} is a random
variable distributed according to the *Binomial* (2*, P*)
distribution. The possible values of *T*(*X*_{1}, *X*_{2})
are 0, 1 and 2. The values of *T*(*X*_{1}, *X*_{2})
which lead to rejection of *H*_{0} constitute the set {1,2}.

But, the critical region is defined by the elements of S corresponding to *T*(*X _{1}*,

Thus, the critical region is {(0,1), (1,0), (1,1)} whose dimension
is 2.

**Note 8: **When the sampling distribution is continuous, the set of values of** ***t*** **(** ***X*)** **corresponding to the
rejection rule will be an interval or union of intervals depending on the
alternative hypothesis. It is empahazized that **these intervals identify the elements of critical region,
but they do not** **constitute the critical region**.

When the sampling distribution of the test
statistic *Z* is a normal distribution,
the critical values for testing *H*_{0}
against the possible alternative hypothesis at two different levels of significance,
say 5% and 1% are displayed in Table 1.6.

**Table 1.6 **Critical values of the Z
statistic

Tags : Definition, Example Solved Problems | Statistics , 12th Statistics : Chapter 1 : Tests of Significance - Basic Concepts and Large Sample Tests

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12th Statistics : Chapter 1 : Tests of Significance - Basic Concepts and Large Sample Tests : Level of Significance, Critical Region and Critical Value(S) | Definition, Example Solved Problems | Statistics

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