In some hypotheses testing problem, elements of the critical region may be identified by a rejection rule of the type t ( X) ≥ c.

**ONE-TAILED AND TWO-TAILED TESTS**

In some hypotheses testing problem, elements of the critical
region may be identified by a rejection rule of the type t ( __X__) ≥ c. In
this case, P( t ( __X__) ≥ c) will be the area, which falls at the right end
(Figure1.1) under the curve representing the sampling distribution of t (__X__).
The statistical test defined by this kind of critical region is called **right-tailed test**.

On the other hand, suppose that the rejection rule t (__X__) ≤
c determines the elements of the critical region. Then, P( t (__X__) ≤ c)
will be the area, which falls at the left end (Figure.1.2) under the curve
representing the sampling distribution of t (__X__) . The statistical test
defined by this kind of critical region is called **left -tailed test**.

The above two tests are commonly known as **one-tailed tests**.

**Note 9: **It should be noted that the sampling distribution of** ***t*** **(* X*)

**Example 1.5**

Suppose a pizza restaurant claims its average pizza delivery time
is 30 minutes. But you believe that the restaurant takes more than 30 minutes.
Now, the null and the alternate hypotheses can be formulated as

*H _{0} *:

Suppose that the decision is taken based on the delivery times of
4 randomly chosen pizza deliveries of the restaurant. Let *X*_{1}
, *X*_{2}, *X*_{3}, and *X*_{4}
represent the delivery times of the such four occasions. Also, let *H _{0}*
be rejected, when the sample mean exceeds 31. Then, the critical region is

Critical
Region =

In
this case, P( > 31) –will
be the area, which fall at the right end under the curve representing the
sampling distribution of . Hence, this test can be categorized
as a right-tailed test.

Suppose
that H_{0} is rejected, when either t(* X*) ≤

*Figure 1.3 Two-tailed Test *

**Example 1.6**

A manufacturer of ball-bearings, which are used in some machines,
inspects to see whether the diameter of each ball-bearing is 5 mm. If the
average diameter of ball-bearings is less than 4.75 mm or greater than 5.10 mm,
then such ball-bearings will cause damages to the machine.

Here the null and the alternate hypotheses are

*H _{0} *:

Suppose that the decision on *H _{0}* is made based on
the diameter of 10 randomly selected ball-bearings. Let

In this case, *P *( <4.75) is the area, which will fall at the left end and *P*(* >* 5.10) is the
area, which will fall at the right end under the curve representing the
sampling distribution of . This kind of test can be categorized
as a two-tailed test (see Figure 1.3).

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 : One-Tailed and Two-Tailed Tests | Definition, Example Solved Problems | Statistics

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