Home | | Statistics 12th Std | One-Tailed and Two-Tailed Tests

# 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.

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) need not be with symmetric shape always. Sometimes, it may be positively or negatively skewed.

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

H0 : μ = 30 minutes and H1: μ > 30 minutes

Suppose that the decision is taken based on the delivery times of 4 randomly chosen pizza deliveries of the restaurant. Let X1 , X2, X3, and X4 represent the delivery times of the such four occasions. Also, let H0 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 H0 is rejected, when either t(X) ≤ a or t(X) ≥ b) holds. In this case, P(t (X) ≤ a)  and P(t (X) ≥ b) will be the areas, which fall respectively at left and right ends under the curve representing the sampling distribution of t ( X) (Figure 1.3). The statistical test defined  with this kind of rejection rule is known as two-tailed test. 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

H0 : μ = 5 and H1: μ ≠ 5.

Suppose that the decision on H0 is made based on the diameter of 10 randomly selected ball-bearings. Let Xi, i = 1, 2, …, 10 represent the diameter of the randomly chosen ball bearings. Then, the critical region is 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
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Statistics : Chapter 1 : Tests of Significance - Basic Concepts and Large Sample Tests : One-Tailed and Two-Tailed Tests | Definition, Example Solved Problems | Statistics