Question

what would the effect be of a one-tailed test and the probability of rejecting the null hypothesis?

Answer 1

This question is explained with the help of an example.

Suppose we are testing for a Mean with population standard deviation known, . So we would be using Normal Distribution and thus Z-Test for the following Hypothesis:

Case 1. Null Hypothesis, Ho: versus Alternative Hypothesis Ha:

Case 2. Null Hypothesis, Ho: versus Alternative Hypothesis Ha:

Tails of a test are dependent on the alternative hypothesis.

Case 1 corresponds to a two tailed test since it has not equals to symbol while Case 2 corresponds to a One-Tailed Test (Right Tailed) since it has a greater than symbol.

Now the effect of the type of Test - One Tailed or Two tailed comes into effect while taking the Decision and concluding for Rejection or Acceptance of Null Hypothesis.

For Case 1.

The Critical Value of Z would correspond to the Z-score of   and the probability for Rejecting Ho will be given as: , Zobserved is the value of Test Statistics.

The Rejection Region is thus,

For Case 2.

The Critical Value of Z would correspond to the Z-score of   and the probability for Rejecting Ho will be given as:

Thus, The Rejection Region is thus, .

Thus, For One Tailed and Two Tailed Tests the Rejection Regions are different and the corresponding probability of rejecting null hypothesis i.e. p-value varies.

If Rejection Regions and p-values are wrong basis the type of test, then we may wrongly Reject Ho when it is True and may wrongly accept Ho when it is false.