Question

A test based on critical region \(W\) is said to be unbiased for testing \(H_0: \theta = \theta_0\) against \(H_1: \theta \ne \theta_0\) if (where \(\alpha\) and \(\beta\) are Type I and Type II errors respectively) 

• A. $\beta \ge \alpha$
• B. $1 - \beta < \alpha$
• C. $\beta \le \alpha$
• D. $1 - \beta \ge \alpha$

Hint:

An "unbiased" test is one that is more likely to give a "correct" rejection (Power) than a "wrong" rejection (Significance). This ensures the test is actually pointing you in the right direction.

Answer 1

The Correct Option is D

This question concerns the definition of an unbiased test in the context of power and significance levels. 

Step 1: Define the Power of a Test
The power of a test, often denoted as $1 - \beta$, is the probability of rejecting the null hypothesis when the alternative hypothesis is true. 

Step 2:Define the Level of Significance
The level of significance ($\alpha$) is the maximum probability of rejecting the null hypothesis when it is actually true. 

Step 3: Understanding Test Unbiasedness
A test is called unbiased if its power under any alternative hypothesis is at least as great as its power under the null hypothesis. 
In other words, you are more likely to reject the null when it is false than when it is true. 

Step 4: Formulate the Inequality
Power under Alternative $\ge$ Power under Null 
$(1 - \beta) \ge \alpha$. 
Therefore, the condition for an unbiased test is $1 - \beta \ge \alpha$.