Question

In a testing of hypothesis problem, which one of the following statements is true?

• A. The probability of the Type-I error cannot be higher than the probability of the Type-II error
• B. Type-II error occurs if the test accepts the null hypothesis when the null hypothesis is actually false
• C. Type-I error occurs if the test rejects the null hypothesis when the null hypothesis is actually false
• D. The sum of the probability of the Type-I error and the probability of the Type-II error should be 1

Hint:

A Type-I error is rejecting a true null hypothesis, while a Type-II error is accepting a false null hypothesis.

Answer 1

The Correct Option is B

Step 1: Recall the meaning of Type-I and Type-II errors. 
In hypothesis testing:

• A Type-I error occurs when the null hypothesis is rejected even though it is actually true.
• A Type-II error occurs when the null hypothesis is not rejected (accepted) even though it is actually false.

Using these definitions, we now evaluate each option.

Step 2: Examine option (A).
The statement claims that the probability of a Type-I error cannot exceed the probability of a Type-II error.
This is incorrect because the two probabilities are not directly constrained by each other.
Their values depend on factors such as the significance level $\alpha$ and the power of the test, and either one can be larger.

Step 3: Examine option (B).
This option states that a Type-II error happens when the test accepts the null hypothesis even though it is false.
This exactly matches the definition of a Type-II error.
Hence, option (B) is correct.

Step 4: Examine option (C).
Here, a Type-I error is described as rejecting the null hypothesis when it is false.
This is incorrect because rejecting a false null hypothesis is actually the correct decision, not an error.

Step 5: Examine option (D).
The claim that the probabilities of Type-I and Type-II errors must add up to 1 is false.
These probabilities are not complements of each other and depend on different aspects of the test design.

Step 6: Final conclusion.
The only statement that correctly describes an error type is:

\[ \boxed{(B)} \]