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

Let's analyze the statements related to Type-I and Type-II errors in the context of hypothesis testing.

1. Type-I Error (False Positives): This occurs when the null hypothesis is rejected, even though it is actually true. The probability of committing this error is denoted by \(\alpha\), which is known as the significance level of the test. In simpler terms, we're mistakenly identifying an effect or difference when there isn't one.
2. Type-II Error (False Negatives): This occurs when we fail to reject the null hypothesis even though it is false. The probability of committing this error is denoted by \(\beta\). Here, we're failing to identify an effect or difference that actually exists.

Now, let's examine the options given in the question:

• The probability of the Type-I error cannot be higher than the probability of the Type-II error.: This statement is incorrect. The probabilities of Type-I and Type-II errors are independent of each other, and one can be higher than the other depending on the context and the test design.
• Type-II error occurs if the test accepts the null hypothesis when the null hypothesis is actually false.: This is the correct statement. As explained, a Type-II error is indeed the failure to reject a false null hypothesis.
• Type-I error occurs if the test rejects the null hypothesis when the null hypothesis is actually false.: This statement is incorrect. A Type-I error occurs when the null hypothesis is rejected though it is actually true.
• The sum of the probability of the Type-I error and the probability of the Type-II error should be 1.: This is incorrect. The probabilities \(\alpha\) and \(\beta\) do not add up to 1 as they are conditional probabilities under different circumstances related to the truth of the null hypothesis.

Therefore, the correct answer is that a Type-II error occurs if the test accepts the null hypothesis when the null hypothesis is actually false.

Answer 2

Step 1: Understanding Type-I and Type-II errors.
In hypothesis testing, a **Type-I error** occurs when the null hypothesis is rejected when it is actually true. In contrast, a **Type-II error** occurs when the null hypothesis is accepted when it is actually false. Therefore, the correct statement will describe the correct definition of these errors.
Step 2: Analyzing the options.
(A) The probability of the Type-I error cannot be higher than the probability of the Type-II error:
This is not true. The probabilities of Type-I and Type-II errors are independent of each other, and neither is bound to be greater than the other. The likelihood of each error depends on the test's significance level (\(\alpha\)) and the power of the test.
(B) Type-II error occurs if the test accepts the null hypothesis when the null hypothesis is actually false:
Correct — this is the definition of a Type-II error. A Type-II error happens when the null hypothesis is incorrectly accepted when it should be rejected because it is actually false.
(C) Type-I error occurs if the test rejects the null hypothesis when the null hypothesis is actually false:
This is incorrect. A Type-I error occurs when the null hypothesis is rejected when it is actually true, not false.
(D) The sum of the probability of the Type-I error and the probability of the Type-II error should be 1:
This is incorrect. The sum of the probabilities of Type-I and Type-II errors is not necessarily 1. They are independent of each other and depend on various factors such as the significance level (\(\alpha\)) and the power of the test.
Step 3: Conclusion.
The correct statement is (B), which accurately describes the Type-II error.