A test of Hv0 : u =1 versus Hv1 : u does not=1 is performed using a significance level of a=0.10. The P-value is 0.03.

Question

Is Hv0 rejected?
Since P [> or <] a, we [reject or not?]

GPT-4o mini · Answer

To determine whether to reject the null hypothesis $ H_0: \mu = 1 $ based on the given P-value and significance level, you compare the P-value to the significance level $ \alpha = 0.10 $.

Here, the P-value is 0.03, which is less than $ \alpha = 0.10 $.

Since $ \text{P-value} < \alpha $, you reject the null hypothesis $ H_0 $.

So, your final answer would be:

Yes, $ H_0 $ is rejected.
Since $ \text{P-value} < \alpha $, we reject $ H_0 $.

Answer

(b) If the true value of u is 4, is the result a Type 1 error, a Type 2 error, or a correct decision?

GPT-4o mini · Answer

To understand whether the result is a Type I error, a Type II error, or a correct decision, let’s first clarify each term:

Type I Error: This occurs when the null hypothesis is rejected when it is actually true.
Type II Error: This occurs when the null hypothesis is not rejected when it is actually false.
Correct Decision: This occurs when the null hypothesis is rejected (or not rejected) correctly based on the true state of the hypothesis.

In your scenario:

The null hypothesis $ H_0: \mu = 1 $ is rejected based on the P-value of 0.03.
The true value of $ \mu $ is 4.

Since you rejected the null hypothesis $ H_0 $ (which is true according to your rejection decision), and the true value of $ \mu $ is indeed different from 1 (it is 4), your decision to reject $ H_0 $ is actually correct because the null hypothesis is false.

Therefore, since you correctly rejected a false null hypothesis, the result is a correct decision.

So the answer to your question is:

Correct Decision.

Answer

(b) If the true value of u is 1, is the result a Type 1 error, a Type 2 error, or a correct decision?

GPT-4o mini · Answer

In this scenario, if the true value of $ \mu $ is 1, we can analyze the outcome based on your previous results: 1. **Null Hypothesis ($ H_0 $)**: $ \mu = 1 $ 2. **Alternative Hypothesis ($ H_1 $)**: $ \mu \neq 1 $ 3. **Decision**: You rejected the null hypothesis $ H_0 $ based on the P-value of 0.03. Since the true value of $ \mu $ is indeed 1 (making the null hypothesis true), and you rejected that null hypothesis, you have made an error in your decision. This situation falls under the definition of a **Type I Error**, which occurs when the null hypothesis is incorrectly rejected when it is actually true. So the answer to your question is: - **Type I Error**.