Question

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.

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

Answer 1

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 2

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

Answer 3

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

1. **Type I Error**: This occurs when the null hypothesis is rejected when it is actually true.
2. **Type II Error**: This occurs when the null hypothesis is not rejected when it is actually false.
3. **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 4

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

Answer 5

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**.