Question

A test of Hv0 : u=9 versus Hv1 : u < 9 is performed using a significance level of a=0.05. The P-value is 0.07.

Is Hv0 rejected?
Since P [< or > ?] a, we [do or dont reject?] Hv0
GPT-4o mini
To determine whether to reject \( H_{0} \) based on the P-value and the significance level, we compare the P-value to the significance level \( \alpha = 0.05 \).

Here, the P-value is 0.07.

Since \( P \text{-value} (0.07) > \alpha (0.05) \), we do not reject \( H_{0} \).

So, in your format:

- Since \( P < \alpha \) or \( P > \alpha \)? We have \( P > \alpha \).
- Therefore, we do not reject \( H_{0} \).
(b) If the true value of u is 9, is the result a Type 1 error, a Type 2 error, or a correct decision?
GPT-4o mini
To assess the decision based on the true value of \( u \):

1. **Type I Error** occurs when we reject the null hypothesis \( H_{0} \) when it is true.
2. **Type II Error** occurs when we fail to reject the null hypothesis \( H_{0} \) when it is false.
3. **Correct Decision** occurs when we correctly do not reject \( H_{0} \) when it is true, or correctly reject it when it is false.

In this case, if the true value of \( u \) is indeed 9, and we have decided not to reject \( H_{0} \), then we have made a correct decision.

So the answer is: it is a correct decision.
If the true value of u is 0, is the result a Type 1 error, a Type 2 error, or a correct decision?
GPT-4o mini
If the true value of \( u \) is 0 and we are testing the null hypothesis \( H_0: u = 9 \) against the alternative hypothesis \( H_1: u < 9 \):

1. Since the true value (0) is less than 9, \( H_0 \) is false.
2. We found that we do not reject \( H_0 \) because the P-value (0.07) was greater than the significance level \( \alpha = 0.05 \).

In this scenario:

- We failed to reject the null hypothesis \( H_0 \) when it is actually false. This situation represents a **Type II error**.

So, if the true value of \( u \) is 0, the result is a **Type II error**.