A test of Hv0 : u=9 versus Hv1 : u

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 · Answer

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} $.

Answer

(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 · Answer

To assess the decision based on the true value of $ u $:

Type I Error occurs when we reject the null hypothesis $ H_{0} $ when it is true.
Type II Error occurs when we fail to reject the null hypothesis $ H_{0} $ when it is false.
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.

Answer

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 · Answer

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**.