To decide whether to reject the null hypothesis $H_0$ based on the given $P$-value and different levels of significance ($\alpha$), follow this rule:

• Reject $H_0$ if $P \leq \alpha$.
• Fail to reject $H_0$ if $P > \alpha$.

Given $P = 0.0596$, let's evaluate for each $\alpha$:

(a) $\alpha = 0.01$

• $P = 0.0596 > \alpha = 0.01$
• Decision: Fail to reject $H_0$.

(b) $\alpha = 0.05$

• $P = 0.0596 > \alpha = 0.05$
• Decision: Fail to reject $H_0$.

(c) $\alpha = 0.10$

• $P = 0.0596 \leq \alpha = 0.10$
• Decision: Reject $H_0$.

Summary of Decisions:

• For $\alpha = 0.01$: Fail to reject $H_0$.
• For $\alpha = 0.05$: Fail to reject $H_0$.
• For $\alpha = 0.10$: Reject $H_0$.

Would you like a detailed explanation of hypothesis testing or more examples?

Here are 5 related questions to expand your understanding:

1. What is the relationship between $P$-value and the strength of evidence against $H_0$?
2. How do you interpret the decision to "fail to reject $H_0$"?
3. Why do different levels of significance change the decision?
4. What happens to the probability of a Type I error as $\alpha$ increases?
5. How would you calculate a $P$-value from test statistics?

Tip: A smaller $\alpha$ reduces the chance of a Type I error but increases the chance of a Type II error, so choose $\alpha$ carefully based on the context of the test.