Here is the analysis of each part based on the provided screenshot and hypothesis testing principles:

Part 1 of 4

Statement: "If $P = 0.05$, the result is statistically significant at the $\alpha = 0.01$ level."

• Explanation: The $P$-value of $0.05$ is greater than the significance level $\alpha = 0.01$. For a result to be statistically significant, the $P$-value must be less than $\alpha$.
• Answer: False

Part 2 of 4

Statement: "If $P = 0.05$, the null hypothesis is rejected at the $\alpha = 0.01$ level."

• Explanation: Similar to Part 1, the $P$-value of $0.05$ exceeds $\alpha = 0.01$. Therefore, the null hypothesis cannot be rejected at the $\alpha = 0.01$ level.
• Answer: False

Part 3 of 4

Statement: "If $P = 0.05$, the result is statistically significant at the $\alpha = 0.10$ level."

• Explanation: Here, the $P$-value $0.05$ is less than the significance level $\alpha = 0.10$. This means the result is statistically significant at this level.
• Answer: True

Part 4 of 4

Statement: "If $P = 0.05$, the null hypothesis is rejected at the $\alpha = 0.10$ level."

• Explanation: Since $P = 0.05$ is less than $\alpha = 0.10$, we reject the null hypothesis at the $\alpha = 0.10$ level.
• Answer: True

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Related Questions

1. What is the significance of a $P$-value in hypothesis testing?
2. How is the $\alpha$-level chosen for statistical tests?
3. What does it mean to reject or fail to reject the null hypothesis?
4. Can a result be statistically significant at one $\alpha$-level but not at another?
5. What are Type I and Type II errors in hypothesis testing?

Tip: Always compare the $P$-value with the chosen $\alpha$-level before interpreting the statistical significance.