Question

If a hypothesis is rejected at the 0.025 level of significance, it:

• A. Must Be Rejected At Any Level
• B. Must Be Rejected At The 0.01 Level
• C. Must Not Be Rejected At The 0.01 Level
• D. May Or May Not Be Rejected At The 0.01 Level

Hint:

If a hypothesis is rejected at a certain $\alpha$, it is guaranteed to be rejected at any larger significance level ($\alpha' > \alpha$). However, its rejection at a smaller significance level ($\alpha'' < \alpha$) is uncertain.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The level of significance ($\alpha$) represents the threshold probability of committing a Type I error (rejecting a true null hypothesis).

Step 2: Detailed Explanation:
When a null hypothesis is rejected at a significance level $\alpha = 0.025$, it implies that the observed $p$-value of the test statistic is less than or equal to $0.025$: \[ p\text{-value} \leq 0.025 \]
To determine the outcome at a more stringent significance level, such as $\alpha = 0.01$:
We must compare the $p$-value directly with $0.01$.
There are two possibilities:
1. If the $p$-value is very small, say $0.005$, then since $0.005 < 0.01$, the hypothesis will also be rejected at the $0.01$ level.
2. If the $p$-value lies between the two thresholds, say $0.02$, then since $0.02 > 0.01$, the hypothesis will not be rejected at the $0.01$ level.
Because the exact $p$-value is unknown, we cannot make a definitive statement.
Thus, the hypothesis may or may not be rejected at the 0.01 level.

Step 3: Final Answer:
The hypothesis may or may not be rejected at the 0.01 level.