Question

In a null hypothesis significance test, the significance level \(\alpha\) was set to 0.01. For which one or more of the following p values would the null hypothesis be rejected?

• A. p = 0.004
• B. p = 0.02
• C. p = 0.10
• D. p = 0.009

Hint:

Reject the null hypothesis whenever the p value is smaller than the significance level \(\alpha = 0.01\). Compare each option directly against 0.01.

Answer 1

The Correct Option is A, D

Step 1: Understanding the Question.
We are given a fixed significance level \(\alpha = 0.01\) for a null hypothesis significance test, and four candidate p values. We need to find which of these p values would lead us to reject the null hypothesis at this significance level.

Step 2: Key Formula or Approach.
The decision rule in a null hypothesis significance test is simple: reject the null hypothesis if the p value is less than the chosen significance level \(\alpha\), and fail to reject it otherwise.
\[ \text{Reject } H_0 \text{ if } p < \alpha \]
Here \(\alpha = 0.01\), so we compare each given p value against 0.01.

Step 3: Detailed Explanation.
Option (A): \(p = 0.004\). Since \(0.004 < 0.01\), this p value is smaller than \(\alpha\), so we reject \(H_0\).
Option (B): \(p = 0.02\). Since \(0.02 > 0.01\), this p value is larger than \(\alpha\), so we do not reject \(H_0\).
Option (C): \(p = 0.10\). Since \(0.10 > 0.01\), this is well above \(\alpha\), so we do not reject \(H_0\).
Option (D): \(p = 0.009\). Since \(0.009 < 0.01\), this p value is smaller than \(\alpha\), so we reject \(H_0\).

Step 4: Final Answer.
The null hypothesis is rejected for p values 0.004 and 0.009, so options (A) and (D) are correct.
\[ \boxed{p = 0.004 \text{ and } p = 0.009 \text{ lead to rejection}} \]