Question

Which of the following p-values will lead us to reject the null hypothesis if the significance level of the test is 5% ?

• A. 0.15
• B. 0.10
• C. 0.06
• D. 0.046

Hint:

A simple rhyme to help you remember the rule:
"If the $p$ is low, the null must go."
If the $p$-value is less than or equal to the significance level ($\alpha$), we reject the null hypothesis.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In hypothesis testing, the $p$-value represents the probability of obtaining test results at least as extreme as the observed results, assuming the null hypothesis ($H_0$) is true.
We compare the $p$-value to the significance level ($\alpha$) to make a decision.
Key Formula or Approach:
The decision rule for hypothesis testing is: \[ \text{If } p\text{-value} \le \alpha \implies \text{Reject } H_0 \] \[ \text{If } p\text{-value} > \alpha \implies \text{Do not reject } H_0 \]

Step 2: Detailed Explanation:
The given significance level is: \[ \alpha = 5\% = 0.05 \] Let us evaluate each $p$-value option against $\alpha = 0.05$:
- Option 1: $0.15 > 0.05 \implies$ Do not reject $H_0$.
- Option 2: $0.10 > 0.05 \implies$ Do not reject $H_0$.
- Option 3: $0.06 > 0.05 \implies$ Do not reject $H_0$.
- Option 4: $0.046 \le 0.05 \implies$ Reject $H_0$.
Therefore, only a $p$-value of 0.046 leads us to reject the null hypothesis.

Step 3: Final Answer:
The correct $p$-value is 0.046, corresponding to Option (D).