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

A p value tells us the probability of observing data at least as extreme as what we got, if the null hypothesis were actually true. The rule for rejecting the null hypothesis is simply to line up the p value against the significance level $\alpha$, which is 0.01 here, and reject $H_0$ whenever the p value falls below it. Let's line up all four values against 0.01.

1. p = 0.004: Placing this on a number line against 0.01, it sits to the left, meaning it is smaller. So it clears the bar for rejection, and $H_0$ is rejected.
2. p = 0.02: This sits to the right of 0.01, meaning it is larger. The evidence against $H_0$ is not strong enough at this significance level, so we fail to reject $H_0$.
3. p = 0.10: This is far to the right of 0.01, clearly larger, so again we fail to reject $H_0$.
4. p = 0.009: This is just barely to the left of 0.01, so it is still smaller than $\alpha$, and $H_0$ is rejected.

So only p = 0.004 and p = 0.009 fall below the 0.01 cutoff, and these are the two values for which the null hypothesis gets rejected.

Let's summarize:

• Reject $H_0$ only when $p < \alpha$; here that means p strictly less than 0.01.
• p = 0.02 and p = 0.10 are both larger than 0.01, so they do not give enough evidence to reject $H_0$.

Therefore, options (A) and (D) are the p values that lead to rejection of the null hypothesis.