Final answer:
In statistics, a smaller p-value is generally considered better as it indicates stronger evidence against the null hypothesis when compared to the significance level. A p-value less than the established alpha (α) leads to rejecting the null hypothesis, while a higher p-value does not.
Step-by-step explanation:
In statistics, the p-value is the probability that an event will happen purely by chance assuming the null hypothesis is true. A smaller p-value indicates stronger evidence against the null hypothesis. When comparing against a significance level (denoted as α), a p-value less than α suggests that we should reject the null hypothesis in favor of the alternative hypothesis.
For example, if α is set at 0.05 and the p-value is 0.0187, this p-value is considerably smaller than α, providing confidence in the decision to reject the null hypothesis. In contrast, if the p-value were 0.5485 and α were 0.01, the p-value would be larger than α, indicating insufficient evidence to reject the null hypothesis.
Therefore, in hypothesis testing, a smaller p-value is usually better because it suggests a lower probability of the results occurring by chance, thereby giving significant evidence to support the alternative hypothesis. A common mnemonic is: 'If the p-value is low, the null must go. If the p-value is high, the null must fly.' This means that a lower p-value (< α) warrants rejecting the null hypothesis, while a higher p-value (≥ α) does not justify rejecting it.