Final answer:
A smaller p-value indicates stronger evidence against the null hypothesis in statistics. The p-value less than 0.05 is commonly used as a threshold for rejecting the null hypothesis. The context of hypothesis testing in mathematics involves comparing the p-value with a significance level to make decisions about the null hypothesis.
Step-by-step explanation:
The p-value is a critical concept in hypothesis testing within the field of statistics, a branch of mathematics. This value represents the probability that a given event will occur purely by chance assuming that the null hypothesis is true. A smaller p-value indicates stronger evidence against the null hypothesis, suggesting that the observed results are unlikely to have occurred by random chance alone.
As a rule of thumb, if the p-value is less than the predetermined significance level (commonly 0.05 or 5%), the evidence against the null hypothesis is strong enough to reject it in favor of the alternative hypothesis. For instance, a p-value of 0.01 (1%) provides very strong evidence against the null hypothesis and typically results in its rejection. The smaller the p-value, the more significant the results are, as it implies the observed data is increasingly inconsistent with what we would expect if the null hypothesis were true.