Why are smaller p-values indicative of stronger evidence against the null hypothesis?

Nice start! As we are discussing probability this week, I’d like to follow up on your statement that “More grounded proof against the null hypothesis is shown by smaller p-values.” Why are smaller p-values indicative of stronger evidence against the null hypothesis? That is, what does the p value represent in terms of probability?

ANSWER
The p-value represents the probability of obtaining the observed results, or more extreme results, assuming that the null hypothesis is true. A smaller p-value means that it is less likely to obtain the observed results if the null hypothesis is true. Therefore, smaller p-values are indicative of stronger evidence against the null hypothesis.

Another way to think about it is that the p-value is a measure of the compatibility of the observed data with the null hypothesis. A smaller p-value means that the observed data is less compatible with the null hypothesis.

Here is an example:

Suppose we are conducting a hypothesis test to determine whether a new drug is effective in treating a certain disease. The null hypothesis is that the drug is not effective. We conduct a clinical trial and find that the drug is significantly more effective than the placebo. This means that the p-value is very small.

The small p-value indicates that it is very unlikely to obtain the observed results if the null hypothesis is true. Therefore, we can conclude that there is strong evidence against the null hypothesis and that the drug is effective.

It is important to note that the p-value is not the same as the probability that the null hypothesis is true. The p-value only tells us how likely it is to obtain the observed results if the null hypothesis is true. It does not tell us whether the null hypothesis is actually true or false.

To decide whether to reject the null hypothesis, we need to consider the p-value and the significance level. The significance level is the maximum p-value that we are willing to tolerate before rejecting the null hypothesis. It is typically set to 0.05.

If the p-value is less than the significance level, then we reject the null hypothesis. This means that we conclude that there is sufficient evidence to reject the null hypothesis and that the alternative hypothesis is more likely to be true.

If the p-value is greater than the significance level, then we fail to reject the null hypothesis. This means that we do not have sufficient evidence to reject the null hypothesis and that the alternative hypothesis is not more likely to be true.