As a statistical expert with a deep understanding of hypothesis testing and statistical inference, I can provide a comprehensive explanation of the relationship between critical values and p-values.
The critical value and the p-value are two distinct concepts in statistical hypothesis testing, but they are related in the context of making decisions about the null hypothesis. Let's delve into what each term means and how they are used in practice.

Critical Value:
A critical value is a threshold that is used to determine whether to reject the null hypothesis in a statistical test. It is derived from the distribution of the test statistic under the assumption that the null hypothesis is true. The critical value is a point on the scale of the test statistic that separates the region that leads to rejection of the null hypothesis from the region that leads to a failure to reject it. For example, in a two-tailed test with a significance level (alpha) of 0.05, the critical value would be the point that corresponds to the 2.5th percentile from each tail of the distribution.

P-value:
The p-value, on the other hand, is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from my sample data, assuming that the null hypothesis is true. It is not a probability that the null hypothesis is true or false; rather, it is a measure of the strength of the evidence against the null hypothesis. A small p-value indicates that the observed data would be unlikely if the null hypothesis were true, suggesting that the null hypothesis might be false.

**Relationship Between Critical Value and P-value:**
The relationship between the critical value and the p-value becomes clear when we consider the decision rule for hypothesis testing. If the p-value is smaller than the significance level (alpha), we reject the null hypothesis. This is equivalent to saying that if the test statistic is greater than the critical value (in a one-tailed test) or falls in the rejection region defined by the critical values (in a two-tailed test), we reject the null hypothesis.

For example, if we set our significance level at 0.05, we are saying that we will reject the null hypothesis if the p-value is less than 0.05. This corresponds to the test statistic exceeding the critical value for a one-tailed test or falling outside the range defined by the critical values for a two-tailed test.

Example:
Let's consider a scenario where we are testing the effectiveness of a new drug. The null hypothesis (H0) might be that the drug has no effect, and the alternative hypothesis (H1) is that the drug does have an effect.

1. We conduct a test and calculate a test statistic, say a t-value.
2. We then compare this test statistic to the critical value from the t-distribution that corresponds to our chosen significance level (alpha).
3. If our test statistic is greater than the critical value, we reject the null hypothesis and conclude that the drug has an effect.
4. Alternatively, we calculate the p-value from our test statistic.
5. If the p-value is less than our significance level (alpha), we again reject the null hypothesis.

In practice, many statisticians prefer to use p-values because they provide a continuous measure of evidence against the null hypothesis, rather than a binary decision based on a critical value. However, the decision to reject or not reject the null hypothesis can be based on either the p-value being smaller than the significance level or the test statistic exceeding the critical value.

In conclusion, while the critical value and the p-value serve different roles in hypothesis testing, they are related in that they both inform the decision about whether to reject the null hypothesis. A small p-value typically leads to the rejection of the null hypothesis if it is below the significance level, which is equivalent to the test statistic exceeding the critical value.

read more >>

A small -value is an indication that the null hypothesis is false. ... For example, we decide either to reject the null hypothesis if the test statistic exceeds the critical value (for = 0.05) or analagously to reject the null hypothesis if the -value is smaller than 0.05.read more >>