What is the difference between P value and confidence interval?

As a statistical expert with a deep understanding of statistical analysis and its applications, I often encounter questions regarding the fundamental concepts of statistical inference, such as the difference between a P-value and a confidence interval. These two concepts are integral to hypothesis testing and estimation in statistics, but they serve different purposes and are interpreted differently.
P-value:
The P-value is a statistic that measures the strength of the evidence against the null hypothesis. Specifically, it answers the question: "Assuming the null hypothesis is true, what is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from my sample data?" A low P-value indicates strong evidence against the null hypothesis, suggesting that the observed data would be unlikely if the null hypothesis were true. The P-value is calculated from the sampling distribution of the test statistic under the assumption that the null hypothesis holds true.

Significance Level (α):
The significance level, often denoted by α (alpha), is a threshold that determines when we reject the null hypothesis. If the P-value is less than α, the result is considered statistically significant, and we reject the null hypothesis in favor of the alternative hypothesis. Commonly used significance levels are 0.05, 0.01, and 0.001.

Confidence Interval:
A confidence interval, on the other hand, provides an estimated range for an unknown population parameter. It is constructed from sample data and gives a range that likely contains the true population parameter with a certain level of confidence. The confidence level, often expressed as a percentage, indicates the frequency (or proportion) of possible confidence intervals that contain the true value of the unknown parameter. For example, a 95% confidence interval suggests that if we were to take many samples and construct a confidence interval from each, approximately 95% of those intervals would contain the true population parameter.

**Relationship Between P-value and Confidence Interval**:
While both the P-value and the confidence interval are related to the concept of statistical significance, they are not the same and do not directly translate into one another. However, there is a connection. If the P-value is less than the significance level (α), it means that the observed data provide significant evidence against the null hypothesis. In such a case, a confidence interval constructed at the same significance level would not include the null hypothesis value. This is because the confidence interval is designed to exclude values that are not supported by the data at the chosen confidence level.

Example:
Let's say we are testing the effectiveness of a new drug with a null hypothesis that the drug has no effect (i.e., the mean difference in health outcomes is zero). If the P-value from our test is less than 0.05, we have significant evidence to reject the null hypothesis. Correspondingly, a 95% confidence interval for the mean difference would not include zero, indicating that the true mean difference is likely not zero.

In summary, the P-value tells us about the strength of the evidence against the null hypothesis, while the confidence interval gives us an estimated range for the population parameter. The significance level is a threshold for determining statistical significance based on the P-value, and the confidence level is a measure of how often confidence intervals constructed in this way would contain the true parameter if the experiment were repeated many times.

So, if your significance level is 0.05, the corresponding confidence level is 95%. If the P value is less than your significance (alpha) level, the hypothesis test is statistically significant. ... If the P value is less than alpha, the confidence interval will not contain the null hypothesis value.Apr 2, 2015