Confidence Intervals and Hypothesis Tests

Descriptive statistics tell you about the data you have; statistical inference tells you about the larger population you cannot fully observe. If you measure the blood pressure of 100 patients, you know the sample mean exactly — but what you really want to know is the true average blood pressure for all patients like these. Inference bridges that gap by using sample data, combined with probability theory, to make rigorous statements about population parameters.

The two workhorses of inference are confidence intervals and hypothesis tests. A confidence interval takes a sample statistic and wraps a margin of error around it, producing a range that is likely to capture the true population parameter. A 95% confidence interval, for example, is constructed by a method that captures the true parameter 95% of the time across many repeated samples. The width of the interval depends on the sample size and the desired level of confidence — more data or less confidence yields a narrower interval, while less data or higher confidence yields a wider one.

Hypothesis testing takes a different approach: you start with a specific claim about the population (the null hypothesis, $H_0$) and ask whether the observed data are compatible with that claim. The p-value quantifies how surprising your sample would be if the null hypothesis were true. A very small p-value suggests the data are unlikely under $H_0$, giving you evidence to reject it in favor of an alternative hypothesis $H_a$. It is critical to remember that the p-value is not the probability that the null hypothesis is true — a common and consequential misinterpretation.

In this lesson you will learn the general form of a confidence interval, practice computing point estimates and margins of error, interpret p-values correctly, and see how confidence level and sample size trade off against interval precision. These concepts underpin virtually all applied statistics, from clinical trials to quality control to social science research.