AI Tutor
A pharmaceutical company claims their new pain reliever works faster than the standard 20-minute benchmark. When their trial shows an average time of 18.5 minutes, how do scientists prove this is genuine progress rather than lucky random variation?
We use hypothesis testing, a mathematical framework designed to decide whether experimental data provides convincing evidence to support a new claim over an established baseline.
Before we can write a testable statement, what mathematical quantities are we actually testing?

Parameters vs. Statistics

A population parameter (such as population mean μ\mu or proportion pp) is a fixed, unknown value describing an entire group, while a sample statistic (such as sample mean xˉ\bar{x} or sample proportion p^\hat{p}) is a numerical value calculated directly from collected sample data.
What baseline starting point must every statistical test establish before examining any data?

The Null Hypothesis (H0H_0)

In 1928, mathematicians Jerzy Neyman and Egon Pearson formalized statistical decision-making around a default 'presumed innocent' assumption: the null hypothesis (H0H_0), which asserts that no effect, no difference, or no change exists.
Because H0H_0 represents the status quo, it is always stated with an exact equality (==), such as H0:ΞΌ=20H_0: \mu = 20 or H0:p=0.5H_0: p = 0.5.
If the null hypothesis represents no change, what statement represents the new claim we want to investigate?

The Alternative Hypothesis (H1H_1)

The alternative hypothesis (H1H_1 or HaH_a) represents the claim we suspect is true and hope to establish using gathered evidence.