Hypothesis testing is a fundamental procedure in statistics used to make decisions based on data. It allows us to test a claim or assumption about a population parameter (like a mean or proportion).

The process involves setting up two competing hypotheses and then using sample data to determine which hypothesis is better supported.

Think of it as a formal scientific method for decision-making. We start with a theory, collect evidence, and then evaluate whether the evidence supports or refutes the theory.

**Null Hypothesis (H₀):** This is the default assumption or the status quo. It typically states that there is no effect, no difference, or no relationship. We assume the null hypothesis is true until the evidence suggests otherwise.

**Alternative Hypothesis (H₁ or Hₐ):** This is the claim we are trying to find evidence for. It is the opposite of the null hypothesis and represents the effect or difference we are investigating.

**Significance Level (α):** This is a pre-determined threshold for 'statistical significance'. It represents the probability of rejecting the null hypothesis when it is actually true. Common values for α are 0.05 (5%), 0.01, and 0.10.

**P-Value:** The p-value is the probability of observing our sample data (or something more extreme) if the null hypothesis were true.

**The Decision Rule:** If the p-value is less than or equal to the significance level (p ≤ α), we reject the null hypothesis. This means our result is statistically significant. If p > α, we fail to reject the null hypothesis.

In hypothesis testing, we can make two types of mistakes:

**Type I Error (α):** Rejecting the null hypothesis when it is actually true. This is a 'false positive'. The probability of a Type I error is equal to the significance level, α.

**Type II Error (β):** Failing to reject the null hypothesis when it is actually false. This is a 'false negative'.

Suppose we want to test if a coin is fair. We flip it 100 times and get 65 heads.

**H₀:** The coin is fair (probability of heads = 0.5).

**H₁:** The coin is not fair (probability of heads ≠ 0.5).

We set our significance level α = 0.05. Using statistical calculations, we find the p-value for getting 65 heads is very small (e.g., p ≈ 0.0018).

Since 0.0018 is less than 0.05, we reject the null hypothesis and conclude that the coin is likely not fair.

Companies use hypothesis testing to compare two versions of a product, like a website or app, to see which one performs better. This is called A/B Testing.

**H₀:** There is no difference in conversion rate between the old website design (A) and the new design (B).

**H₁:** There is a difference in conversion rate between the two designs.

By showing each version to a different group of users and analyzing the data, the company can make a data-driven decision about which design to launch.