Statistics and Probability - Hypothesis Testing (Chi-squared test, t-tests)

Null and Alternative Hypotheses ($H_0$ and $H_1$): The Null Hypothesis ($H_0$) represents the status quo or a statement of 'no effect' or 'independence'. The Alternative Hypothesis ($H_1$) is the claim we are testing for. Visually, $H_0$ is represented by the central portion of a probability distribution curve, while $H_1$ points toward the 'rejection regions' or tails of the distribution.

Significance Level ($\alpha$) and P-value: The significance level $\alpha$ (typically $0.05$ or $5\%$) is the probability threshold for rejecting $H_0$. The p-value is the probability of observing results as extreme as the sample data, assuming $H_0$ is true. Visually, the p-value is the area under the curve in the tail(s); if this area is smaller than the area defined by $\alpha$, we reject $H_0$.

Chi-squared ($\chi^2$) Test for Independence: This test evaluates whether two categorical variables are independent based on a contingency table. Visually, imagine a table comparing 'Observed' ($O$) frequencies against 'Expected' ($E$) frequencies. The $\chi^2$ distribution curve is positively skewed (starting at zero and trailing to the right), and a high $\chi^2$ value moves the result into the shaded rejection region in the right tail.

Degrees of Freedom ($df$): This parameter defines the specific shape of the $\chi^2$ or t-distribution curve. For a $\chi^2$ test of independence, $df = (r - 1)(c - 1)$ where $r$ is rows and $c$ is columns. Visually, as degrees of freedom increase, the peak of the $\chi^2$ curve shifts to the right and becomes more symmetrical, resembling a normal distribution.

One-sample and Two-sample t-tests: These tests compare means. A one-sample t-test compares a sample mean to a hypothesized population mean, while a two-sample t-test compares the means of two independent groups. Visually, the t-distribution looks like a bell-shaped Normal curve but has 'fatter tails' to account for greater uncertainty in smaller samples.

Critical Values and Rejection Regions: The critical value is the 'cut-off' point on the x-axis of a distribution. Visually, this value separates the 'fail to reject' region (the main body of the curve) from the 'rejection region' (the tails). If your calculated test statistic (t or $\chi^2$) falls beyond this value into the tail, the result is statistically significant.

One-tailed vs. Two-tailed Tests: A one-tailed test looks for a change in a specific direction ($>$ or $<$), while a two-tailed test looks for any difference ($\neq$). Visually, a one-tailed test places the entire significance level $\alpha$ in one tail, whereas a two-tailed test splits $\alpha$ into two equal areas of $\frac{\alpha}{2}$ at both ends of the distribution.