1. Let's start by stating the problem: You want to understand when to reject the null hypothesis ($H_0$) in a t-test and a chi-squared test, and whether the rejection criteria follow the same format as "$p < \alpha$, therefore reject $H_0$". 2. The general rule for hypothesis testing is: If the p-value is less than the significance level $\alpha$ (e.g., 0.05), we reject the null hypothesis $H_0$. This applies to both t-tests and chi-squared tests. 3. For a t-test: - You calculate the test statistic $t_{stat}$. - You find the critical value $t_{crit}$ from the t-distribution table based on degrees of freedom and $\alpha$. - The rejection rule depends on the test type: - For a two-tailed test: reject $H_0$ if $|t_{stat}| > t_{crit}$. - For a left-tailed test: reject $H_0$ if $t_{stat} < -t_{crit}$. - For a right-tailed test: reject $H_0$ if $t_{stat} > t_{crit}$. 4. For a chi-squared test: - Calculate the test statistic $\chi^2_{stat}$. - Find the critical value $\chi^2_{crit}$ from the chi-squared distribution table based on degrees of freedom and $\alpha$. - Since the chi-squared distribution is always right-skewed, reject $H_0$ if $\chi^2_{stat} > \chi^2_{crit}$. 5. Summary: - Both tests use the p-value rule: reject $H_0$ if $p < \alpha$. - The test statistic comparison to critical value depends on the test and tail: - t-test: reject $H_0$ if $t_{stat}$ is in the critical region (depends on tail). - chi-squared test: reject $H_0$ if $\chi^2_{stat} > \chi^2_{crit}$. 6. So, the format "$t.stat < t.crit = \text{reject } H_0$" is not always correct; it depends on the test direction. The p-value comparison is the most consistent and recommended approach.