Test Statistic vs Standardized Test Statistic

A test statistic compresses a whole sample into one number whose behavior under the null hypothesis you already know, which is what lets you convert it into a p-value. The common ones in an introductory course are z, t, and $\chi^2$ (chi-square), and the first two share the single recipe described in standardized test statistic. For example, $z = 2.10$ says the sample proportion sits 2.10 standard errors above the value the null hypothesis claims. The larger the magnitude, the further the data are from the null, and the smaller the p-value.

That one formula, $\frac{\text{statistic} - \text{parameter}}{\text{standard error of the statistic}}$, generates every z and t statistic you will compute, and it appears in exactly that form on the AP formula sheet. It counts how many standard errors separate what you observed from what the null hypothesis assumed. For example, testing $H_0: p = 0.50$ with p-hat (the sample proportion) equal to 0.60 and n = 100, the standard error is $\sqrt{0.50(0.50)/100} = 0.05$, so $z = (0.60 - 0.50)/0.05 = 2.00$. Notice the standard error uses the null value 0.50 rather than p-hat, because the calculation assumes the null hypothesis is true.