## Setting up a Statistical Model **Description:** To setup a [[Statistical Model]], requires defining the population from which the sample is drawn and the underlying distribution that characterizes this population. It includes specifying the population parameters like the mean, variance, etc. **Example:** Suppose you are testing the effectiveness of a new drug. You assume that the population of interest (e.g., patients with a specific condition) follows a [[Gaussian Distribution]] for a particular health metric (like blood pressure levels). Mathematically, this can be expressed as: $ X \sim \mathcal N(\mu, \sigma^2) $ where $X$ represents the blood pressure levels, $\mu$ is the mean blood pressure level, and $\sigma^2$ is the variance. ## Defining a Null Hypothesis **Description:** The [[Hypothesis Tests|null hypothesis]] $H_0$ is a statement of no effect or no difference, which you assume to be true until evidence suggests otherwise. The alternative hypothesis $H_A$ is what you aim to support, indicating the presence of an effect or a difference. **Example:** $H_0:\mu=\mu_0$ where $\mu_0$ is the mean blood pressure level without the drug, and $H_A: \mu \not = \mu_0$ indicating the mean blood pressure levels are different with the drug. This can be set up as: $ H_0:\mu=\mu_0 \quad \text{vs.} \quad H_A: \mu\not= \mu_0 $ ## Defining a Test Statistic **Description:** A [[Statistical Test]] defines the quantitative decision rule to either reject or fail to reject the null hypothesis. The test statistic is a is a function of the sample data and should be chosen based on the statistical model and the nature of the hypothesis. **Example:** If you're testing the mean blood pressure level, a common test statistic is the t-statistic, which measures how much the sample mean deviates from the null hypothesis mean. It is computed as: $ T=\frac{\bar X-\mu_0}{s/\sqrt n} $ ## Setting the Significance Level **Description:** The significance level $\alpha$ is the probability of rejecting $H_0$ when it is actually true (Type I error). **Example:** You might set $\alpha=0.05$, indicating that you are willing to accept a 5% chance of incorrectly rejecting the null hypothesis (i.e., concluding the drug is effective when it actually is not).