HYPOTHESIS-TESTING.EXCALIDRAW

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Hypothesis-Testing.excalidraw

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◇Overview

This visual reference summarizes the Statistical Hypothesis Testing Decision Framework: partitioning the sampling distribution of a test statistic into the Acceptance (Non-Rejection) Region and the Critical (Rejection) Region under the Null Hypothesis $H_0$, along with the trade-off between Type I Error ($\alpha$) and Type II Error ($\beta$).

◇Key Definitions

• Test Statistic ($Z$ or $T$): Standardized distance between sample estimate $\bar{X}$ and hypothesized parameter $\mu_0$:
  $Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}, \qquad T = \frac{\bar{X} - \mu_0}{S / \sqrt{n}} \sim t_{n-1}$
• Decision Rule ($p$-value vs. $\alpha$): Reject $H_0$ if and only if $p\text{-value} \le \alpha$ (equivalently, if the test statistic falls inside the critical region).

◇Worked Example

Problem: Test $H_0: \mu = 50$ vs. $H_a: \mu \ne 50$ (two-tailed) at significance level $\alpha = 0.05$ ($z_{\alpha/2} = \pm 1.96$) when a sample of size $n = 36$ with known $\sigma = 12$ yields $\bar{x} = 54.5$.

Solution:

• Standard error: $\sigma_{\bar{X}} = \frac{12}{\sqrt{36}} = 2.0$.
• Test statistic: $z = \frac{54.5 - 50}{2.0} = +2.25$.
• Since $|z| = 2.25 > 1.96$ (and two-tailed $p\text{-value} = 2(1 - \Phi(2.25)) \approx 0.0244 < 0.05$), reject $H_0$.

◇References & Navigation

• Related Notes: ◇Hypothesis Testing