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Hypothesis Testing Visualization | EasyCFA | Wall Street Uni

Quantitative Methods

Hypothesis Testing: Rejection Regions & Decision Logic

Interactive hypothesis testing visualization showing sampling distribution, rejection regions, critical values, test statistic, and decision logic. Understand how significance level α defines rejection regions and how to interpret test results.

Subtopic: Hypothesis Testing
Level: Applied
Status: Featured
Views: 297

Parameters

Distribution

Test Type

Formula

\begin{aligned} z &= \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \\ z &= \frac{52 - 50}{10 / \sqrt{30}} \\ z &= 1.095 \end{aligned}

Interactive Chart

Hypothesis testing uses a test statistic to determine whether to reject the null hypothesis (H₀). The test statistic measures how many standard errors the sample mean is from the hypothesized population mean. For a Z-test: z = (x̄ - μ₀) / (σ / √n). For a T-test: t = (x̄ - μ₀) / (s / √n). The distribution of this test statistic under H₀ is either standard normal (Z) or t-distributed.

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Study aids

Working with this chart

The errors candidates make on this topic, and what to carry into the exam.

Common mistakes

Confusing "fail to reject H₀" with "accept H₀" (we never accept, only fail to reject)
Using wrong tails: two-tailed for ≠, right-tailed for >, left-tailed for <
Forgetting to divide α by 2 for two-tailed tests when using tables
Using z-test when σ is unknown and n is small (should use t-test)
Confusing significance level α with p-value (α is predetermined, p-value is calculated)
Thinking statistical significance means practical importance
Not adjusting for multiple testing (increases Type I error)
Confusing one-sided and two-sided critical values

Exam tips

MEMORIZE: Z-critical values: ±1.645 (10%), ±1.96 (5%), ±2.58 (1%) for two-tailed
MEMORIZE: For one-tailed at 5%: Z-critical = 1.645 (or -1.645 for left tail)
Test statistic in rejection region → Reject H₀
p-value < α → Reject H₀ (both methods give same answer)
Small p-value = strong evidence against H₀
CFA loves: "Is this result statistically significant at 5% level?"
Watch for: z-test (σ known) vs t-test (σ unknown, or small sample)
Common setup: Given sample mean, hypothesized mean, std dev, sample size → calculate test statistic
Decision wording: "Reject H₀" or "Fail to reject H₀" (NEVER "accept")
Quick check: If |test statistic| > |critical value|, reject H₀ (for two-tailed)

Parameters

Distribution

Test Type

alphaSignificance Level

0.010.1

xbarSample Mean

4060

mu0Hypothesized Mean

4060

nSample Sizeobservations

10500

sigmaStandard Deviation

120

Formula

\begin{aligned} z &= \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \\ z &= \frac{52 - 50}{10 / \sqrt{30}} \\ z &= 1.095 \end{aligned}

Interactive Chart

Hypothesis Test

H₀:μ = 50

H₁:μ ≠ 50

Test Type:two-tailed

Distribution:Z (normal)

Sampling Distribution Under H₀

Sample Statistics

Sample Mean (x̄):52

Sample Size (n):30

Std Dev (σ):10

Std Error:1.83

Test Results

Significance (α):5.0%

Test Statistic:1.095

p-value:0.2733

p-value vs α:p ≥ α

Decision

Fail to reject H₀

Insufficient evidence against H₀

Student Instructions:

Change α and observe how rejection regions widen or narrow
Move the sample mean and see when H₀ is rejected
Switch between one-tailed and two-tailed tests
Compare critical-value and p-value decision rules