This chapter introduces the fundamental concepts of hypothesis testing, a core tool in inferential statistics. It focuses on testing claims about a population mean or proportion based on sample data. The chapter explains how to formulate null and alternative hypotheses, choose the appropriate significance level, and compute test statistics using either the z-test (when population standard deviation is known) or t-test (when it is unknown). It also covers hypothesis testing for population proportions using the z-test under specific sample conditions. Key ideas such as Type I and Type II errors, p-values, and one- vs. two-tailed tests are discussed. The goal is to help students make informed decisions based on data evidence and statistical reasoning.

📘 Part 1: Key Terms & Explanations

🔑 Keyword	📖 Explanation
Hypothesis	A statement about a population parameter (e.g., mean or proportion).
Null Hypothesis (H₀)	The default assumption, usually includes "=" (e.g., \( \mu = 50 \)).
Alternative Hypothesis (H₁)	The opposite of H₀; uses ≠, >, or <.
Significance Level (α)	Probability of Type I error (usually 0.05 or 0.01).
Critical Value	Threshold value to reject H₀ based on α.
Test Statistic	Value calculated from the sample to test H₀.
p-value	Probability of observing the sample result (or more extreme) assuming H₀ is true.
Type I Error (α)	Rejecting H₀ when it is true (false positive).
Type II Error (β)	Failing to reject H₀ when it is false (false negative).
Power (1−β)	The ability to correctly reject a false H₀.
Z-test	Used when population standard deviation (\( \sigma \)) is known.
t-test	Used when \( \sigma \) is unknown and sample size is small.
Two-tailed test	Tests for any significant difference (≠).
One-tailed test	Tests for deviation in one direction (greater or less).
Proportion Test	Used to test population proportions (binary outcomes).

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📐 Part 2: Test Formulas & Procedures

✅ A. One-Sample Mean Test

1. Z-test (σ known)

Formula:

$$ Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} $$

• \( \bar{x} \): sample mean
• \( \mu_0 \): hypothesized mean
• \( \sigma \): population standard deviation
• \( n \): sample size

2. t-test (σ unknown)

Formula:

$$ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} $$

• \( s \): sample standard deviation
• \( df = n - 1 \): degrees of freedom

✅ B. One-Sample Proportion Test

Conditions: \( np \geq 5 \) and \( n(1 - p) \geq 5 \)

Formula:

$$ Z = \frac{\hat{p} - p_0}{\sqrt{p_0(1 - p_0)/n}} $$

• \( \hat{p} \): sample proportion
• \( p_0 \): hypothesized population proportion

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🔁 Hypothesis Testing Steps

1. State H₀ and H₁ clearly.
2. Select significance level α (e.g., 0.05).
3. Choose test type: Z or t, one- or two-tailed.
4. Calculate test statistic (Z or t).
5. Compare with critical value or compute p-value.
6. Make a conclusion:
   • If p-value < α → reject H₀.
   • Otherwise → fail to reject H₀.