Hypothesis Testing: Concepts, Types, and Procedures

Hypothesis Testing

Definition and Purpose

Hypothesis testing is a statistical decision-making process used to determine whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves formulating two competing hypotheses and using sample data to decide which hypothesis is more plausible.

• Null Hypothesis (H0): The statement being tested, usually representing the status quo or no effect.

• Alternative Hypothesis (H1 or HA): The statement we want to test for, indicating the presence of an effect or difference.

Types of Hypotheses

Null and Alternative Hypotheses

• Null Hypothesis (H0): Assumes no change or effect. For example, H0: μ = μ0.

• Alternative Hypothesis (H1): Represents a new claim or effect. For example, H1: μ ≠ μ0, μ < μ0, or μ > μ0.

Types of Tests Based on Hypothesis Form

Test Directionality

The form of the alternative hypothesis determines the type of test:

• Two-tailed test (양측검정): H0: θ = θ0 vs. H1: θ ≠ θ0

• Left-tailed test (좌측검정): H0: θ ≥ θ0 vs. H1: θ < θ0

• Right-tailed test (우측검정): H0: θ ≤ θ0 vs. H1: θ > θ0

Key Terms in Hypothesis Testing

Test Statistic and Significance Level

• Test Statistic: A standardized value calculated from sample data, used to decide whether to reject H0. Common examples include t, Z, χ2, and F statistics.

• Significance Level (α): The probability of rejecting H0 when it is actually true (Type I error). Typical values are 0.01, 0.05, or 0.10.

Errors in Hypothesis Testing

Type I and Type II Errors

• Type I Error (α): Rejecting H0 when it is true.

• Type II Error (β): Failing to reject H0 when H1 is true.

There is a trade-off between α and β; decreasing one typically increases the other.

Steps in Hypothesis Testing

Classical and p-value Approaches

The hypothesis testing procedure can be summarized as follows:

1. State the hypotheses (H0 and H1).

2. Choose the significance level α.

3. Select the appropriate test statistic.

4. Determine the critical region (classical) or calculate the p-value.

5. Make a decision: reject or fail to reject H0.

p-value and Decision Rule

Definition and Calculation

• p-value: The probability, under H0, of obtaining a result as extreme or more extreme than the observed result. If p-value < α, reject H0.

• Calculation: For a Z-test, $Z_s = \frac{\overline{X} - \mu_0}{\sigma / \sqrt{n}}$

Hypothesis Test for Population Mean (Z-test)

Large Sample or Known Population Variance

• Hypotheses: H0: μ = μ0, HA: μ ≠ μ0

• Test Statistic: $Z = \frac{\overline{X} - \mu_0}{\sigma / \sqrt{n}}$

• Critical Values: $-z_{\alpha/2}, z_{\alpha/2}$ for two-tailed tests

Hypothesis Test for Population Proportion

Test Statistic and Decision Rule

• Hypotheses: H0: p = p0, HA: p ≠ p0

• Test Statistic: $Z = \frac{\hat{p} - p_0}{\sqrt{p_0 q_0 / n}}$

• Critical Values: $-z_{\alpha/2}, z_{\alpha/2}$ for two-tailed tests

Summary Table: Types of Errors

	H0 True	H1 True
Do Not Reject H0	Correct Decision	Type II Error (β)
Reject H0	Type I Error (α)	Correct Decision

Additional info: The notes also emphasize the importance of expressing results as "fail to reject H0" rather than "accept H0" to reflect the logic of statistical inference.