Chapter 8: Hypothesis Testing – Structured Study Notes

스터디 가이드 - 스마트 노트

자료에 맞춘 맞춤형 노트, 핵심 정의, 예시, 맥락을 확장해 제공합니다.

Hypothesis Testing

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental statistical procedure used to evaluate claims about population parameters. It provides a structured method for determining whether sample data provide enough evidence to reject a stated hypothesis about a population.

Hypothesis: A claim or statement about a property of a population.
Hypothesis Test (Test of Significance): A procedure for testing a claim about a property of a population.

Definitions of hypothesis and hypothesis test

Null and Alternative Hypotheses

Every hypothesis test involves two competing hypotheses: the null hypothesis and the alternative hypothesis. These form the basis for statistical decision-making.

Null Hypothesis (H0): States that the value of a population parameter is equal to some claimed value.
Alternative Hypothesis (H1, Ha): States that the parameter differs from the null hypothesis, using symbols such as <, >, or ≠.
Hint: The null hypothesis acts as a "working assumption" for the test.

Definitions of null and alternative hypotheses

Significance Level (α)

The significance level, denoted by α, is the probability threshold used to determine when sample evidence is considered significant enough to reject the null hypothesis.

Common values: 0.05, 0.01, 0.10 (most common is 0.05).
Interpretation: α is the probability of mistakenly rejecting the null hypothesis when it is actually true.

Formula:

Definition of significance level

Test Statistics and Sampling Distributions

Test statistics are calculated from sample data and are used to make decisions about the null hypothesis. The choice of test statistic depends on the parameter being tested and the sampling distribution.

Test Statistic: Converts the sample statistic to a score (z, t, χ2) under the assumption that H0 is true.
Sampling Distribution: The probability distribution of the test statistic under H0.

Definition of test statistic

Table: Parameters and Corresponding Test Statistics

Parameter	Sampling Distribution	Requirements
Proportion p	Normal (z)	np ≥ 5 and nq ≥ 5
Mean μ	t	σ not known and normally distributed population or n > 30
Mean μ	Normal (z)	σ known and normally distributed population or n > 30
Std. dev. or variance σ2	χ2	Strict requirement: normally distributed population

Table of parameters and test statistics

Critical Region and Types of Tests

The critical region (or rejection region) is the set of values for the test statistic that leads to rejection of the null hypothesis. The location of the critical region depends on the type of test: two-tailed, left-tailed, or right-tailed.

Two-tailed test: Critical region in both tails.
Left-tailed test: Critical region in the left tail.
Right-tailed test: Critical region in the right tail.

Definition of critical region Explanation of two-tailed, left-tailed, and right-tailed tests Critical region in two-tailed, left-tailed, and right-tailed tests

P-Value Method

The P-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. The P-value method compares this probability to the significance level to make a decision.

P-value: If P-value ≤ α, reject H0. If P-value ≥ α, fail to reject H0.
Calculation: Depends on the type of test (left-tailed, right-tailed, two-tailed).

Definition of P-value Flowchart for finding P-values

Critical Value Method

The critical value method compares the test statistic to predetermined critical values. If the test statistic falls in the critical region, the null hypothesis is rejected.

Critical value(s): Separates the critical region from non-rejection region.
Decision: If test statistic is in the critical region, reject H0; otherwise, fail to reject H0.

Definition of critical value

Decision and Conclusion

After calculating the test statistic and comparing it to the P-value or critical value, a decision is made to either reject or fail to reject the null hypothesis. The final conclusion should be stated in simple, nontechnical terms.

P-value method: If P-value ≤ α, reject H0; if P-value ≥ α, fail to reject H0.
Critical value method: If test statistic is in the critical region, reject H0; otherwise, fail to reject H0.
Final conclusion: State whether there is sufficient evidence to support or reject the original claim.

Flowchart for wording final conclusion

Procedure for Hypothesis Tests

The hypothesis testing process follows a structured sequence of steps:

Identify the claim and express it in symbolic form.
Identify the null and alternative hypotheses.
Select the significance level (α).
Identify the relevant test statistic and its sampling distribution.
Calculate the test statistic and find the P-value or critical value(s).
Make a decision to reject or fail to reject H0.
Restate the decision in nontechnical terms.

Procedure for hypothesis tests

Examples of Hypothesis Testing

Example 1: Mean Wait Time

Claim: The mean wait time at Space Mountain is equal to 40 minutes.
Null hypothesis: H0: μ = 40
Alternative hypothesis: H1: μ ≠ 40
Possible conclusions: Reject H0 or fail to reject H0.
Note: Cannot conclude there is sufficient evidence to support the claim; only reject or fail to reject.

Example 2: Proportion of Homes with Landline Only

Claim: Fewer than 10% of homes have only a landline telephone.
Symbolic form: p < 0.10
Null hypothesis: H0: p = 0.10
Alternative hypothesis: H1: p < 0.10
Test statistic: Calculated using sample data.

Example 3: Standard Deviation of Systolic Blood Pressure

Claim: Healthy adults have systolic blood pressure standard deviation greater than 5 mm Hg.
Symbolic form: σ > 5
Null hypothesis: H0: σ = 5
Alternative hypothesis: H1: σ > 5
Test statistic: Calculated using sample data.

Example 4: Proportion Test with z = 2.00

Claim: p > 0.25
Test type: Right-tailed
P-value: Calculated from z = 2.00
Decision: If P-value < 0.05, reject H0; otherwise, fail to reject.
Critical value: zcritical for α = 0.05

Example 5: Proportion Test with z = 1.80

Claim: p ≠ 0.314
Test type: Two-tailed
P-value: Calculated from z = 1.80
Decision: If P-value < 0.05, reject H0; otherwise, fail to reject.
Critical value: zcritical for α = 0.05

Example 6: Proportion Test with P-value = 0.3257

Claim: More than 58% of adults would erase all personal information online.
P-value: 0.3257
Decision: Fail to reject H0 (since P-value > 0.05)
Conclusion: There is not sufficient evidence to support the claim.

Example 7: Proportion Test with P-value = 0.00001

Claim: More than 35% of air travelers would choose another airline for inflight Wi-Fi.
P-value: 0.00001
Decision: Reject H0 (since P-value < 0.05)
Conclusion: There is sufficient evidence to support the claim.

Additional info: These examples illustrate the application of hypothesis testing to real-world claims, using both the P-value and critical value methods.