Statistical Inference

4.

Interval Estimation and Confidence Intervals

4.1.

The Concept of Interval Estimation

4.1.1.

Point Estimate vs Interval Estimate

4.1.2.

Quantifying Uncertainty

4.1.3.

Margin of Error

4.2.

Definition and Interpretation of Confidence Intervals

4.2.1.

Formal Definition

4.2.2.

Interpretation in Repeated Sampling

4.2.3.

Correct vs Incorrect Interpretations

4.2.4.

Coverage Probability

4.3.

The Confidence Level

4.3.1.

Meaning of Confidence Level

4.3.2.

Common Confidence Levels

4.3.3.

Relationship to Significance Level

4.3.4.

Choosing Appropriate Confidence Level

4.4.

General Construction of Confidence Intervals

4.4.1.

Pivotal Quantity Method

4.4.2.

Inversion of Hypothesis Tests

4.4.3.

General Formula Structure

4.4.4.

Components of Margin of Error

4.5.

Confidence Intervals for Single Population Parameters

4.5.1.

Confidence Intervals for Population Mean

4.5.1.1.

Population Variance Known (Z-Interval)

4.5.1.1.1.

Formula and Assumptions

4.5.1.1.2.

Use of Standard Normal Distribution

4.5.1.1.3.

Critical Values

4.5.1.2.

Population Variance Unknown (t-Interval)

4.5.1.2.1.

Formula and Assumptions

4.5.1.2.2.

Use of t-Distribution

4.5.1.2.3.

Degrees of Freedom

4.5.1.2.4.

Robustness to Normality Assumption

4.5.2.

Confidence Intervals for Population Proportion

4.5.2.1.

Large-Sample Wald Interval

4.5.2.1.1.

Formula and Conditions

4.5.2.1.2.

Normal Approximation

4.5.2.2.

Wilson Score Interval

4.5.2.2.1.

Improved Coverage Properties

4.5.2.2.2.

Formula and Application

4.5.2.3.

Agresti-Coull Interval

4.5.2.3.1.

Adjustment for Small Samples

4.5.2.3.2.

Rationale and Procedure

4.5.2.4.

Exact Binomial Intervals

4.5.2.4.1.

Clopper-Pearson Method

4.5.2.4.2.

4.5.3.

Confidence Intervals for Population Variance

4.5.3.1.

Use of Chi-Squared Distribution

4.5.3.2.

Formula and Assumptions

4.5.3.3.

Sensitivity to Normality Assumption

4.6.

Confidence Intervals for Comparing Two Populations

4.6.1.

Difference Between Two Means

4.6.1.1.

Independent Samples

4.6.1.1.1.

Equal Variances Assumed (Pooled t-Interval)

4.6.1.1.2.

Unequal Variances (Welch's Method)

4.6.1.1.3.

Assumptions and Robustness

4.6.1.2.

4.6.1.2.1.

Paired Data Structure

4.6.1.2.2.

Confidence Interval for Mean Difference

4.6.1.2.3.

4.6.2.

Difference Between Two Proportions

4.6.2.1.

Large-Sample Interval

4.6.2.2.

Pooled vs Unpooled Methods

4.6.2.3.

Assumptions and Conditions

4.6.3.

Ratio of Two Variances

4.6.3.1.

Use of F-Distribution

4.6.3.2.

Formula and Assumptions

4.6.3.3.

Interpretation of Ratio

4.7.

Sample Size Determination

4.7.1.

Factors Affecting Sample Size

4.7.1.1.

Desired Margin of Error

4.7.1.2.

Confidence Level

4.7.1.3.

Population Variability

4.7.2.

Sample Size for Estimating Means

4.7.2.1.

Known Population Variance

4.7.2.2.

Unknown Population Variance

4.7.2.3.

4.7.3.

Sample Size for Estimating Proportions

4.7.3.1.

Conservative Approach

4.7.3.2.

Using Prior Information

4.7.3.3.

Finite Population Correction

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5. Hypothesis Testing Framework