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Lab 6: Confidence Intervals and Statistical Inference

Student Name:

Introduction: Confidence Intervals and Statistical Inference

This laboratory explores confidence intervals as a fundamental tool for statistical inference. We will use an interactive Shiny application to investigate how confidence intervals behave under different conditions, including varying population distributions, sample sizes, and confidence levels. This hands-on approach will deepen understanding of interval estimation and its reliability.

Learning Objectives

Upon completion of this laboratory, students will be able to:

• Understand the conceptual foundation of confidence intervals
• Distinguish between point estimators and interval estimators
• Calculate confidence intervals using both z-distribution and t-distribution
• Analyze how sample size affects confidence interval width and coverage
• Evaluate how confidence level impacts interval precision and reliability
• Investigate the effects of different population distributions on CI performance
• Interpret confidence interval coverage and width in practical contexts

Time Allocation (Total: 100 minutes)

• Part 1: CI Fundamentals and Parameter Effects (35 minutes)
• Part 2: Distributional Effects and Assumption Violations (35 minutes)
  
• Part 3: Practical Applications and Synthesis (30 minutes)

Part 1: CI Fundamentals and Parameter Effects

1.1 Confidence Interval Theory

Confidence intervals provide a range of plausible values for population parameters with a specified level of confidence. The key formulas are:

When population standard deviation (σ) is known: $$\overline{X} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$$

When population standard deviation is unknown: $$\overline{X} \pm t_{n-1,\alpha/2} \frac{s}{\sqrt{n}}$$

1.2 Using the CI Shiny Application

To run the confidence interval simulation application, click on the following Shiny app:

Access the CI simulation app here: https://math214.netlify.app/ciapp

App Interface Overview:

• Population Distribution: Normal, Uniform, Right Skewed, Left Skewed, Bimodal, Cauchy
• Population Variance Known: Yes/No toggle
• Confidence Level: 0.01 to 0.99
• Sample Size: 5 to 500 observations per sample
• Number of Samples: 10 to 5000 replications
• Random Seed: For reproducible results

1.3 Key CI Concepts

• Coverage Probability: Proportion of intervals containing the true parameter
• Interval Width: Precision of the estimate (narrower = more precise)
• Reliability Trade-off: Higher confidence → wider intervals → less precision
• Sample Size Effect: Larger samples → narrower intervals → better precision

Part 2: Distributional Effects and Assumption Violations

2.1 Normal Distribution Performance

Normal populations provide the ideal case where confidence interval assumptions are perfectly satisfied.

2.2 Non-Normal Distributions

Various non-normal distributions test the robustness of confidence interval procedures:

• Uniform: Symmetric but not normal
• Skewed Distributions: Test asymmetry effects
• Bimodal: Complex distribution shapes
• Cauchy: Heavy-tailed distribution violating CLT assumptions

2.3 Assumption Violations

The Cauchy distribution serves as a critical test case where fundamental assumptions (finite mean and variance) are violated, demonstrating the limits of standard confidence interval methods.

Part 3: Practical Applications and Synthesis

3.1 Real-world Applications

Confidence intervals are used across various fields:

• Healthcare: Clinical trial results and treatment effects
• Social Sciences: Survey results and population estimates
• Quality Control: Process monitoring and specification limits
• Economics: Economic indicators and policy impacts

3.2 Interpretation Guidelines

Proper interpretation of confidence intervals requires understanding that:

• The interval either contains the parameter or it doesn’t
• The confidence level refers to the long-run success rate of the procedure
• Results should be considered in context of the study design and assumptions

Assessment (Total: 50 points)

Section A: Formative Understanding (15 points)

A1. Explain the conceptual difference between point estimators and interval estimators. Discuss the advantages and disadvantages of each approach, and provide examples of situations where confidence intervals are particularly valuable. (4 points)

A2. Describe the trade-off between confidence level and interval precision. Why can’t we have both high confidence and high precision simultaneously? Provide mathematical reasoning to support your explanation. (4 points)

A3. Compare and contrast the z-interval and t-interval procedures. When should each be used, and what are the key assumptions underlying each method? (4 points)

A4. Discuss the practical interpretation of a 95% confidence interval. What common misconceptions exist about confidence intervals, and how would you explain the correct interpretation to someone without statistical training? (3 points)

Section B: Application and Analysis (20 points)

B1. Using the CI app with a normal population, σ known, 95% confidence level, n=30, and 100 samples: - Report the proportion of intervals covering the true mean - Report the average interval width - Analyze the balance of non-covering intervals (left vs right) - Discuss the reliability of the procedure (5 points)

# Your code and analysis here

B2. Repeat B1 with σ unknown (using sample standard deviation). Compare the results and explain any differences observed between the known and unknown variance cases. (5 points)

# Your code and analysis here

B3. Investigate the effects of confidence level by testing 80%, 90%, 95%, and 99% levels with n=40 and normal population. Create a table summarizing coverage proportions and average widths. Analyze how confidence level affects interval performance. (5 points)

# Your code and analysis here

B4. Examine sample size effects by testing n=10, 40, 90, 160, 250 with 95% confidence and normal population. Create a table and analyze how sample size impacts both coverage and precision. (5 points)

# Your code and analysis here

Section C: Statistical Synthesis (15 points)

C1. Write a comprehensive report (200-250 words) discussing the robustness of confidence interval procedures to violations of distributional assumptions. Include examples from different distribution types and provide guidance for practitioners on when standard methods can be safely used versus when alternatives are needed. (7 points)

C2. Design a complete simulation study to evaluate the minimum sample size required for satisfactory confidence interval performance with a specific non-normal distribution of your choice. Outline the methodology, evaluation criteria, and practical implications of your findings. (4 points)

C3. Reflect on the pedagogical value of interactive confidence interval simulations. Discuss how visualizing interval coverage and width enhances conceptual understanding compared to traditional formula-based approaches. (4 points)

Submission Guidelines:

• Complete R Markdown document with all code and analysis
• Screenshots or detailed descriptions of app outputs for each analysis
• Professional writing with clear analytical narrative
• Properly formatted tables and visualizations
• Knitted HTML document submitted via designated platform
• All analysis should demonstrate use of the CI application for exploration