📝 Quiz 2 Details

When:
- 📅 Date: Friday, July 25
- ⏰ Window: 7 AM – 12 AM
- ⏳ Duration: 1 hour once started

Where: 💻 Online via Canvas

Covers: Material from Weeks 3-4

📚 What to Expect

• Discrete & continuous distributions
• Probability calculations
• Expected value & variance
• Normal distribution applications
• Note: Upload photos of written work for calculation problems

🎯 Learning Objectives

• Know the difference between z and t distributions
• Understand when to use each distribution
• Learn to find critical values from tables and plots
• Practice calculating confidence intervals step-by-step
• Interpret results correctly in context

📋 What We’ll Cover

1. Review: Confidence interval basics
2. The t-Distribution: When and why we use it
3. Critical Regions: Finding the right values
4. Practical Examples: z and t calculations
5. Common Mistakes: What to avoid
6. Real Applications: Making it meaningful

🎯 The Big Idea

A confidence interval (CI) takes a single sample statistic and turns it into a range that is likely to contain an unknown population parameter; most often the mean \mu.

CI template

\underbrace{\text{Point estimate}}_{\color{blue}{(e.g., \bar{x})}} \;\pm\; \underbrace{\text{(critical value) $\times$ (standard error)}}_{\color{red}{\text{Margin of Error (ME)}}}

For the mean

Key points

• We never know the true mean \mu in practice, that’s exactly what the CI estimates.
  
• Use the population SD σ only when it is genuinely known (e.g., industrial process with long‑term QC).
  
• Otherwise substitute the sample SD s and switch to the t‑distribution, which is wider to reflect that extra uncertainty.

Key Formula: \bar{x} \pm z^* \cdot \frac{s}{\sqrt{n}} (when using z-distribution)

🎯 Problem Setup

Research Question: What is the average SAT score of students at UCSB?

Given Information:

• Sample size: n = 50 students
• Sample mean: \bar{x} = 1180
• Population standard deviation: \sigma = 120 (known from past data)
• Confidence level: 95\%

Question: Construct a 95\% confidence interval for the population mean SAT score.

🎯 Problem Setup

Research Question: What is the average daily coffee consumption at our office?

Given Information:

- Sample size: n = 16 employees

- Sample mean: \bar{x} = 2.8 cups

- Sample standard deviation: s = 0.9 (\sigma unknown)

- Confidence level: 90%

Question: Construct a 90% confidence interval for the population mean daily coffee consumption.

🎯 Your Turn!

Problem: A researcher wants to estimate the average time students spend studying per day.

Given:

• Sample size: n = 25 students
  
• Sample mean: \bar{x} = 3.2 hours
• Sample standard deviation: s = 1.1 hours
• Confidence level: 95\%

Questions:

1. Should you use z or t-distribution? Why?
2. What are the degrees of freedom?
3. What is the critical value?
4. Calculate the 95% confidence interval
5. Interpret your result in context

🤔 Think About It…

• Why is the t-distribution appropriate here?
• How would the interval change if n = 100?
• What if we wanted 99% confidence instead?

🧠 Core Concepts

1. Distribution Choice - σ known → z-distribution - σ unknown + n ≥ 30 → z-distribution
- σ unknown + n < 30 → t-distribution

2. t-Distribution Properties - Heavier tails than z - Depends on degrees of freedom (df = n-1) - Approaches z as df increases

3. Critical Regions - α/2 in each tail for two-sided CI - Critical values from tables or software - Larger confidence → larger critical values

🛠️ Practical Skills

4. Calculation Steps 1. Check conditions (σ known?, sample size?) 2. Choose distribution (z or t) 3. Find critical value 4. Calculate standard error 5. Compute margin of error
6. Construct interval 7. Interpret in context

5. Interpretation - “We are C% confident…” - Focus on the process, not individual interval - Consider practical significance

6. Common Pitfalls to Avoid - Wrong distribution choice - Incorrect degrees of freedom - Using α instead of α/2 - Misinterpreting the interval