Outline

1. Introduction
2. Central Limit Theorem (CLT)
3. Applications of CLT
4. Introduction to Hypothesis Testing
5. Steps in Hypothesis Testing
6. Types of Errors
7. Conclusion

1. Introduction

• Statistical inference: Drawing conclusions about populations from samples
• Two key concepts: Central Limit Theorem and Hypothesis Testing
• Fundamental to many fields: science, economics, psychology, and more

2. Central Limit Theorem (CLT)

• Definition: The distribution of sample means approximates a normal distribution as the sample size becomes larger
• Conditions:
  • Independent observations
  • Sufficiently large sample size (usually n ≥ 30)
• Key points:
  • Mean of sample means = Population mean
  • Standard deviation of sample means = Population standard deviation / √n

3. Applications of CLT

• Enables statistical inference about population parameters
• Justifies the use of many statistical methods (t-tests, ANOVA, etc.)
• Allows for the construction of confidence intervals
• Crucial in quality control and process improvement

Example: Estimating average customer wait time in a call center

4. Introduction to Hypothesis Testing

• Definition: A method for using sample data to decide between two competing claims about a population parameter
• Components:
  • Null hypothesis (H₀): The status quo claim
  • Alternative hypothesis (H₁ or Hₐ): The claim to be tested

Example: Testing if a new teaching method improves test scores

• H₀: μ = 70 (The new method does not change the average score)
• H₁: μ > 70 (The new method improves the average score)

5. Steps in Hypothesis Testing

1. State the hypotheses (H₀ and H₁)
2. Choose a significance level (α)
3. Collect sample data
4. Calculate the test statistic
5. Determine the p-value or critical value
6. Make a decision: Reject H₀ or Fail to reject H₀
7. Interpret the results in context

6. Types of Errors

• Type I Error: Rejecting H₀ when it’s actually true
  • Probability = α (significance level)
• Type II Error: Failing to reject H₀ when it’s actually false
  • Probability = β
• Power of a test = 1 - β
  • Probability of correctly rejecting a false null hypothesis

7. Conclusion

• CLT is the foundation for many statistical methods
• Hypothesis testing provides a framework for making decisions based on data
• Understanding these concepts is crucial for:
  • Designing experiments
  • Analyzing data
  • Drawing valid conclusions from research

Questions?

Thank you for your attention!