Lecture Notes: Difference in Means and Confidence Intervals

Introduction

• Topic: Continuing discussion on confidence intervals, focusing on the difference in means (Chapter 9).
• Previous Topic: Confidence intervals for one sample mean.
• Current Focus: Estimating the difference between two sample means.

Key Questions for Analyzing Difference in Means

1. Are the samples independent?
   • Answer: Yes or No.
2. Do you know the population variance?
   • Answer: Yes or No.
   • Note: Knowing population standard deviation implies knowledge of population variance.

Four Scenarios Based on Key Questions

Scenario 1: Independent Samples, Known Variances

• Conditions: Yes, samples are independent; Yes, variances are known.
• Equation: ( \bar{x}1 - \bar{x}2 = Z{\alpha/2} \frac{\sigma}{\sqrt{n}} )

Scenario 2: Independent Samples, Unknown But Assumed Equal Variances

• Conditions: Yes, independent; No, variances unknown; Assume equal if sample variance ratio is between 1/2 and 2.
• Equation: Uses pooled variance approach.

Scenario 3: Independent Samples, Variances Unknown and Assumed Different

• Conditions: Yes, independent; No, variances unknown; Assume different.
• Equation: Separate sample variance usage.

Scenario 4: Dependent Samples (Paired Observations)

• Conditions: No, samples are not independent.
• Equation: ( \bar{D} - t_{\alpha/2} \frac{s_D}{\sqrt{n}} < \mu_D < \bar{D} + t_{\alpha/2} \frac{s_D}{\sqrt{n}} )

Analysis and Interpretation

• Variance Known vs Unknown:
  • Use Z if variances are known.
  • Use T if variances are unknown.
• Degrees of Freedom:
  • Important for T-distribution.
  • Complex calculation, ensure correct understanding.
• Pooled Variance:
  • Calculate when variances are assumed equal.

Conclusion from Confidence Intervals

• Zero in Interval:
  • If confidence interval spans zero, possible that ( \mu_1 - \mu_2 = 0 ), indicating no significant difference.
• Negative/Positive Limits:
  • Negative Limits: Indicates ( \mu_2 > \mu_1 ).
  • Positive Limits: Indicates ( \mu_1 > \mu_2 ).

Class Preparation

• Next Steps: Examples will be tackled in class.
• Action Items: Bring questions to class for discussion.