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# Probability and Statistics IV: Confidence Intervals and Hypothesis Tests

## Overview

This course covers two important methodologies in statistics – confidence intervals and hypothesis testing.

Confidence intervals are encountered in everyday life, and allow us to make probabilistic statements such as: “Based on the sample of observations we conducted, we are 95% sure that the unknown mean lies between A and B,” and “We are 95% sure that Candidate Smith’s popularity is 52% +/- 3%.” We begin the course by discussing what a confidence interval is and how it is used. We then formulate and interpret confidence intervals for a variety of probability distributions and their parameters.

Hypothesis testing allows us to pose hypotheses and test their validity in a statistically rigorous way. For instance, “Does a new drug result in a higher cure rate than the old drug” or “Is the mean tensile strength of item A greater than that of item B?” The second half the course begins by motivating hypothesis tests and how they are used. We then discuss with the types of errors that can occur with hypothesis testing, and how to design tests to mitigate those errors. Finally, we formulate and interpret hypothesis tests for a variety of probability distributions and their parameters.

Hypothesis testing allows us to pose hypotheses and test their validity in a statistically rigorous way. For instance, “Does a new drug result in a higher cure rate than the old drug” or “Is the mean tensile strength of item A greater than that of item B?” The second half the course begins by motivating hypothesis tests and how they are used. We then discuss with the types of errors that can occur with hypothesis testing, and how to design tests to mitigate those errors. Finally, we formulate and interpret hypothesis tests for a variety of probability distributions and their parameters.

## Syllabus

“FCPS” refers to the free text, A First Course in Probability and Statistics

Module 1: Confidence Intervals
• Lesson 1: Introduction to Confidence Intervals (FCPS §6.1)
• Lesson 2: Normal Mean (variance known) (FCPS §6.2)
• Lesson 3: Difference of Two Normal Means (variances known) (FCPS §6.3)
• Lesson 4: Normal Mean (variance unknown) (FCPS §6.4)
• Lesson 5: Difference of Two Normal Means (unknown equal variances) (FCPS §6.5.1)
Module 1 (cont’d): Confidence Intervals
• Lesson 6: Difference of Two Normal Means (variances unknown) (FCPS §6.5.2)
• Lesson 7: Difference of Paired Normal Means (variances unknown) (FCPS §6.5.3)
• Lesson 8: Normal Variance (FCPS §6.6)
• Lesson 9: Ratio of Variances of Two Normals (FCPS §6.7)
• Lesson 10: Bernoulli Proportion (FCPS §6.8)
Module 2: Hypothesis Testing
• Lesson 1: Introduction to Hypothesis Testing (FCPS §7.1)
• Lesson 2: The Errors of Our Ways (FCPS §7.1)
• Lesson 3: Normal Mean Test with Known Variance (FCPS §7.2.1)
• Lesson 4: Normal Mean Test with Known Variance: Design (FCPS §7.2.2)
• Lesson 5: Two-Sample Normal Means Test with Known Variances (FCPS §7.2.3)
• Lesson 6: Normal Mean Test with Unknown Variance (FCPS §7.3.1)
• Lesson 7: Two-Sample Normal Means Tests with Unknown Variances (FCPS §7.3.2)
Module 2 (cont’d): Hypothesis Testing
• Lesson 8: Two-Sample Normal Means Test with Paired Observations (FCPS §7.3.2)
• Lesson 9: Normal Variance Test (FCPS §7.4.1)
• Lesson 10: Two-Sample Normal Variances Test (FCPS §7.4.2)
• Lesson 11: Bernoulli Proportion Test (FCPS §7.4.3)
• Lesson 12: Two-Sample Bernoulli Proportions Test (FCPS §7.4.4)
• Lesson 13: Goodness-of-Fit Tests: Introduction (FCPS §7.5.1)
• Lesson 14: Goodness-of-Fit Tests: Examples (FCPS §7.5.2)
• Lesson 15 [OPTIONAL]: Goodness-of-Fit Tests: Honors Example (FCPS §7.5.3)

David Goldsman

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