Lecture_7-mas

Introduction

  • Course Information: BB1719, Spring 2024, Introduction to Data Analysis

  • Instructor: Dr. Michelle Sahai

  • Lecture Focus: Probabilities, P-value, and Hypothesis Testing

Learning Outcomes

  • Understand key concepts, including:

    • Null hypothesis (H₀) and alternative hypothesis (H or H₁)

    • Type 1 error vs. Type 2 error

    • Power of a test

    • P-value and its interpretation

Hypothesis Testing Steps

  1. Set the Hypothesis

    • Null Hypothesis (H₀): Assumes no effect or difference.

    • Alternative Hypothesis (H₁): Assumes some effect or difference exists.

  2. Set the Significance Level (α)

    • Determines the criteria for a decision (commonly 0.05).

  3. Compute the Test Statistics

  4. Make a Decision

    • Determine if there is a statistically significant difference between the two groups.

    • Example Groups: Control group vs. Intervention group.

Null and Alternative Hypothesis

  • Null Hypothesis (H₀): States there is no difference between groups.

    • Example: Mean control = Mean intervention group.

  • Alternative Hypothesis (H₁): States there is a significant difference.

    • Example: There is a difference in blood sugar levels.

Type 1 and Type 2 Errors

  • Type 1 Error: Incorrectly rejecting H₀ when it is true (False Positive).

    • Probability of committing a Type 1 error is represented by α.

  • Type 2 Error: Failing to reject H₀ when it should be rejected (False Negative).

    • Probability of committing a Type 2 error is represented by β.

P-value and Interpretation

  • Definition: The probability of observing results as extreme as those in the data, assuming H₀ is true.

  • Interpreting P-values:

    • Low P-value (< α): Reject H₀ (suggests a statistically significant effect).

    • High P-value (> α): Fail to reject H₀ (suggests no significant effect).

Factors Impacting P-value

  • The significance level (α), typically set at 5%.

  • A p-value measures the evidence against the null hypothesis.

Assessing Statistical Significance

  • A comparison between two groups can help determine if a significant difference exists based on test results.

Types of Statistical Tests

  • t-Test: Compares the means of two groups.

  • ANOVA: Compares means across three or more groups.

  • Chi-Squared Test: Compares categorical variables and proportions.

  • Regression Analysis: Analyzes the relationship between two or more variables.

Power of a Test

  • Definition: The probability of correctly rejecting H₀ when it is false (i.e., detecting a true effect).

  • Calculation: Power = 1 - β (where β is the probability of Type 2 error).

  • Desirable Power Level: Studies should aim for a power of ≥ 0.8 (or 80%) for reliable results.

    • Influencers of power include sample size, effect size, and precision of results.

Summary of Key Concepts

  • Understand the relationship and differences between:

    • Null Hypothesis (H₀)

    • Alternative Hypothesis (H₁)

    • Errors (Type 1 and Type 2)

    • Interpretation of p-value

    • Concept of Power in hypothesis testing.