Introduction to Hypothesis Testing

Introduction

Key Concepts

1. What is Hypothesis Testing?

2. Types of Hypotheses

• Null Hypothesis ($H_0$): This is the default or skeptical assertion that there is no effect or no difference. It represents a statement of no change or no association.
• Alternative Hypothesis ($H_a$): This hypothesis contradicts the null hypothesis. It represents the outcome that the researcher aims to support, indicating an effect or a difference.

3. Steps in Hypothesis Testing

1. State the Hypotheses: Clearly define the null and alternative hypotheses based on the research question.
2. Select the Significance Level ($\alpha$): Common choices are 0.05, 0.01, or 0.10, representing the probability of rejecting the null hypothesis when it is true.
3. Choose the Appropriate Test: Depending on the data type and sample size, select tests such as z-test, t-test, chi-square test, or ANOVA.
4. Compute the Test Statistic: Calculate the test statistic using relevant formulas. For example, the z-test statistic is computed as:

4. Types of Errors

• Type I Error ($\alpha$): Rejecting the null hypothesis when it is actually true. The probability of committing a Type I error is equal to the significance level.
• Type II Error ($\beta$): Failing to reject the null hypothesis when the alternative hypothesis is true.

5. One-Tailed vs. Two-Tailed Tests

• One-Tailed Test: Used when the alternative hypothesis is directional, specifying that a parameter is either greater than or less than the null hypothesis value.
• Two-Tailed Test: Used when the alternative hypothesis is non-directional, stating that a parameter is simply different from the null hypothesis value without specifying the direction.

6. P-Value Interpretation

7. Power of a Test

8. Effect Size

9. Assumptions of Hypothesis Testing

• Independence of observations
• Normality of the data distribution
• Homogeneity of variances (for certain tests like ANOVA)

10. Example of Hypothesis Testing

1. State the Hypotheses:
   • $H_0$: $\mu = 75$
   • $H_a$: $\mu \neq 75$
2. Significance Level: $\alpha = 0.05$
3. Choose the Test: Two-tailed t-test (assuming the population standard deviation is unknown)
4. Compute the Test Statistic: $$ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} $$ Suppose $\bar{x} = 78$, $s = 10$, and $n = 30$: $$ t = \frac{78 - 75}{10 / \sqrt{30}} \approx 0.82 $$
5. Determine the P-Value: Using t-distribution tables or software, the p-value for $t = 0.82$ with 29 degrees of freedom is approximately 0.42.
6. Decision: Since $0.42 > 0.05$, fail to reject the null hypothesis.
7. Conclusion: There is insufficient evidence to conclude that the average score differs from 75.

11. Common Mistakes in Hypothesis Testing

• Confusing Significance Level and P-Value: The significance level is a threshold, while the p-value is a calculated probability.
• Ignoring Assumptions: Not verifying whether the data meets the assumptions required for the chosen test can invalidate the results.
• Misinterpreting the P-Value: A p-value does not measure the probability that the null hypothesis is true.
• Multiple Comparisons: Conducting multiple tests increases the likelihood of Type I errors if adjustments are not made.

12. Applications of Hypothesis Testing

• Medicine: Testing the efficacy of new drugs or treatments.
• Business: A/B testing for marketing strategies.
• Education: Evaluating the impact of teaching methods on student performance.
• Engineering: Quality control and process improvement.

13. Advanced Topics

• Confidence Intervals: Provide a range of plausible values for population parameters, complementing hypothesis testing.
• Non-Parametric Tests: Used when data does not meet the assumptions required for parametric tests, such as the Mann-Whitney U test.
• Bayesian Hypothesis Testing: Incorporates prior beliefs and evidence to update the probability of hypotheses.

Comparison Table

Summary and Key Takeaways