Hypothesis Testing Notes

Breakdown of Topics in Hypothesis Testing Seminar

• Hypothesis Testing: Proportions
  • Background of hypothesis testing
  • Steps to complete a hypothesis test
  • Examples
• Inferences About Means
  • Similar four steps to complete a hypothesis test
  • Examples
  • Confidence intervals for means

Hypothesis Testing: Proportions

• Contextual Example: Testing the effectiveness of a new COVID vaccine claimed to be 95% effective.
  • Objective: Determine if the claim is accurate based on sample data.
• Gather a sample, administer the vaccine, and measure infection rates.
• Expectation: If the vaccine is truly 95% effective, the sample's infection rate should be close to this figure.

Key Concept of Hypothesis Testing

• Hypothesis testing is a statistical method used to determine the validity of a claim based on sample data.
• Statistical Importance: Evaluates the likelihood that observed results could occur under the hypothesis.

Steps of Hypothesis Testing

1. Set Up Hypotheses

   • Null Hypothesis (H₀): Assumes no effect or difference.
   • Alternative Hypothesis (H₁): Assumes there is an effect or difference.
   • Example: H₀: p = 0.95 (the vaccine is effective), H₁: p ≠ 0.95 (the vaccine is not effective).

2. Specify the Appropriate Model

   • Determine the test statistic for the data.
   • Use sampling distributions to understand how data behaves if H₀ is true.

3. Mechanics: Calculate P-value

   • Quantifies evidence against H₀.
   • Measures the probability of observing the data or something more extreme if H₀ is true.

4. Draw Conclusions

   • Reject H₀ if P-value is less than significance level (α, commonly set at 0.05).
   • Fail to reject H₀ if P-value is greater than α.

Illustrative Examples

• If in a sample of 100, 78 are protected, is this close enough to 95%? Determine using statistics.
• Test Statistic (Z):
  • Z = \frac{p - p0}{\sqrt{\frac{p0(1 - p0)}{n}}} where p is the sample proportion and $p0$ is the hypothesized population proportion.
  • Large discrepancies from H₀ suggest rejecting the null hypothesis.

Understanding P-values

• Relationship between test statistics and P-values:
  • High P-value indicates the sample proportion is close to the hypothesized proportion.
  • Low P-value suggests a significant difference.
• Interpretation of P-values:
  • P-value < α: strong evidence against H₀, reject H₀.
  • P-value > α: insufficient evidence against H₀, fail to reject H₀.

Types of Errors

• Type I Error (α): Rejecting H₀ when it is actually true.
• Type II Error (β): Failing to reject H₀ when it is false.
• Balance between Type I and Type II errors based on chosen significance level (α).

Example Application

• Olivia's Free-Throw Accuracy: Testing if summer practice led to improved success rates:
  • Previous success: 38.4% (H₀: p = 0.384) vs. current success rate of 62.5% (25 made/40 attempts).
  • Execute hypothesis testing steps to find P-value and draw conclusions.

Confidence Intervals and Hypothesis Tests

• Both methods utilize similar calculations and assumptions.
• 95% confidence intervals can approximate hypothesis tests, associating with a significance level of 0.05.

Conclusion of Testing Procedure

• Ensure the understanding of setting hypotheses, computing test statistics, determining p-values, and making decisions are solidified for exam preparation.
• Past exam questions provide invaluable practice for grasping concepts and methods.