Lecture Notes on Hypothesis Testing

Alpha Level and Medical Questions

• Default Alpha Levels:

  • 0.05 for medical questions.

  • 0.01 for non-medical questions is suggested, but not strictly enforced.

  • Any deviation from these must be explained.

Testing Hypotheses

• Alpha and H0 (Null Hypothesis):

  • Alpha signifies the probability that H0 is true when data suggests otherwise.

  • Set an alpha level for the hypothesis test and use that for decision-making.

Example of Coin Flipping

• Fair Coin Test:

  • Experiment with 100 flips.

  • If results fall between a specific range (e.g., 45 heads), consider it fair, otherwise conclude it's not.

Sample Mean and Proportions

• Statistical Symbols:

  • U = population mean; Pi (π) = proportion of successes.

  • Initial belief about a proportion (e.g., 15% left-handed) is expressed as Pi = 0.15.

• Estimating Success:

  • Use x̄ (sample mean) and π̂ (sample proportion) as estimates.

Hypothesis Testing

• Purpose:

  • To determine if collected data supports or contradicts initial beliefs.

• Conclusion Outcome:

  • Accept or reject the null hypothesis based on statistical calculations.

  • Essential to understand what data indicates regarding initial beliefs (H0).

Data Validity and Confidence

• Data Assumptions:

  • Default expectation is a simple random sample unless stated otherwise.

  • Any biases must be documented.

Test Statistic and P-Value

• Calculating P-Value:

  • Test statistics lead to the calculation of p-value.

  • Compare p-value against alpha to decide on H0.

  • Reject H0 if p < alpha.

• Court Analogy:

  • A jury might not assert clear guilt or innocence but rather determine if the evidence is sufficient or not to conclude.

Reporting Conclusions

• Conclusion Writing:

  • Write conclusions comprehensibly for non-experts.

  • Avoid statistical jargon in conclusions: no references to null hypothesis or p-values should be present.

• Clear Statements:

  • Conclusions should directly answer the research question in simple language (e.g., "the insurance company underestimated the average claim amounts").

Assumptions and Concerns

• Key Assumptions:

  • Random sampling is a default assumption unless proven otherwise.

  • Test for normality is crucial when sample sizes are small.

• Concerns:

  • Document any concerns arising from data collection methods.

Example Case Studies

Insurance Claims

• • Original belief: average claim = $1,800.

  • New sample data resulting in a mean of $1,950, indicating potential miscalculation.

  • Calculate test statistics and compare with alpha to decide on conclusions.

Parking Time Study

• • Question posed against university’s claim of 30 minutes on average.

  • Sample data shows lesser time (25 minutes).

  • Analyze using appropriate test statistics for a valid conclusion.

Quality Control Case

• • Mean weight of objects supposed to be 16 ounces.

  • Sample recorded at 15.8 ounces.

  • Calculate test stats; conclusion indicates potential regulatory actions necessary.

Election Polling

• • Assessing the likelihood of candidate winning with proportions from surveys.

  • P-value calculates relation to alpha (0.01) to support or reject the winning prediction.

Final Remarks

• Understand the implications of not rejecting the null hypothesis and how it correlates with real-world situations.

• Ensure clarity and comprehension in reporting results to facilitate understanding for all audiences, regardless of their statistical knowledge.