Test of Significance

Administrative Announcements

Welcome back from Thanksgiving break.
This session covers the last chapters, specifically Chapter 7 and Chapter 23.
Exam Schedule:
- Final exam review session on Monday next week.
- Final exam will occur on December 12, opening at 3:30 PM and accessible for 24 hours until December 13, 3:30 PM.
Exam 3 scores have been finalized, and extra credit points added to Canvas.
Average score for Exam 3 was approximately 83-84%.
No significant issues were reported with exam item statistics; majority of items were answered correctly (above 50% for participation).
Outstanding tests & quizzes remaining:
- Two Learning Curve quizzes for Chapters 7 and 23 (only Chapter 23 has a post-class quiz).

Exam 4 will include a mix of new content and cumulative items from Exams 1, 2, and 3.
Expected format: 60-item exam, approximately 15 items will be cumulative.
Larry mentioned using Respondus LockDown Browser for online administration.
Exam availability: December 12, accessible for 24 hours.

Importance of setting up a null hypothesis (H₀) and an alternative hypothesis (H₁).
- Example: Testing preference for fresh brewed versus instant coffee versus skepticism (H₀: population proportion = 0.5).
- H₁: population proportion > 0.5 (people prefer fresh brewed coffee).
P-value: Probability of obtaining a sample statistic as extreme as the observed value, given that the null hypothesis is true.
- Defined as ( P(X \geq x | H₀ \text{ is true}) ).
- In significance tests, lower p-values (e.g., <0.05) are indicative of evidence against the null hypothesis.

Set up null and alternative hypotheses.
Determine the sample statistic (e.g., sample proportion or mean).
Compute the standard error (SE).
- For sample proportions: ( SE = \sqrt{\frac{p(1-p)}{n}} ).
- For sample means: ( SE = \frac{s}{\sqrt{n}} ) where s is sample standard deviation.
Convert the sample statistic into a standard score (z-score or t-test statistic).
- Standard score formula: ( Z = \frac{x - \mu}{SE} ) or for t-tests ( T = \frac{X̄ - µ}{SE} ).
Look up the p-value in statistical tables based on the calculated standard score.
Compare p-value to significance level (usually 0.05) to decide whether to reject H₀.

Previous example involved determining the p-value for a null hypothesis about coffee preference.
Calculated:
- Sample proportion p̂ = 0.72.
- Null hypothesis proportion H₀ = 0.5 leading to SE = 0.0707.
- Converted p̂ to z-score, found it to be a test statistic of 3.1.
- Retrieved percentile from statistical tables corresponding to p-value (0.001).

Introduced t-tests for scenarios requiring a sample mean rather than proportion.
Reviewed a pregnancy length example:
- Null Hypothesis: ( H₀: \mu = 280 ) days.
- Alternate Hypothesis: ( H₁: \mu < 280 ) days.
- Sample mean of 275 days with sample standard deviation of 10 days translates to:
- SE calculation = ( \frac{10}{\sqrt{95}} \approx 1.03 ).
- Calculated t score leads to a standard score leading to a very low p-value (<0.0003).

T-test mechanics:
- Formula for t-test statistic: [ T = \frac{X̄ - µ}{s / \sqrt{n}} ]
- Explains determination of statistical significance based on observed versus population means.
Importance of recognizing sample size in detecting significant differences; increasing sample sizes lowers SE and narrows the distribution peak aiding significance detection.

Described distinctions in testing based on directional hypotheses (one-tailed) versus non-directional hypotheses (two-tailed).
- One-tailed tests concentrate critical value to one side; allows for lower test statistic needed for significance.
- Example shown delineating critical value changes affecting what constitutes evidence against H₀.

Emphasis on central limit theorem and significance testing authority in how we frame hypotheses and interpret results.
Reminder about applying statistical inference to make robust conclusions from sample data while recognizing limitations and variability.