Confidence Intervals (Sample Means)
Explain bootstrapping as resampling from the sample to approximate the sampling distribution, and distinguish this from repeated sampling from the population
Identify standard error as a measure of sampling variability and interpret its relationship to sample standard deviation, sample size, and precision of point estimates
Explain the Central Limit Theorem and explain when sampling distributions of means are approximately normal even when populations are not
Interpret confidence intervals correctly, including what "95% confidence" means and does NOT mean in terms of repeated sampling, and identify common interpretation errors
Confidence Intervals (Sample Proportions)
Distinguish between the sampling distribution of a sample proportion and the population distribution
Apply the success-failure condition (np ≥ 10 and n(1-p) ≥ 10) to evaluate when normal approximation is appropriate for proportions
Identify alternative approaches (i.e., logit transformation) when the success-failure condition is violated
Understanding Odds and Transformations
Explain odds and and how log odds (i.e., logit) transformations address boundary constraints for proportions
Recognize logit transformation in published methods and understand why it's appropriate for proportions with boundary concerns
Comparisons of Proportions
Distinguish between risk ratios and odds ratios, including when they are approximately equal (rare outcome assumption, <10%) and when it is appropriate to interpret each
Critical Evaluation of Published Research
Evaluate confidence interval interpretations for common errors including probability statements about parameters, inappropriate use of normal approximation, and overstated precision
Apply integrated understanding of all concepts to assess validity, precision, and appropriate interpretation of findings in published research
Bootstrap Vocabulary:
Sampling with replacement - Each observation can be selected multiple times in a single bootstrap sample
Bootstrap sample - A sample drawn with replacement from the original sample, same size as original
Bootstrap distribution - The distribution of a statistic across many bootstrap samples; approximates the sampling distribution
Bootstrap standard error - Standard deviation of the bootstrap distribution; estimates SE without parametric assumptions
Standard Error and CI Vocabulary:
Precision - How narrow the confidence interval is; narrower intervals = more precise estimates
Standard error (SE) - Measure of how much sample statistics vary across repeated samples; quantifies sampling variability
Standard deviation vs. Standard error - SD describes variability in data; SE describes variability in statistics
Margin of error - The "plus or minus" value added/subtracted from the sample statistic to create a confidence interval
Confidence level - The proportion of confidence intervals that would contain the true parameter if we repeated the study many times (e.g., 95%)
Confidence interval - A range of plausible values for the population parameter, constructed from sample data and accounting for sampling variability
Coverage probability - The actual long-run proportion of CIs that capture the true parameter
Normal Approximation Vocabulary:
Central Limit Theorem (CLT) - The sampling distribution of the sample mean becomes approximately normal as sample size increases, regardless of population distribution
Normal approximation - Using the normal distribution to approximate the sampling distribution when CLT conditions are met
Success-failure condition - For proportions: normal approximation is valid when np ≥ 10 AND n(1-p) ≥ 10
Boundary/edge effects - Problems that arise when proportions are near 0 or 1, causing the normal approximation to perform poorly
Odds and Transformations:
Odds - The ratio of the probability of an outcome occurring to the probability of it not occurring; odds = p/(1-p)
Odds transformation - Converting proportions from (0,1) scale to (0,∞) scale, removing upper boundary constraint
Log-odds (logit) - Natural logarithm of odds; transforms proportions to (-∞,+∞) scale, removing all boundary constraints
Logit transformation - Mathematical transformation used in logistic regression and CI construction for proportions; handles boundary issues
Risk vs. Odds Vocabulary:
Risk (probability) - The proportion of people in a group who experience an outcome; P(outcome)
Risk ratio (relative risk, RR) - The ratio of risk in one group to risk in another group
Odds ratio (OR) - The ratio of odds in one group to odds in another group
Rare outcome assumption - When outcome is rare (typically <10%), odds ratio ≈ risk ratio numerically
Exaggeration bias - When odds ratios are interpreted as risk ratios for common outcomes, effects appear larger than they are
Terms to Connect from Last Week:
Sampling distribution → Now we approximate it via bootstrap or assume it's normal via CLT
Sampling variability → Quantified by standard error
Parameter vs. Statistic → CI estimates parameter using statistic
Repeated sampling → Bootstrap approximates this; CI interpretation refers to this