Z-Intervals (Known Sigma)
| Item | Description |
|---|
CI = xbar +/- z*(sigma/sqrt(n)) | For population mean when sigma is known — normally distributed OR n>30 |
z = 1.645 for 90% CI | Standard normal critical values |
z = 1.96 for 95% CI | Most common — 95% confidence level |
z = 2.576 for 99% CI | Wider interval for higher confidence |
Margin of Error = z*sigma/sqrt(n) | Half-width of confidence interval |
Sample Size for desired MOE: n = (z*sigma/MOE)^2 | How many observations needed for given precision |
T-Intervals (Unknown Sigma)
| Item | Description |
|---|
CI = xbar +/- t*(s/sqrt(n)) | Use when population std dev is unknown — most real-world situations |
t depends on degrees of freedom (n-1) | Wider than z for small n, converges to z as n grows |
When n > 30 | t approaches z — z-interval is acceptable approximation |
CI for Proportion: phat +/- z*sqrt(phat*(1-phat)/n) | For binomial proportions — requires np>=10, n(1-p)>=10 |
CI for Difference of Means | Independent groups with t — Welch's df recommended |
Bootstrap Confidence Intervals
| Item | Description |
|---|
Bootstrap Principle | Resample from data with replacement — no distribution assumptions needed |
Percentile Method | Take 2.5th and 97.5th percentiles of bootstrap distribution |
BCa (Bias-Corrected Accelerated) | Adjusts for bias and skewness — more accurate than percentile |
When to Use Bootstrap | Non-normal data, small samples, complex statistics (medians, ratios) |
Bootstrap Replicates | Typically 1,000-10,000 resamples — more is better but computationally costly |
Pitfall | Bootstrap cannot fix a bad sample — GIGO (Garbage In, Garbage Out) |
Interpretation Guide
| Item | Description |
|---|
Correct Interpretation | We are 95% confident the true parameter is between [L, U] |
Wrong Interpretation | There is a 95% probability the parameter is between [L, U] |
CI Width Factors | Larger n = narrower; larger confidence = wider; larger variance = wider |
CI Overlap Rule of Thumb | Rough guide: if 95% CIs don't overlap, difference is significant |
Non-significance | CI containing the null value (0 for difference, 1 for ratios) = not significant |
Pro Tip: A 95% confidence interval does NOT mean there's a 95% chance the true parameter is in the interval. It means: in repeated sampling, 95% of intervals constructed this way capture the true value.