Descriptive Statistics
Describe a Distribution: SOCS
- Shape (symmetric, skewed L/R, unimodal, bimodal)
- Outliers / unusual features
- Center (mean or median)
- Spread (SD, IQR, range)
Key Formulas
z = (x - mean) / SD
IQR = Q3 - Q1
Outlier if: x < Q1 - 1.5(IQR) or x > Q3 + 1.5(IQR)
68-95-99.7 Rule (Normal Distributions)
- 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD
Skewed right: mean > median | Skewed left: mean < median
Linear Regression
y-hat = a + bx
b = r(sy / sx) a = y-bar - b(x-bar)
Residual = observed - predicted = y - y-hat
r² = proportion of variation in y explained by x
Interpretation Templates
Slope: For each additional [unit x], the predicted [y] increases/decreases by [b] [units y].
r²: [r²]% of the variability in [y] is explained by the linear relationship with [x].
Residual plot: Good model = random scatter. Pattern = bad fit.
Study Design
| Feature | Observational | Experiment |
|---|
| Treatment? | No | Yes (imposed) |
| Causation? | NO (association only) | YES (if well-designed) |
| Key mechanism | Random selection | Random assignment |
Sampling Methods
- SRS: Every group of n has equal chance
- Stratified: Divide into strata → SRS from each
- Cluster: Randomly select whole clusters
- Systematic: Every k-th individual
Bias Types
- Undercoverage: Some groups excluded
- Nonresponse: Selected people don't respond
- Response: Inaccurate answers (wording, lying)
- Voluntary response: Self-selection
Probability
P(A or B) = P(A) + P(B) - P(A and B)
P(A and B) = P(A) × P(B|A)
If independent: P(A and B) = P(A) × P(B)
P(A|B) = P(A and B) / P(B)
Random Variables
μ(aX+b) = aμX + b σ(aX+b) = |a|σX
If independent: Var(X±Y) = Var(X) + Var(Y)
Always ADD variances, even when subtracting!
Binomial: X ~ B(n, p)
μ = np σ = √(np(1-p))
Geometric: trials until first success
μ = 1/p
Sampling Distributions
| p-hat (proportion) | x-bar (mean) |
|---|
| Mean | p | μ |
| SD | √(p(1-p)/n) | σ/√n |
| Normal when | np≥10, n(1-p)≥10 | Pop Normal OR n≥30 (CLT) |
CLT: For large n, x-bar is approximately Normal regardless of population shape.
Inference Framework (4 Steps)
- STATE: Parameter, hypotheses (or confidence level)
- PLAN: Name test, check conditions (Random, 10%, Large Counts or Normal)
- DO: Calculate statistic, p-value (or interval)
- CONCLUDE: Decision + interpretation in context
Conditions Checklist
- Random sample or random assignment
- 10% condition: n ≤ 10% of population
- Large Counts (proportions): np≥10, n(1-p)≥10
- Normal/Large Sample (means): n≥30 or population Normal
Inference Formulas
Proportions (z-procedures)
CI: p-hat ± z*√(p-hat(1-p-hat)/n)
Test: z = (p-hat - p0) / √(p0(1-p0)/n)
Means (t-procedures, df = n-1)
CI: x-bar ± t*(s/√n)
Test: t = (x-bar - μ0) / (s/√n)
Chi-Square (df varies)
χ² = Σ(O-E)²/E
Expected = (row total × col total) / grand total
GOF: df = categories - 1
Two-way: df = (r-1)(c-1)
Slope (df = n-2)
t = b / SE(b) CI: b ± t* × SE(b)
Common z* and Interpretation Templates
| Confidence Level | z* |
|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
Interpretation Templates
CI: We are [C]% confident that the true [parameter] of [context] is between [lower] and [upper].
p-value: Assuming [H0 is true], the probability of getting a result as extreme as or more extreme than what we observed is [p-value].
Conclusion (reject): Since p-value < α, we reject H0. There IS convincing evidence that [Ha in context].
Conclusion (fail to reject): Since p-value > α, we fail to reject H0. There is NOT convincing evidence that [Ha in context].
Which Test Do I Use? (Decision Guide)
| Data Type | Scenario | Procedure |
|---|
| Categorical (proportions) | One proportion | 1-prop z-test / z-interval |
| Compare two proportions | 2-prop z-test / z-interval |
| Distribution / association of categories | Chi-square (GOF / Independence / Homogeneity) |
| Quantitative (means) | One mean | 1-sample t-test / t-interval |
| Two independent means | 2-sample t-test / t-interval |
| Paired data (before/after, same subject) | Matched pairs t-test |
| Linear relationship (slope) | t-test for slope / CI for slope |
Type I/II Errors & Power
| H0 True | H0 False |
|---|
| Reject H0 | Type I Error (α) | Correct! (Power) |
| Fail to reject H0 | Correct! | Type II Error (β) |
- Power = 1 - β (prob of correctly rejecting false H0)
- Power increases with: larger n, larger α, larger true effect
Never Say These on the Exam
- NEVER: "Accept the null hypothesis" → Say "fail to reject"
- NEVER: "Prove" anything → Say "convincing evidence"
- NEVER: "There is a [C]% probability the true value is in the CI"
- NEVER: "Correlation implies causation" (unless experiment)
- NEVER: "The data is Normal" for checking CLT → Say "sampling distribution is approx. Normal"
- NEVER: Forget CONTEXT in interpretations
Conditions Quick Reference
Proportions
- Random, 10%, Large Counts: np≥10 and n(1-p)≥10
- For tests: use p0. For intervals: use p-hat.
Means
- Random, 10%, Normal/Large Sample (n≥30 or no strong skew)
Chi-Square
- Random, 10%, ALL expected counts ≥ 5
Regression Slope (LINER)
- Linear, Independent, Normal residuals, Equal variance, Random