Statistics Key Concepts: Population, Sample, Variables, and Hypothesis Testing

Population

Entire group of interest.

Sample

Subset of the population used to estimate parameters.

Parameter

Numerical summary of a population.

Statistic

Numerical summary of a sample.

Quantitative variables

Continuous and discrete variables, e.g., height.

Categorical variables

Nominal and ordinal variables, e.g., gender, rankings.

5-number summary

Minimum, Q1, Median, Q3, Maximum.

Empirical Rule

68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD for a normal distribution.

Z confidence interval formula

x̄ ± z(σ/√n).

t confidence interval formula

x̄ ± t(s/√n).

Confidence interval for a proportion

p̂ ± z√[p̂(1-p̂)/n].

Hypothesis testing steps

1) State H₀, Hₐ; 2) Choose α; 3) Choose test; 4) Compute statistic & p-value; 5) Compare p-value to α; 6) Conclude.

Paired t-test

Used when comparing means from the same subjects or matched pairs.

Wilcoxon Rank Sum test

Used to compare medians of two independent samples without assuming normality.

ANOVA F-test formula

F = MSbetween / MSwithin.

Chi-Square formula

χ² = Σ((O - E)² / E).

Assumptions of simple linear regression

Linearity, independence, homoscedasticity, and normally distributed residuals.

Kruskal-Wallis test

Used when comparing more than two groups without assuming normality.

Sample size requirements for proportion z-test

nπ ≥ 10 and n(1-π) ≥ 10.

Pooled p̂ for two-proportion z-test

p̂pooled = (x₁ + x₂) / (n₁ + n₂).

Decision rule if a CI contains the null value

Fail to reject H₀.

Factors increasing power of a statistical test

Larger sample size, larger effect size, smaller σ, larger α.

Sign Test

Used for median tests when data is non-normal and small sample size.