Elementary Statistical Theory Comprehensive
Guide
Table of Contents
1. Sampling Theory
2. Sampling Distributions
3. Confidence Intervals
4. Hypothesis Testing
5. Chi-Square Tests
6. ANOVA
7. Correlation
1. Sampling Theory
Key Concepts
● Population: Complete set of items of interest
● Sample: Subset of the population used to make inferences
● Sampling Frame: List of all population members
Sampling Methods
Method Description When to Use Formula
Simple Every member has equal When population is N/A
Random chance homogeneous
Systemati Select every k-th item Large, ordered k = N/n
c populations
Stratified Divide population into strata, When subgroups differ nᵢ =
sample from each significantly (Nᵢ/N)×n
, Cluster Divide into clusters, randomly Geographically dispersed N/A
sample clusters populations
Multistage Combination of methods Large, complex N/A
populations
Example Problem
Scenario: You need to survey 1,000 households in a city of 100,000. How would you
sample?
Solution:
1. Obtain a sampling frame (e.g., voter registry)
2. Use systematic sampling: k = 100,000/1,000 = 100
3. Select every 100th household after random start
2. Sampling Distributions
Central Limit Theorem
For large samples (n ≥ 30), the sampling distribution of the mean is approximately
normal regardless of population distribution.
Formulas:
● Mean of sampling distribution: μₓ̄ = μ
● Standard error (infinite pop.): σₓ̄ = σ/√n
● Finite population correction: σₓ̄ = (σ/√n)×√[(N-n)/(N-1)]
t-Distribution
Use when:
● Sample size small (n < 30)
Guide
Table of Contents
1. Sampling Theory
2. Sampling Distributions
3. Confidence Intervals
4. Hypothesis Testing
5. Chi-Square Tests
6. ANOVA
7. Correlation
1. Sampling Theory
Key Concepts
● Population: Complete set of items of interest
● Sample: Subset of the population used to make inferences
● Sampling Frame: List of all population members
Sampling Methods
Method Description When to Use Formula
Simple Every member has equal When population is N/A
Random chance homogeneous
Systemati Select every k-th item Large, ordered k = N/n
c populations
Stratified Divide population into strata, When subgroups differ nᵢ =
sample from each significantly (Nᵢ/N)×n
, Cluster Divide into clusters, randomly Geographically dispersed N/A
sample clusters populations
Multistage Combination of methods Large, complex N/A
populations
Example Problem
Scenario: You need to survey 1,000 households in a city of 100,000. How would you
sample?
Solution:
1. Obtain a sampling frame (e.g., voter registry)
2. Use systematic sampling: k = 100,000/1,000 = 100
3. Select every 100th household after random start
2. Sampling Distributions
Central Limit Theorem
For large samples (n ≥ 30), the sampling distribution of the mean is approximately
normal regardless of population distribution.
Formulas:
● Mean of sampling distribution: μₓ̄ = μ
● Standard error (infinite pop.): σₓ̄ = σ/√n
● Finite population correction: σₓ̄ = (σ/√n)×√[(N-n)/(N-1)]
t-Distribution
Use when:
● Sample size small (n < 30)
