Module 5 Quick Reference Card

1. Notation

2. Key Definitions

Parameter: Fixed value describing population (usually unknown)

Statistic: Calculated from sample data (varies from sample to sample)

Sampling Distribution: Distribution of a statistic across all possible samples of size n

Standard Error (SE): Standard deviation of a sampling distribution

3. Central Limit Theorem

Statement: For samples of size n from a population with μ and σ, the sampling distribution of x̄ is approximately normal with:

When CLT Applies:

• Random sampling
• Independent observations
• n ≥ 30 OR population is normal

4. Standard Error Formulas

For Sample Means:

For Sample Proportions:

5. Sample Size Effects

Key Rule: To cut SE in half, multiply n by 4

6. Sampling Dist of Proportions

Sample Proportion:

where x = number of successes

Properties:

Success-Failure Condition:

For normal approximation:

• np ≥ 10
• n(1-p) ≥ 10

7. z-score Formulas

For Sample Means:

For Sample Proportions:

Then use z-table to find probabilities

8. Step-by-Step Procedure

Finding P(x̄) or P(p̂):

1. Check conditions
   • For x̄: n ≥ 30 or normal pop
   • For p̂: np ≥ 10 and n(1-p) ≥ 10
2. Find mean and SE
   • μₓ̄ = μ, σₓ̄ = σ/√n
   • μₚ̂ = p, σₚ̂ = √(p(1-p)/n)
3. Calculate z-score
4. Find probability (z-table)

9. Common Scenarios

10. Quick Checks

Before Using CLT:

• Is sample random?
• Are observations independent?
• Is n ≥ 30 (or population normal)?

Before Normal Approx for p̂:

• Is np ≥ 10?
• Is n(1-p) ≥ 10?

Red Flags:

• Using σ instead of SE for sample means
• Forgetting √n in denominator
• Not checking CLT conditions
• Confusing parameters and statistics