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📊 SAMPLING - Theory & Formulas

Cambridge AS & A Level Mathematics

📖 Part 1: Introduction to Sampling

Key Definitions

Population: Complete set of ALL items of interest

Sample: Part of the population (size = n)

Representative Sample: Accurately reflects population characteristics

Biased Sample: Does NOT properly represent population

Random Sample: ALL possible samples of size n have equal probability of selection

💡 Why Use Samples?

Reason	Example
💰 Cost-Effective	Test 50 products vs 10,000
⏰ Time-Saving	Survey 100 people vs millions
🔨 Destructive Testing	Crash testing helmets
🌍 Impossible to Survey All	All fish in the ocean

🎲 Random Sampling Methods

Using Random Number Tables:

• Number population: 000 to 499 (for 500 items)
• Pick starting point in table
• Read digits matching your numbering
• Ignore numbers outside range
• Ignore repeats

Using Excel:

=RAND() → Random number 0 to 1
=INT(250*RAND())+1 → Random integer 1 to 250

⚠️ Types of Bias

Type	Example
Location Bias	Survey only at gym about exercise
Time Bias	Survey Monday afternoon only
Leading Questions	"Don't you agree that...?"
Small Sample	Ask only 10 people

📊 Part 2: Distribution of Sample Means

Sample Mean (X̄)

Definition: Average of all observations in sample

X̄ = (x₁ + x₂ + ... + xₙ) / n

⚡ Different samples → Different sample means!

🔑 FUNDAMENTAL FORMULAS

1. Expected Value of Sample Mean

E(X̄) = μ

Sample mean equals population mean!

2. Variance of Sample Mean

Var(X̄) = σ² / n

Variance decreases as sample size increases!

3. Standard Deviation of Sample Mean

SD(X̄) = σ / √n

Also called: Standard Error (SE)

⭐ THE CENTRAL LIMIT THEOREM (CLT)

Most Important Theorem in Statistics!

X̄ ~ N(μ, σ²/n)

When n is large (usually n ≥ 30)

What it means:

• Sample means follow NORMAL distribution
• Mean = μ (population mean)
• Variance = σ²/n
• Works EVEN IF original population is NOT normal!

📏 How Large Should n Be?

Original Population	Minimum n
Normal Distribution	n ≥ 5
Approximately Symmetric	n ≥ 20
Skewed Distribution	n ≥ 30
Any Distribution (Safe)	n ≥ 50

📋 Complete Formula Summary

Concept	Formula
Population Mean	μ = E(X)
Population Variance	σ² = Var(X)
Sample Mean	X̄ = Σxᵢ / n
Expected Value	E(X̄) = μ
Variance	Var(X̄) = σ²/n
Standard Error	SE = σ/√n
Distribution (CLT)	X̄ ~ N(μ, σ²/n)
Z-Score	Z = (X̄ - μ)/(σ/√n)

🔢 Working with Sample Totals

If T = sample total of n observations:

T = n × X̄

E(T) = nμ
Var(T) = nσ²
T ~ N(nμ, nσ²) when n is large

⚡ Continuity Correction

For DISCRETE distributions (Binomial, Poisson):

Continuity Correction = ± 1/(2n)

NOT ± 1/2

Probability	Correction
P(X̄ < a)	P(X̄ < a - 1/(2n))
P(X̄ ≤ a)	P(X̄ < a + 1/(2n))
P(X̄ > a)	P(X̄ > a + 1/(2n))
P(X̄ ≥ a)	P(X̄ > a - 1/(2n))

📝 Problem Solving Steps

Step 1: Identify μ, σ² (or σ), n

Step 2: Check if CLT applies (n ≥ 30 or population normal)

Step 3: Write distribution: X̄ ~ N(μ, σ²/n)

Step 4: Calculate SE: σ/√n

Step 5: Find Z-score: Z = (X̄ - μ)/(σ/√n)

Step 6: Use normal tables to find probability

Step 7: Apply continuity correction if discrete

💡 Example 1: Pears in Bags

Problem: Pears: μ=45g, σ²=52g², n=6. Find P(Total > 300g)

Solution: Total > 300g means X̄ > 50g

X̄ ~ N(45, 52/6) = N(45, 8.67)
SE = √8.67 = 2.94
Z = (50-45)/2.94 = 1.70
P(Z > 1.70) = 1 - 0.9554 = 0.0446

Answer: 4.46%

💡 Example 2: Water for Exercise

Problem: μ=500ml, σ=50ml, n=25. 13L available. Enough?

Solution: Need X̄ < 520ml (13000/25)

X̄ ~ N(500, 100) (σ²/n = 2500/25)
SE = 10
Z = (520-500)/10 = 2.0
P(Z < 2.0) = 0.9772

Answer: 97.72% probability

💡 Example 3: Binomial (with Continuity Correction)

Problem: X ~ B(60, 0.25), n=50, Find P(X̄ ≤ 16)

Solution:

μ = 60×0.25 = 15
σ² = 60×0.25×0.75 = 11.25
X̄ ~ N(15, 11.25/50) = N(15, 0.225)
Correction: +1/(2×50) = +0.01
P(X̄ ≤ 16) = P(X̄ < 16.01)
Z = (16.01-15)/√0.225 = 2.13
P(Z < 2.13) = 0.983

Answer: 98.3%

🎯 Quick Reference Card

If you know...	You can find...
μ, σ², n	E(X̄) = μ, Var(X̄) = σ²/n
Population normal	X̄ is normal for ANY n
n ≥ 30	X̄ ~ N(μ, σ²/n) by CLT
Discrete distribution	Use continuity correction ±1/(2n)
Sample total T	T ~ N(nμ, nσ²)

✅ Key Takeaways

• Random sampling: Everyone has equal chance
• E(X̄) = μ: Sample mean targets population mean
• Var(X̄) = σ²/n: Bigger sample = smaller variance
• CLT: X̄ is approximately normal when n is large
• Works for ANY distribution!
• Continuity correction: ±1/(2n) for discrete