Normal approximation (as applicable) - Statistics AP Study Notes
Overview
# Normal Approximation to the Binomial Distribution ## Key Learning Outcomes Students learn to approximate binomial distributions B(n,p) using the normal distribution N(np, np(1-p)) when n is large and p is not close to 0 or 1, typically applying the rule that np > 5 and n(1-p) > 5. The continuity correction is essential, adding or subtracting 0.5 to discrete values when converting to the continuous normal distribution (e.g., P(X ≤ 8) becomes P(Y < 8.5)). This approximation simplifies calculations for large-sample binomial problems that would otherwise be computationally intensive. ## Exam Relevance This topic frequently appears in AS/A-Level examinations, requiring students to justify when the approximation is valid, apply continuity corrections correctly, and use normal distribution
Core Concepts & Theory
Normal Approximation to the Binomial Distribution allows us to use the continuous normal distribution to approximate discrete binomial probabilities when certain conditions are met. This technique is essential when dealing with large sample sizes where binomial calculations become impractical.
Key Conditions for Approximation:
- For a binomial distribution X ~ B(n, p), we can approximate with N(μ, σ²) when np > 5 AND nq > 5 (where q = 1 - p)
- These conditions ensure the distribution is sufficiently symmetric and bell-shaped
Critical Parameters:
- Mean: μ = np
- Variance: σ² = npq
- Standard deviation: σ = √(npq)
Continuity Correction is the most vital adjustment when using normal approximation. Since we're approximating a discrete distribution with a continuous one, we must account for the "gaps" between discrete values:
- P(X = r) becomes P(r - 0.5 < X < r + 0.5)
- P(X ≤ r) becomes P(X < r + 0.5)
- P(X < r) becomes P(X < r - 0.5)
- P(X ≥ r) becomes P(X > r - 0.5)
- P(X > r) becomes P(X > r + 0.5)
Memory Aid (COIN): Continuity correction Obligatory, Inequalities Need adjustment. Always add/subtract 0.5 based on whether you're including or excluding the boundary value.
Cambridge Note: The syllabus explicitly requires understanding when approximation is appropriate and proper application of continuity correction—this appears in nearly every exam paper.
Detailed Explanation with Real-World Examples
Why Normal Approximation Matters:
Calculating P(X ≤ 80) when X ~ B(200, 0.35) directly requires summing 81 individual binomial probabilities—practically impossible without technology. Normal approximation provides an elegant, accurate alternative.
Real-World Application 1: Quality Control
A factory produces 10,000 light bulbs daily with a 2% defect rate. Finding the probability that fewer than 180 bulbs are defective using binomial formula requires calculating thousands of terms. Instead: n = 10,000, p = 0.02, so np = 200 (>5) and nq = 9,800 (>5). We approximate with N(200, 196), apply continuity correction for "fewer than 180" as P(X < 179.5), then standardize.
Real-World Application 2: Election Polling
Imagine 500 voters surveyed, 48% support a candidate. To find the probability that exactly 250 support them, we'd use X ~ B(500, 0.48). Check: np = 240 (>5), nq = 260 (>5) ✓. Approximate with N(240, 124.8). For "exactly 250", apply P(249.5 < X < 250.5).
The Factory Analogy:
Think of discrete binomial values as individual boxes on a warehouse floor. The normal curve is like draping a smooth sheet over these boxes. The continuity correction ensures we're covering the entire box when we want to include it, not just its center point. Without correction, we'd be measuring from box-center to box-center, missing half of each end box!
Important: Normal approximation is an estimate. It works brilliantly for large n but becomes less accurate as n decreases or p approaches 0 or 1.
Worked Examples & Step-by-Step Solutions
**Example 1:** A biased coin shows heads with probability 0.4. It's tossed 100 times. Find P(35 ≤ X ≤ 45) where X is the number of heads. **Solution:** *Step 1 – Check conditions:* np = 100(0.4) = 40 > 5 ✓ nq = 100(0.6) = 60 > 5 ✓ Approximation valid. *Step 2 – Calculate parameters:* μ = np = 40 σ...
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Key Concepts
- Normal Distribution: A special bell-shaped curve that describes many natural phenomena and is easy to work with mathematically.
- Binomial Distribution: Describes the number of 'successes' in a fixed number of independent trials, where each trial has only two outcomes.
- Normal Approximation: Using the Normal distribution to estimate probabilities for a Binomial distribution when certain conditions are met.
- Conditions for Normal Approximation: Rules that must be satisfied (n*p ≥ 10 and n*(1-p) ≥ 10) for the approximation to be valid.
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Exam Tips
- →Always state and check the conditions (n*p ≥ 10 and n*(1-p) ≥ 10) for using Normal approximation on your exam.
- →Clearly define the mean (μ = n*p) and standard deviation (σ = √(n*p*(1-p))) you are using for the Normal distribution.
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