Normal Approximation to the Binomial

This lesson covers the normal approximation to the binomial distribution as required by the Edexcel A-Level Mathematics specification (9MA0), Paper 3 Section A -- Statistics. You need to understand when and why this approximation is used, how to apply it with a continuity correction, and how to calculate probabilities.

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Why Approximate?

When n is large, calculating binomial probabilities involves large factorials and summing many terms. The normal distribution provides a simpler approximation.

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When to Use the Normal Approximation

The approximation is appropriate when:

• n is large
• np > 5 and n(1 - p) > 5

These conditions ensure the binomial is roughly symmetric and bell-shaped.

Exam Tip: Always check np > 5 and n(1 - p) > 5 and state your conclusion.

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Setting Up the Approximation

If X ~ B(n, p) and conditions are met:

X is approximately N(mu, sigma²) where:

• mu = np
• sigma² = np(1 - p)
• sigma = sqrt(np(1 - p))

Example

X ~ B(100, 0.4): np = 40 > 5, n(1-p) = 60 > 5. So X approximately N(40, 24).

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The Continuity Correction