Normal Approximation to Binomial

Statistics · Axiom Academy

LESSON Normal Approximation to Binomial Understanding when and how to use the normal distribution to approximate binomial probabilities, including continuity correction 1. When to Use Normal Approximation The normal approximation to the binomial is valid when the sample size is large enough that the binomial distribution becomes approximately symmetric. The rule of thumb is: Use normal approximation when: np ≥ 10 (expected number of successes is at least 10) n(1-p) ≥ 10 (expected number of failures is at least 10) When both conditions are met, the binomial distribution is sufficiently symmetric for normal approximation. For example, if n = 100 and p = 0.15: n(1-p) = 100(0.85) = 85 ≥ 10 ✓ Normal approximation is appropriate! 2. Setting Up the Approximation Once we've verified the conditions are met, we convert the binomial distribution to a normal distribution using the same mean and standard deviation: Standard Deviation: σ = √(np(1-p)) X ~ N(μ, σ²) where μ = np and σ = √(np(1-p)) Example: Suppose X ~ Binomial(n=100, p=0.3). Then: σ = √(100(0.3)(0.7)) = √21 ≈ 4.58 We approximate with X ~ N(30, 21) Since binomial is discrete and normal is continuous, we need a continuity correction. We adjust discrete values by ±0.5 to account for the area under the continuous curve: P(X = k) → P(k - 0.5 < X < k + 0.5) Example: For X ~ Binomial(100, 0.3), find P(X ≤ 35): Check: np = 30 ≥ 10 ✓, n(1-p) = 70 ≥ 10 ✓ Apply continuity correction: P(X ≤ 35) → P(X < 35.5)

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