Exploring the Normal Approximation to the Binomial Distribution, this section will delve into its conditions, application, and practical problem-solving.

Normal Approximation to Binomial Distribution

1. Conditions for Normal Approximation:

• np > 5 : Ensures sufficient number of successes.
• n(1-p) > 5 : Ensures sufficient number of failures.
• Purpose: To ensure a symmetric distribution suitable for normal approximation.

2. Applying Continuity Correction Factor:

• For $P(X \leq x)$: Use $P(X \leq x + 0.5)$.
• For $P(X \geq x)$: Use P(X > x - 0.5).
• For $P(X = x)$: Use P(x - 0.5 < X < x + 0.5).
• Purpose: To align the discrete binomial distribution with the continuous normal distribution for more accurate results.

Examples

Example 1: $P(X \geq 10)$ for Binomial Distribution $n = 30, p = 0.2$

• Conditions Check: $np = 6 , n(1-p) = 24$ (Both > 5, condition met).
• Continuity Correction: Adjust to P(X > 9.5).
• Z-Score Calculation: $Z = \frac{9.5 - 6}{\sqrt{4.8}} \approx 1.60$.
• Probability: $P(X \geq 10) \approx 0.0551$ (5.51% chance for 10+ successes)
• Graph:

Example 2: $P(3 \leq X \leq 8)$ for Binomial Distribution $n = 50, p = 0.1$

• Conditions Check: $np = 5 , n(1-p) = 45$ (Both > 5, condition met).
• Continuity Correction: Adjust to $P(2.5 \leq X \leq 8.5)$.
• Z-Scores: $Z_1 = -1.18$ (for X = 2.5), $Z_2 = 1.65$ (for X = 8.5).
• Probability: $P(3 \leq X \leq 8) \approx 0.8312$ (83.12% chance for 3-8 successes).
• Graph: