Approximating the Binomial with the Normal Distribution

Key Concepts

• Approximation Equation: Use the normal distribution to approximate the binomial distribution with ( \mu = np ) and ( \sigma = \sqrt{npq} ).
• Z Formula Adaptation: By substituting ( \mu ) and ( \sigma ) into the Z formula, we can transition from a discrete to a continuous function.
• Continuity Correction: Adjust by ±0.5 to account for the discrete nature of the binomial distribution.

When to Use

• Accurate when ( n ) is large and ( p ) is not close to 0 or 1.
• Decent Approximation even when ( n ) is small and ( np ) is reasonably close to 1.

Example Problem

• Given: Binomial distribution with 15 trials (( n = 15 )), ( p = 0.4 ), find ( P(X = 4) ).
• Find: ( q = 0.6 ) (since ( q = 1 - p )).

Binomial Probability Calculation

• Use the binomial distribution table for cumulative probability calculation:
  • ( P(X \leq 4) - P(X \leq 3) = 0.2173 - 0.095 )
  • Result: ( P(X = 4) = 0.1268 ).

Normal Approximation

• Parameters:
  • ( \mu = np = 6 )
  • ( \sigma^2 = npq = 3.6 )
  • Continuity correction for ( P(X = 4) ): evaluate ( 3.5 \leq X \leq 4.5 )
• Z Values Calculation:
  • ( Z_1 = \frac{3.5 - 6}{\sqrt{3.6}} = -1.32 )
  • ( Z_2 = \frac{4.5 - 6}{\sqrt{3.6}} = -0.79 )
• Probability from Z Table:
  • ( P(Z < -0.79) = 0.2148 )
  • ( P(Z < -1.32) = 0.0934 )
  • Result: ( P(X = 4) = 0.1214 ) or 12.14%.

Conclusion

• Comparison: Binomial result (12.68%) and Normal approximation (12.14%) are similar.
• Purpose: Simplifies calculations using the well-known normal distribution for discrete scenarios like binomial problems.