Normal Approximation To Binomial

Normal Approximation to the Binomial

Overview

  • A Binomial random variable X counts the number of successes in n independent trials, each with success probability p.

  • Binomial probabilities can be calculated using exact formulas:

    • P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

  • For cumulative probabilities, use: P(a ≤ X ≤ b) = Σ P(X = k) from a to b, often calculated using tables or statistical software.

  • When n is large and p is not near 0 or 1, binomial probabilities can be approximated using the normal distribution.

Normal Approximation Conditions

  • Works well when:

    • n is large

    • p is not too close to 0 or 1.

Visual Representation

  • The binomial distribution for large n will resemble a normal distribution.

  • Example: For Binomial(n = 100, p = 0.5), the distribution is bell-shaped like the normal curve.

Choosing the Correct Normal Distribution

Parameters of the Normal Distribution

  • A binomial random variable has parameters:

    • Mean (μ): μ = n * p

    • Variance (σ²): σ² = n * p * (1 - p)

  • Standard deviation (σ): σ = √(n * p * (1 - p)).

From Discrete to Continuous

  • Binomial random variables are discrete, with integer outcomes (e.g., P(X = 50)).

  • Normal distributions are continuous, making P(Y = 50) = 0.

  • To tackle this, we approximate using intervals (like 49.5 ≤ X ≤ 50.5) to define continuous probabilities.

Effectiveness of the Approximation

  • Example:

    • For Binomial(n=100, p=0.5): P(X = 50) is approximately 0.0796.

    • For Normal(μ = 50, σ = 5): P(49.5 ≤ Y ≤ 50.5) ≈ 0.0797.

Key Rules for Using the Approximation

  • Continuity Correction:

    • Include the entire rectangle for values of x in the interval of interest.

  • Standardization:

  • If using a table, standardize the values of X: Z = (X - μ) / σ.

  • Conditions on np and n(1 - p):

    • Ensure both np and n(1 - p) > 5 for accurate approximations.

Importance of the Last Rule

  • Example of poor approximation:

    • For n = 50 and p = 0.05, the binomial distribution looks skewed.

    • A normal approximation yields P(X ≤ 1) which differs significantly from actual value, leading to inaccuracies.

Examples

  • Example 1:

    • X is a binomial random variable with n = 30 and p = 0.4.

    • Use normal approximation for P(X ≤ 10).

    • Calculate: np = 12, n(1 - p) = 18.

    • Approximation is satisfactory.

  • Example 2: Production Line of Batteries

    • 95% reliability rate, n = 200.

    • Find P(X ≥ 195):

      • Success (working battery) = 195.

      • np = 190, n(1 - p) = 10.

    • Normalize: P(X ≥ 195) ≈ P(Z ≥ (194.5 - 190) / √[200(0.95)(0.05)]).

      • Result: P(Z ≥ 1.46) = 1 - 0.9278 = 0.0722.