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3.4.4 Normal Approximation to the Binomial Distribution

3.4.4 Normal Approximation to the Binomial Distribution

Key Concepts

  • Binomial Distribution: Discrete distribution useful for modeling the number of successes in a fixed number of independent Bernoulli trials.
  • Normal Distribution: Continuous distribution often used to approximate the binomial distribution under certain conditions.

Detailed Explanation

Conditions for Normal Approximation

  • The sample size n should be large enough such that both np and n(1-p) are greater than or equal to 5.
  • This condition ensures that the distribution of the sample proportion is sufficiently symmetrical.

Matching Parameters

  • The mean (μ) and variance (σ^2) of the normal distribution are set to np and np(1-p) respectively, to match those of the binomial distribution.

Continuity Correction

  • When using the normal approximation, a continuity correction is applied to adjust for the approximation of a discrete distribution by a continuous one. This involves adjusting the binomial variable x by 0.5 in calculations.

Practical Application

Example Calculation:

  • Suppose you flip a fair coin 100 times. What is the probability of getting exactly 50 heads?
  • Using the normal approximation:
    • Mean (μ) = 50, Variance (σ^2) = 25 (since p=0.5)
    • To find P(X=50), use P(49.5 < X < 50.5) with the normal approximation.

Using Templates for Calculation:

  • Binomial probabilities and their normal approximations can be computed using statistical software or Excel templates, which simplify the process of determining these probabilities.
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