Normal approximation to the binomial

Definition

Normal approximation to the binomial is a method used to approximate the probabilities of a binomial distribution using the normal distribution when the sample size is large and the probability of success is neither very close to 0 nor 1.

5 Must Know Facts For Your Next Test

1. The normal approximation can be applied when both $np \geq 10$ and $n(1-p) \geq 10$, where $n$ is the number of trials and $p$ is the probability of success.
2. To use the normal approximation, continuity correction is applied by adjusting the discrete binomial variable by ±0.5.
3. The mean ($\mu$) of the approximating normal distribution is given by $np$, and its standard deviation ($\sigma$) is given by $\sqrt{np(1-p)}$.
4. This approximation simplifies complex binomial probability calculations, making them more feasible for large samples.
5. Normal approximation becomes more accurate as the sample size increases.

Review Questions

• When can you use normal approximation for a binomial distribution?
• What are mean ($\mu$) and standard deviation ($\sigma$) in terms of $n$ and $p$ for the approximating normal distribution?
• Why do we apply continuity correction in normal approximation?

Related terms

Central Limit Theorem: A theorem that states that, given a sufficiently large sample size from a population with a finite level of variance, the sample means will be approximately normally distributed.

Binomial Distribution: A discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials with constant probability of success.

Continuity Correction: An adjustment made when approximating a discrete distribution with a continuous one, often adding or subtracting 0.5 to improve accuracy.