Approximate a Binomial Upper Tail with a Normal Distribution

Question

Explain how to approximate the upper tail of a binomial distribution using a normal distribution.

Codemia · Accepted Answer

To approximate the upper tail of a binomial distribution using a normal distribution, we first need to understand the characteristics of the binomial distribution. The binomial distribution is defined by two parameters: the number of trials (n) and the probability of success (p). We are particularly interested in the scenario where we want to find the probability of obtaining a number of successes greater than a certain threshold.

The normal approximation to the binomial distribution is applicable when n is large and p is not too close to 0 or 1. According to the Central Limit Theorem, as n increases, the distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution. For a binomial distribution, the mean (μ) is np and the variance (σ²) is np(1-p). Therefore, we can approximate the binomial distribution with a normal distribution N(μ, σ²).

To approximate the upper tail, we calculate the z-score, which is given by:

$$ z = \frac{x - \mu}{\sigma} $$

where x is the threshold we are interested in. We then use standard normal distribution tables or software to find the probability associated with this z-score. It is important to apply a continuity correction by adjusting x by 0.5, especially when x is not an integer, to improve the accuracy of the approximation.

One major trade-off when using the normal approximation is accuracy. The approximation is generally good when np and n(1-p) are both greater than 5. If these conditions are not met, the approximation may lead to significant errors. Additionally, for very small or very large values of p, alternative methods such as the exact binomial test may be more appropriate. It is crucial to validate the approximation against actual binomial probabilities in critical applications.

For example, suppose we have a binomial distribution with n = 100 and p = 0.5. We want to find the probability of getting more than 60 successes. First, we calculate the mean and variance:
- Mean (μ) = np = 100 * 0.5 = 50
- Variance (σ²) = np(1-p) = 100 * 0.5 * 0.5 = 25

Next, we find the z-score for x = 60:
$$ z = \frac{60 - 50}{\sqrt{25}} = 2 $$

Using a standard normal distribution table, we find that the probability of z > 2 is approximately 0.0228. Thus, the probability of getting more than 60 successes is about 2.28%.

Approximate a Binomial Upper Tail with a Normal Distribution

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