Normal approximation is a statistical method that allows us to approximate complex probability distributions using the normal (bell curve) distribution. This can be useful in situations where we want to make predictions or calculations about a population, but the exact distribution of the data is unknown or too complex to work with directly.

Here are two examples of how normal approximation can be used in practice:

Example 1: Estimating the mean and standard deviation of a population

Imagine that we are studying the heights of students in a high school. We want to know the mean and standard deviation of the student heights, but we don’t have the resources to measure the heights of every single student in the school. Instead, we decide to take a sample of 100 students and measure their heights.

Since our sample is a random subset of the entire population, we can use the sample data to estimate the mean and standard deviation of the entire population. However, our sample may not be a perfect representation of the population, so we need to use statistical techniques to account for this uncertainty.

One way to do this is to use normal approximation. We can assume that the distribution of heights in the population is normal, and use the mean and standard deviation of our sample to estimate the mean and standard deviation of the population. This allows us to make predictions about the population using the normal distribution, even though we don’t have complete information about the population.

Example 2: Estimating the probability of a rare event

Suppose that we are studying the failure rates of a particular type of car battery. We want to know the probability that a battery will fail within the first year of use. However, the failure rate of these batteries is very low, and we don’t have enough data to calculate the probability directly.

In this case, we can use normal approximation to estimate the probability of a failure. First, we collect data on the failure rates of a large sample of batteries. We then assume that the distribution of failure rates in the population is normal, and use the mean and standard deviation of our sample to estimate the mean and standard deviation of the population.

Using these estimates, we can calculate the probability that a battery will fail within the first year of use using the normal distribution. This allows us to make predictions about the probability of a rare event, even though we don’t have complete data on the entire population.

Overall, normal approximation is a powerful tool that allows us to make predictions and calculations about complex probability distributions using the normal distribution. It can be especially useful when we don’t have complete information about a population, or when the distribution of the data is too complex to work with directly.