A standard normal distribution is a probability distribution that follows a bell-shaped curve and has a mean of 0 and a standard deviation of 1. It is often used in statistics to model natural phenomena and is also known as the Gaussian distribution or the bell curve.
Observations from a standard normal distribution are randomly selected by using a random number generator. This generator generates numbers that are uniformly distributed between 0 and 1. These numbers are then transformed using a mathematical formula to follow a standard normal distribution.
Three observations are a commonly chosen number because it provides enough data points to make meaningful conclusions but is also manageable for analysis. Additionally, the Central Limit Theorem states that as the sample size increases, the sampling distribution of the mean approaches a normal distribution, so three observations can provide a good approximation of the standard normal distribution.
The purpose of selecting 3 observations from a standard normal distribution is to understand the characteristics and properties of the distribution. This can help in making predictions, drawing conclusions, and performing statistical tests. It also allows for the comparison of different distributions and to determine if a dataset follows a normal distribution.
No, the 3 randomly selected observations from a standard normal distribution will not always be the same. This is because each time the observations are randomly selected, a different set of numbers will be generated from the random number generator. However, as the sample size increases, the sampling distribution of the mean will become more consistent and approach the standard normal distribution.