Question

Explain how collecting mean values (X-bar) from samples drawn from a dataset, regardless of the statistical distribution of the dataset, results in a distribution of those values that is centered on the mean value of the original dataset and is approximately normally distributed depending upon the sample size used. Include the name the theorem that describes this phenomenon, and discuss what it means to treat the X-bar data points derived from sampling as a new random variable. Why is this important in engineering?

Answer 1

Answer:

answer is given below

Explanation:

Central Limit Theorem: The Central Limit Theorem (CLT) is a statistical theory that gives a sufficiently large sample size with limited variations from the population, the average of all samples from the same population is approximately the same. . In addition, all models follow a nearly normal distribution model.

The given phenomenon is described in the central limit theory. In other words, if we repeatedly take independent random samples of size n from any population, when n is large, the sample distribution is the normal distribution pattern.

mean of the sample means

                    $\mu _\bar x = \mu$        .............1

and here standard deviation of the sample means is

                      $\mu _\bar x = \frac{\sigma }{\sqrt{n}}$      .........2

and This theory is found elsewhere in the field of statistics. Although the central limit theory may seem abstract and devoid of any application, this theory is actually important for statistical practice.