# Interval Estimation

Pocket

Statistical Inference estimates unique metric features (e.g. mean, standard deviation) for a large population using a small sample.

Small sample (it can be analysed easily) -> Distinct feature of large population ( it is difficult to be analysed)

There are several terms that will be used frequently hereafter therefore we will define uniquely as follows:

The population mean is the average of the population. The population standard deviation is the standard deviation of the population.

On the other hand, the sample mean is the average of the sample. The sample standard deviation is the standard deviation of the sample.

The purpose of the Interval Estimation is to estimate the population mean and population standard deviation from sample mean and sample standard deviation.

Sample mean and sample standard deviation [sample] -> Population mean and population standard deviation [Population].

In other words, we infer information from the small set to the large set.

There are four patterns for Interval Estimation analysis.

（１）Estimation of the population mean

1. In case we know the population standard deviation as prior knowledge [pattern 1]
2. In case the population standard deviation is unknown（large sample：More than 30 elements）[pattern 2]
3. In case the population standard deviation is unknown（small sample：Less than 30 elements）[pattern 3]

（２）Estimation of the population standard deviation 　[pattern 4]

From logical point of view, we should start explaining from pattern 1 to pattern 4. Especially, to understand pattern 1 and pattern 2 the required statistical knowledge is the Normal distribution and the central limit theorem. For pattern 3, the required statistical knowledge is the t-distribution. For pattern 4, the required statistical knowledge is Chi-square distribution.

The pattern 1 assumes that the population standard deviation is known and it is not realistic. However, to understand pattern 2, it is necessary to explain first pattern 1. If you understand pattern 1, it is easy to understand pattern 2.

For statistical inference, as a newcomer in the field, you should first understand pattern 1 and 2 at least. If you get used to pattern 1 and 2, you can challenge yourself with the more complicated pattern 3 and 4.