Interval Estimation for Population Mean（Case when population standard deviation is known）

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Inference is the technique to learn about large population using small sample.

The sample mean $\bar{X}$ and sample standard deviation $s$ (sample)　⇒　The population mean $\mu$ and population standard deviation $\sigma$ (population).

First, you have to understand pattern 1, which is the estimation of population mean $\mu$ in case that the population standard deviation $\sigma$ is known.

This setting of pattern 1, when we assume that the population standard deviation $\sigma$ is known is very unnatural as realistic prior knowledge. Mainly, the statistical inference purpose is to infer the information of vast population by computing information a small sample. However, population standard deviation $\sigma$ does not belong to sample information. Instead it belongs to population information. Naturally, we do not know the population standard deviation $\sigma$ from the beginning. *smiles*

Of course, we can arbitrarily assume the population standard deviation $\sigma$.

However, in fact, if we do not understand this pattern (which will not be used in real situation), we cannot understand other more realistic patterns that can be applied to real problems. Therefore, the study of this unrealistic pattern is necessary to understand the more realistic patterns with real-world applications.

The main knowledge necessary to understand the present case is the normal distribution and the central limit theorem. By combining these two crucial statistical concepts, we can derive the next formula for estimating the population mean at the confidence level of 95%. $\displaystyle P(\bar{X} - 1.96\times \frac{\sigma}{\sqrt{n}} < \mu<\bar{X}+1.96\times\frac{\sigma}{\sqrt{n}})=0.95$

Here, we explain the meaning of the parameters shown in the above equation. $\mu$ : Population mean. ( This is what we want to estimate !) $\sigma$ : Population standard deviation.（This is what we already known as assumption of pattern 1！） $\bar{X}$ : Sample mean　（This is easy to compute from sample ！） $n$ : Elements of the sample (size)　（We know that obviously ！） $P(w)=0.95$ : This denotes that the probability to become the state of $w$ is 0.95 (95%).

If you look carefully to the above equation, we know all the values of variables except the population mean $\mu$. Therefore, if we input all the information we already know into the equation, then we can estimate the population mean (Interval Estimation).

Let’s try an example:

Example: Let’s estimate the average height of all 20 years-old male in a country. Then, we can pick up 900 people as a sample ( $n$), and compute the sample average which gives 170cm $\bar{X}$. Here, although it is not realistic, let’s assume that due to some reason we know that the population standard deviation is 10cm ( $\sigma$). Then, please estimate the population mean of the height ( $\mu$) for all the 20 years-old males of the country at the confidence level of 95%.

(Comment: In this problem, it is not realistic in principle to know the population standard deviation as prior knowledge. However, in Japan, the difference between the highest height and lowest height can be easily estimated around 10cm. Therefore, technically, it is reasonable to assume this value for the population standard deviation in case of Japan. But, even if you do not know the population standard deviation, it is enough to estimate roughly a value for the population standard deviation by taking a larger value than the real one. Then, by using this approximate value, it is at least possible to obtain a reasonable estimation. )

Answer: Let’s substitute all the values into the equation. $\displaystyle \bar{X} -1.96\times \frac{\sigma}{\sqrt{n}}=170-1.96\times\frac{10}{\sqrt{30}}=170-0.65=169.35$ $\displaystyle \bar{X} +1.96\times \frac{\sigma}{\sqrt{n}}=170+1.96\times\frac{10}{\sqrt{30}}=170+0.65=170.65$

Therefore, $\displaystyle P(169.35< \mu<170.65)=0.95$

The result shows that the probability that the population mean ( $\mu$) of all 20 years-old male of the country is between the displayed ranges $169.35cm< \mu<170.65cm$ is 95%.

As a comment, it is worth remarking that if the country has a million of 20 years-old male and we pick up a very small sample of 900 people, we can really know the population mean within an error of ±0.65cm. This result is really remarkable and impressive !!.