A sample from the population of 'Statistics'

So somewhere in the corner of the world exists a huge kingdom with say a million people. And teh King is well known as the care taker and enforcing rules such that no class of the society is left behind with dissatisfaction. One fine day he wakes up with a new thought and after brainstorming with his 'Wise Ministers', he enforces the new law. But how does he ensure people are satisfied with this decision? Well a smart minister suggested 2 ways to get the feedback:

1. Send the raiders across the country, asking each and every person how they feel about the newly enforced law. (Say am talking about ancient times where everything doesn't happen in one click). Which is practically impossible ( its indeed possible if accuracy isnt a big deal but i want to focus on my second point so making this one look complicated)

2. Choose a 'SAMPLE', sufficiently large,  from each region of the whole 'POPULATION', take the feedback  and throw a banquet if its positive. If not, you know how crazy ancient kings were.

Now as simple as it sounds, this process is called 'INFERENTIAL STATISTICS'. And we are going to try to understand a few concepts which will help us in the future readings.

Inferential statistics involves "Estimating" a parameter and performing "Significance Test".

Sampling: 

 

A portion (randomly chosen and unbiased) from the POPULATION is a SAMPLE and the process is called SAMPLING. The number of elements in the population is represented by 'N' while in the sample its 'n'. Rest of the symbols are shown in the diagram above. 

_{Population Standard Deviation}  

         _N

σ =  ∑ ( X_i - μ )²  / N

          ⁱ⁼¹

_{Sample Standard Deviation 
         n}

s =  ∑ ( X_i - xbar )²  / (n -1)

          ⁱ⁼¹

<sub>Sampling Distribution:

Suppose we collect infinite random samples from the population of a particular size and compute the mean of each sample, the distribution of such means is called "Sampling Distribution of Sample Means"  and the sample mean will be approximately equal to the population mean. The distribution will be approximately NORMAL provided the number of sample is large. 

Central Limit Theorem:</sub>

Sampling Distribution of sample means is almost NORMAL provided the number of samples is relatively large. Best practice is to choose n > 30. In such case:

mean of sample means will be approximately equal to the population mean : xbar = μ

standard deviation of the sampling distribution will be : σ / √n