Selecting a sample

In the previous post we have discussed about Sampling errors and In today's session we are going to discuss about Selecting a sample.
Selecting a sample:
In this session, we will talk about selecting a sample from a group of samples for grade VI.
In terms of probability, sampling is a method that aids in random selection process.
The important condition is that all the events to be chosen must have same probability.
For example, the random numbers chosen by a computer program is a random selection process as all the numbers can be chosen with equal probability.
Sample size: sample size is the total sum of sizes of all its cases.
Before we continue, let us understand some basic notations,
N=total number of cases in the sampling space.
n=number of cases in a given sample
f=sampling fraction=n/N
Simple random sampling:
We have to select n units from N.
N is the total cases in sampling space.
All the n must have the equal probability to be chosen.
nCN=total number of subsets of n from N.
Let us make this concept clearer through an example.
Let us take an example that we have to do a survey of a company by selecting its past clients. We want 100 clients to be a part of the survey.
Company provides a list of 1000 employees whose records are in the database of the company.
Sampling fraction=n/N=100/1000=0.10
Hence sampling fraction is 10%.
We have several ways to select the 100 clients. An easy way is to put all the 1000 clients in a group, subdivide the group and randomly select any subgroup. This is a mechanical way to select but it is not as efficient as the quality of samples would depend on how we have subdivided the group and how randomly we choose the subgroups. (know more about  Sampling, here)
The computerized method is more efficient as compared to the mechanical method.
Principally, our main motive is to understand the selection of sample out of many. To know more about cbse syllabus.

The main point here to keep in mind is that the samples must have equal probability to be chosen.
If the selecting probability depends on the previous selection, then it is said to be conditional probability.
In the next topic, we are going to discuss Sampling errors.

Constructing sample spaces

Constructing sample spaces:
Sample space is generally denoted by S, Ω or U. Fundamentally it’s the set of all random trials; like in tossing a coin the sample space is (head, tail). If we toss two coins then sample space for this random trial is (head, head), (head, tail), (tail, head), (tail, tail). In the trial of tossing a single sided die sample space s is (1, 2, 3, 4, 5, and 6). It’s not necessary that one trial should have only one sample space; like in a trial of drawing a card from a standard deck of 52 playing cards one probability of outcome is could be the rank (ace through king) and the another sample space could be suits i.e. (club, diamond, hearts or spades). The subsets of sample space are called event. Problems would be critical after increasing the number of sample spaces. So it would be problematic when there are infinite sample spaces means there are infinite events.
Sample space construction:
Sample space is the set of all possible outcomes and it’s necessary to consider all possibilities. It may be a difficult task and for this purpose counting principle can be used. If there is more than one event it’s important to determine all possibilities that exist. It can be stated that:
“If there are A ways of an event to occur and B ways for occurrence of second event then there are A.B ways for both to occur”. This is the concept of the counting principle.
An example is given below which can help to understand sample space better. (know more about cbse sample papers, here)
Example: one jar contains 1 red, 3 green, 2 blue or 4 yellow balls. Then what would be the probability of each outcome if a single ball is chosen from the jar.
Solution: Sample space for this random trial is
S = (red, green, blue, yellow)
Probability can be calculated as:
Probability:  P (red ball) = 1/10
P (green ball) = 3/10
P (blue ball) = 2/10 = 1/5
P (yellow ball) = 4/10 = 2/5
In the next session we will discuss about Sampling errors.