P&SIA L1 C4 S2
Intro Stats Islamic Approach -- Part 2 Prob & Stats
Lecture 1 Random Samples, Concept 4: Simple Random Samples
S2: Simple Random Samples are REPRESENTATIVE
The main reason for our interest in SRS is that, under suitable conditions, these samples provide an accurate representation of the population. We can take a survey of a SRS of 3000 people drawn from the population and learn about the entire 160 million population of Pakistan. The question arises: HOW are SRSs representative? HOW can a sample of 3000 people represent a population of 160 million people? The two are clearly VERY different. This goes back to the question of MODEL versus REALITY. The sample is LIKE the population in SOME important ways, but not in all ways -- Just as the MODEL is LIKE Reality in some ways.
We need to LEARN what are the ways in which Samples are like the Population?
Suppose E is a subset of the population P, Let #E,#P be the number of elements in set E and in the total population. Then the Probability of E is the ratio #E/#P=P(E)
This may be called the POPULATION PROBABILITY or the TRUE PROBABILITY.
Let S=(S1, S2, ... , S(N)) be a simple random sample of size N from population P. Count the number K of elements within the SRS which belong to the set E. Then K/N is the SAMPLE FREQUENCY of occurrence of E. This sample frequency -- the occurence of elements of E within the random sample -- RESEMBLES the occurence of E within the population
The Law of Large Numbers: If size of the Simple Random Sample N is LARGE, then the SAMPLE FREQUENCY of E within SRS is close to the true probability of E within the Population.
The SAMPLE is similar to the population in that proportions of elements with the sample match the proportion of elements within the POPULATION