Selection of samples:SRSWR vs SRSWOR(2048114)

 

Selection of samples(SRSWR VS SRSWOR)

Anindita Sarkar(2048114)

INTRODUCTION-

SIMPLE RANDOM SAMPLING(with replacement Vs without replacement) :-

The simplest of the methods of probability sampling which is usually called the method of random sampling. In this method, an equal probability of selection is assigned to each available units of the population at the first and each subsequent draw. Thus, if the number of units in the population is N, then the probability of selection of any unit at first draw is 1/N and at the second draw is 1/N-1 etc, which are ultimately equal to 1/N. The sample obtained using the above method is called “ Simple Random Sampling”. Since this result is independent of the specified unit it follows that every one of the units in the population has the same chance of being included in the sample under the procedure of simple random sampling.

Difference Between SRSWOR and SRSWR:

(i)                  If the selected units are not being replaced back in the population before the second draw, it is called SRSWOR and if the selected units are being replaced back in the population before the second draw, it is called SRSWR

(ii)                In SRSWOR, at each draw ,new information on the units will be generated while it may be possible to have the same kind of information on the units in SRSWR.

(iii) SRSWOR method will cover the whole population units while it is not true in the case of SRSWR.

Steps in Selecting a Simple Random Sample-

1. Define the target population.

2. Identify an existing sampling frame of the target population or develop a new one.

 3. Evaluate the sampling frame for undercoverage, overcoverage, multiple coverage, and clustering, and make adjustments where necessary.

4. Assign a unique number to each element in the frame.

 5. Determine the sample size.

 6. Randomly select the targeted number of population elements.

Samples of size m without replacement  from Sn are all possible combinations of m distinct elements from Sn. There are nC m such samples. Samples are unordered, that is, different orderings of the same elements are considered the same sample. There are m! ordered samples for each unordered  sample as this is the number of ways to permute m items.

 

Samples of size m with replacement (WR) from Sn can have any of the n elements in each of the m positions. The elements are not required to be distinct. There are nm such possible samples. Note that WR samples can have the same element in more than one position.

Real world examples of simple random sampling include:

• At a birthday party, teams for a game are chosen by putting everyone's name into a jar, and then choosing the names at random for each team.
• On an assembly line, each employee is assigned a random number using computer software. The same software is used periodically to choose a number of one of the employees to be observed to ensure they are employing best practices.
• A restaurant leaves a fishbowl on the counter for diners to drop their business cards. Once a month, a business card is pulled out to award one lucky diner with a free meal.
• At a bingo game, balls with every possible number are placed inside a mechanical cage. The caller rotates the cage, tumbling around the balls inside. Then, she selects one of the balls at random to be called, like B-12 or O-65.
• 

SIMPLE RANDOM SAMPLING WITH REPLACEMENT (SRSWR): SRSWR is a method of selection of n units out of the N units one by one such that at each stage of selection, each unit has an equal chance of being selected, i.e., 1/ N

SIMPLE RANDOM SAMPLING WITHOUT REPLACEMENT (SRSWOR): SRSWOR is a method of selection of n units out of the N units one by one such that at any stage of selection, any one of the remaining units have the same chance of being selected, i.e. 1/N .

When drawing a sample from a population, there are many different combinations of people that could be selected. This formula is used to derive the number of possible samples drawn with replacement :

N^n

where N is the number in the total population and n is the number of units being sampled.

This Formula is used to calculated the number of possible samples that can be drawn without replacement, disregarding order,

N!/n!(N-n)!

where N is the number of people in the population, n is the number of sampled persons, and ! is the factorial notation for the sequential multiplication of a number times a number minus 1, continuing until reaching 1.

 ANALYSIS USING R CODE----

 

CONCLUSION—In the analysis part I have taken a dataset and I have selected a sample of 15 units from the population using both SRSWR and SRSWOR. When we sample with replacement, the two sample values are independent. Practically, this means that what we get on the first one doesn't affect what we get on the second. Mathematically, this means that the covariance between the two is zero. In sampling without replacement, the two sample values aren't independent.