**Sampling**

A process used in statistical analysis in which a predetermined number of observations will be taken from a larger population. When taking a sample from a larger population, it is important to consider how the sample will be drawn. To get a representative sample, the sample must be drawn randomly and encompass the entire population.

For example, a lottery system could be used to determine the average age of students in a University by sampling 10% of the student body, taking an equal number of students from each faculty.

There are three types of sampling:**1. Probability sampling:** it is the one in which each sample has the same probability of being chosen.**2. Purposive sampling:** it is the one in which the person who is selecting the sample is who tries to make the sample representative, depending on his opinion or purpose, thus being the representation subjective.**3. No-rule sampling:** we take a sample without any rule, being the sample representative if the population is homogeneous and we have no selection bias.

We will always make probability sampling, because in case we choose the appropriate technique, it assures us that the sample is representative and we can estimate the errors for the sampling. There are different types of probability sampling:

• Random sampling with and without replacement.

• Systematic sampling.

• Stratified sampling.

• Cluster sampling.

• Other types of sampling techniques

**Random sampling with and without replacement**

When a certain element is selected and we have measured the variables needed in a certain study and it can be selected again, we say that we make sampling with replacement. This sampling technique is usually called simple random sampling.

In the case that the element cannot be selected again after being selected once, we say that we have obtained the sample through a random sampling without replacement.

**Systematic Sampling**

In systematic sampling, elements are selected from the population at a uniform level that is measured in time, order, or space. If we wanted to interview every twentieth student on a college campus, we would choose a random starting point in the first twenty names in the student directory and then pick every twentieth name thereafter.

**Stratified Sampling**

To use stratified sampling, we divide the population into relatively homogenous groups, called strata. Then we use one of two approaches. Either we select at random from each stratum a specified number of elements corresponding to the proportion of that stratum in the population as a whole or we draw an equal number of elements from each stratum and give weight to the results according to the stratum's proportion of total population.

**Cluster Sampling**

In cluster sampling, we divide the population into groups or clusters and then select a random sample of these clusters. We assume that these individual clusters are representative of the population as a whole. If a market Research team is attempting to determine by sampling the average number of television sets per household in a large city, they could use a city map and divide the territory into blocks and then choose a certain number of blocks (clusters) for interviewing. Every household in each of these blocks would be interviewed. A well designed cluster sampling procedure can produce a more precise sample at considerably less cost than that of simple random sampling.

**Sampling distribution**

Sampling distribution is the distribution of all possible values of a statistic from all possible samples of a particular size drawn from the population.

**Standard Error**

Standard deviation of the distribution of the sample means is called the standard error of the mean.

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**Numerical on Sampling**

A jar contains 3 red marbles, 7 green marbles and 10 white marbles. If a marble is drawn at random, what is the probability that marble drawn is white?

a. 2/5

b. 1/2

c. 3/8

d. 10/13

Ans – b**Solution :**

Here Red = 3

Green = 7

White = 10

Hence total sample space is (3+7+10)= 20

Out of 20 one ball is drawn n(S) = {c(20,a.} = 20

To find the probability of occurrence of one White marble out of 10 white ball

n(R)={c(10,a.} = 10

Hence P(R) = n(R)/n(S)

= 10/20 = 1/2

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A sack contains 4 black balls 5 red balls. What is probability to draw 1 black ball and 2 red balls in one draw?

a. 12/21

b. 9/20

c. 10/21

d. 11/20

Ans – c**Solution :**

Out of 9, 3 (1 black & 2 red) are expected to be drawn)

Hence sample space

n(S) = 9c3

= 9!/(6!×3!)

= 362880/4320

= 84

Now out of 4 black ball 1 is expected to be drawn hence

n(B) = 4c1

= 4

Same way out of 5 red balls 2 are expected be drawn hence

n(R) = 5c2

= 5!/(3!×2!)

= 120/12

= 10

Then P(B U R) = n(B)×n(R)/n(S)

i.e 4×10/84 = 10/21

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