B. Com. I (OE) Basic Statistics Practical 6. Simple random sampling (with and without replacement) and stratified random sampling.

 

DEPARTMENT OF STATISTICS

B. Com. I: Practical - I

Expt. No. 6                                                                                       Date:    /    / 2024

Title: Sampling: Simple random sampling (with and without replacement) and stratified random sampling.

Q.1 In population of size N=20, the observation were 4, 7, 10, 14, 17, 18, 20, 24, 25, 27, 29, 30, 34, 37, 39, 40, 42, 45, 47, 48. Draw Random Sample of size 5 form given population using SRSWOR.

Q.2 In population of size N= 10, the observation were 1, 4, 7, 12, 15, 17, 20, 24, 27, 29. Draw Random Sample of size 5 form given population using SRSWR.

Q.3 Population size of 3 Strata A, B, C are 100, 200, 300 respectively. Draw a sample of size 30 from an entire Population.

i) Obtain how many units are to be selected from strata A, B, C.

ii) Using Random numbers, draw random sample of above obtained size from each strata.

Q.4 Draw a random sample of size 50 from a population of 1000 workers in a company, with 300 from the production line, 200 from maintenance, and 100 from the warehouse, using stratified random sampling

 

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Sampling Methods: 

 

Census and Sampling Method:

 

        There are two methods of data collection: 

i. Census Method        ii. Sampling Method

but we first define the same terms as

 

Population: In Statistics, the group of individuals under study is called 'population'. The number of individuals in belonging to a population is known as Population Size and it is denoted as "N".

e.g. if we are interested to study the income of male in Sangli district then the population will be the all males in Sangli. therefore the population may be group of individual or object like animate or inanimate or peoples or cars. the population is may be  finite or infinite.

now we see the Census Method: 

 

Definition of Census Method: The process of collecting data or information from every member of the population. i.e. we collect data from entire population is called the Census Method. or 100% inspection or complete enumeration. the Census Method is suitable when the population is limited. or when the greater accuracy is expected that case we use Census Method to collect Data.

 

There are some limitations of Census Method: 

 

i. Census Method provide Reliable result.: In Census method we study each and every individual in Population. the data collected from every individual it is expensive and time consuming and it required large amount of time and manpower. 

 

ii. There is in some situation where Census Method is possible but impracticable. e.g. blood test. 

 

iii. If the population is infinite then census method can't used.   

these are the limitations of Census Method that can be overcome using Sampling Method

 

Sampling Method: 

First we see the sample means : 

 

Sample : A finite Sub-Group of population is called a sample, and the number of individuals in Sample is called the sample size and it is denoted as "n".  

 

Sampling :  Sampling is a   Process of collecting data from selected sample. it is also called sampling method. A part of population is studied is called sampling. from this sample we draw a inference about the entire population. for that  the selected sample is unbiased and sufficiently large. 

 

Advantages  of Sampling Method :

 

Followings are the advantages of sampling method to overcome the drawback of Census method.

 

i. Less Time: If the population is large and the study of population required a lot of time. not only for collection but also for analyzing the data. as compared to sample. therefore the sample is required the less time as compared to the population. 

 

ii. Less Cost : The cost of collection of data on each unit in case of population is likely to be more as compared to sampling method. 

 

iii. Reliability: The collection of data in sample survey is more reliable than that of complete enumeration. 

 

iv. Detailed Information:  The sample contain the small size of members therefore we studied it carefully and detailed information can be collected. 

 

v. Necessity:  Some situation where the sampling is necessity, when  we study the destructive sampling where the quality of an object can be determined only by destroying the object in the process of testing.  testing the explosive.

 

vi. if the population is very large or spread over the large geographical areas. when we use only the sampling methods.

 

Remark: there are some limitations of sampling 

i. proper care should be taken in planning of the sample survey, other wise the result may be might bee inaccurate. 

ii. if the time and money is not important factor then Census method is better than sampling method.

 

Following are the Methods of Sampling. 

 

i. Simple Random Sampling 

ii. Stratified Random Sampling 

iii. Systematic Sampling

iv.  Multistage Sampling

v. Cluster Sampling

the Selection of  sampling method dependence on the available information about the Population and Nature of data. 

 

 Now we Discuss only two methods. i. Simple Random Sampling and ii. Stratified Random Sampling.

 

i. Simple Random Sampling (SRS): 

Simple Random Sampling it is denoted as SRS. SRS is the easiest and most commonly used method of sampling. in this method each unit of the population has equal chance to select in sample. i.e. 1/N. the simple random sampling method is divided into two types. due to selection procedure of elements in population.

a) Simple Random Sampling with replacement (SRSWR).

b) Simple Random Sampling without replacement (SRSWOR).

 

a) Simple Random Sampling with replacement (SRSWR): In Simple Random Sampling with replacement (SRSWR). element or unit are selected one by one from the population in such a way that after each drawing the unit is studied completely and then return back to the population before the next unit being drawn.  therefore in SRSWR method the population size is remains the same at every draw. this method of sampling called simple random sampling with replacement (SRSWR) this method is used when the population is finite. but in this method the same unit is selected more than once in sample. it is drawback of that sampling method.

 

b) Simple Random Sampling without replacement (SRSWOR). :  It  is the another method of sampling in which unit are selected one by one  from population without replacement i.e. the unit selected once it not replaced back to the population. this method of selecting the sample is called the Simple Random Sampling without replacement (SRSWOR). in this method the population size is decreases at each draw. this method is used when the population is infinite. and the drawback of SRSWR method is overcome in SRSWOR method.

 

Let N be the population size and n be the sample size n be following methods are used to drawing sample from population. i. Lottery Method, ii. Random Number.

 

i. Lottery Method: 

Suppose we want to select the sample of size n out of the Population Size N. In this method we write the name or number of all N units on the slip of paper  ( or small size of paper having same size, same colour, same shape. and fold it and collect all N chits in box then select the n chits of paper from the box. ( when we select the chits there is no idea to which number in that chits so it is random sample) this method used for prizes of lottery so it is called Lottery Method.

 

ii. Method Of Random Number: 

In this method we give number to each unit in population from 1 to N. ( if the N<99 then  we use two digit number as 01, 02, ......,99)  then we use random number number book to select random number and the this numbered unit selected form population as sample. 

 

Merit and Demerit of Simple Random Sampling. 

Merit: 

Sample unit Randomly selected hence each unit has equal chance to select in sample so person bias is removed.

Demerit: Population is large then work for giving number is tedious, and the size of sample in this method is required to be large.

 

ii. Stratified Random Sampling: 

 

when the is consist of different groups or classes, then  simple random sampling does not give proper representation  of sample in that case we use Stratified Random Sampling.

Stratification means data divided into classes, e.g age , gender.

In the Stratified Random Sampling items of each group are include into the right proportion, when the total population is known then we divided the population N into K strata's or groups of size N1, N2, N3, ............NK. respectively.

such that  ∑Ni = N, we want to take sample of size n units then we select simple random sample without replacement method to select sample of size n1, n2,.... nk. units from the respective group of population. such that ∑ni = n, 

here the unit of size ∑ni = n, selecting using stratified random sample, hence the method is known as stratified random sampling. 

 

now the sample selected from each group using following formulae.

there are two formula to select sample i. Proportional Allocation  and  ii. Optimum Allocation

 

i. Proportional Allocation :

 

ni = (n/N) Ni , i = 1, 2, ....k

 

where ni - i th sample size

N - Population Size 

Ni - i th Population Size

 

ii. Optimum Allocation:

 

ni = n {(NiSi) / ∑ (NiSi)

 

where ni - i th sample size

N - Population Size 

Ni - i th Population Size

 

Example: There are 1000 students in college, out of which 500 from commerce, 200 from arts and 300from science. we want to select sample of 100 students.

 

Solution: Given N= 1000, N1 = 500, N1 =200, N3 =300

n = 100

for finding the sample size we use the Proportional allocation.

 

n1 =  (n/N) N1  =  (500/1000) x100

n1 =  50  i.e 50 students from commerce selected from sample.

 

n2 =  (n/N) N2  =  (200/1000) x100

n2 = 20 i.e 20 students from commerce selected from sample.

 

n3 =  (n/N) N3  =  (300/1000) x100

n3 = 30  i.e 30 students from commerce selected from sample.

 

n =  n1 + n2 +n3 = 50+30+20 = 100 sample size.

 

 

Q. 1 In population of size N=20, the observation were 4, 7, 10, 14, 17,18, 20, 24, 25, 27, 29, 30, 34, 37, 39, 40, 42, 45, 47, 48. Draw Random Sample of size 5 form given population using SRSWOR.

Answer:

Aim: To Draw SRSWOR of size 5.

          The population size is 20 so we have to draw a random sample of size 5 from given population.

Here we use random number method or lottery method to draw a random sample.

Here first we give a number to each population unit from 1 to 20 as:

Number	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Observation	4	7	10	14	17	18	20	24	25	27	29	30	34	37	39	40	42	45	47	48

  

Method i) we use random number between 1 to 20 to choose population unit in to a sample. The we give first five random number  to select the sample of 5. The first 5 random number are 10, 5, 4, 17, 12

Therefore we select 10^th, 5^th, 4^th, 17^th, 12^th. Observation in given observations is 27, 17, 14, 42, 30.

Random No.	10	5	4	17	12
Population Observation	27	17	14	42	30

i.e.

Number	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Observation	4	7	10	14	17	18	20	24	25	27	29	30	34	37	39	40	42	45	47	48

 

Result:

The Random Sample of size 5 by SRSWOR method is given by

27, 17, 14, 42, 30.

Q.3 Population size of 3 Strata A, B, C are 100, 200, 300 respectively. Draw a sample of size 30 from an entire Population.

i) Obtain how many units are to be selected from strata A, B, C.

ii) Using Random numbers, draw random sample of above obtained size from each strata.

Answer:

Aim: To Obtain how many units are to be selected and draw random sample of above obtained size from each strata.

Procedure:

Population size of 3 Strata A, B, C are given as 100, 200, 300 respectively.

i.e. Population size of strata A is 100 , is denoted as N₁=100

Population size of strata B is 200 , is denoted as N₂=200

Population size of strata C is 300 , is denoted as N₃=300

Total Population size is  N = N₁+ N₂ + N₃  =100+200+300 = 600

And  sample of size n= 30

For selecting sample form each strata we use Proportional allocation method. Is given by

n_i = (n/N) N_i where i=1,2, 3.

 Where n_i :- is sample size of i^th Strata,

          n:- is total sample size

          N:- total population size

          N_i :- Population size of i^th Strata,

i)                   The sample size to be selected form Strata 1(i.e. A) is:

n₁ = (n/N) N₁

 n₁ = (30/600) 100

n₁ = 5

Therefore form strata A we Draw sample of size 5,

We draw Random sample of size 5 from strata A by using Simple random sampling method without replacement (i.e. SRSWOR)

We assign a number 1 to 100 for each Population unit in strata A, Then we select 5 random number using random number table or calculator.

The random numbers are: 36, 30, 67, 46, 77.

Now we select the 36^th, 30^th,67^th,46^th,77^th Population unit in Strata A.

Units are Selected in sample form strata A are: 36, 30, 67, 46, 77

ii)                The sample size to be  selected form  Strata 2 (i.e. B)  is:

n₂ = (n/N) N₂

 n₂ = (30/600) 200

n₂ =10

Therefore form strata B we Draw sample of size 10,

We draw Random sample of size 10 from strata B by using Simple random sampling method without replacement (i.e. SRSWOR)

We assign a number 1 to 200 for each Population unit in strata B, Then we select 10 random number using random number table or calculator.

The random numbers are: 112, 5, 34, 195, 157, 100, 1, 95, 162, 82.

Now we select the 112^th, 5^th,34^th,195^th,157^th , 100^th, 1^th,95^th,162^th,82^th Population unit in Strata B.

Units are Selected in sample form strata B are: 112, 5, 34, 195, 157, 100, 1, 95, 162, 82.

iii)              The sample size to be  selected form  Strata 3 (i.e. C)  is:

n₃ = (n/N) N₃

n₃ = (30/600) 300

n₃ =15

Therefore form strata C we Draw sample of size 15,

We draw Random sample of size 15 from strata C by using Simple random sampling method without replacement (i.e. SRSWOR)

We assign a number 1 to 300 for each Population unit in strata C, Then we select 15 random number using random number table or calculator.

The random numbers are: 214, 253, 123, 36, 141, 169, 115, 124, 81, 40, 38, 207, 128, 299, 45.

Now we select the 214^th, 253^th,123^th,36^th,141^th , 169^th, 115^th,124^th,81^th,40^th ,38^th, 207^th,128^th,299^th,45^th Population unit in Strata C.

Units are Selected in sample form strata C are: 214, 253, 123, 36, 141, 169, 115, 124, 81, 40, 38, 207, 128, 299, 45.

Result: 

i) From Strata A, B and C We select sample of 5, 10, and 15 units respectively.

Selected in sample form strata A are: 36, 30, 67, 46, 77

Selected in sample form strata B are: 112, 5, 34, 195, 157, 100, 1, 95, 162, 82.

Selected in sample form strata C are: 214, 253, 123, 36, 141, 169, 115, 124, 81, 40, 38, 207, 128, 299, 45.