GSEB Solutions Class 11 Statistics Chapter 7 Sampling Methods Ex 7

Gujarat Board Statistics Class 11 GSEB Solutions Chapter 7 Sampling Methods Ex 7 Textbook Exercise Questions and Answers.

Gujarat Board Textbook Solutions Class 11 Statistics Chapter 7 Sampling Methods Ex 7

Section – A

Choose the correct option from those given below each question:

Question 1.
A sample selected from a population consists which of the following?
(a) All units of the population
(b) Only 50 % of the units of the population
(c) Only 15 % of the units of the population
(d) Some units of the population
Answer:
(d) Some units of the population

Question 2.
Which of the following statements is true?
(a) A sample in which a unit is selected after replacing the unit selected earlier in the population is called a sample without replacement.
(b) If a unit is to be destroyed during an inquiry then sample inquiry is not only necessary but also inevitable.
(c) In any sampling method, the sample size is larger than the population size.
(d) Stratified random sampling is best if the population is homogeneous.
Answer:
(b) If a unit is to be destroyed during an inquiry then sample inquiry is not only necessary but also inevitable.

Question 3.
Which of the following statements is true?
(a) In stratified random sampling, all the units of the population have equal chance of being selected in the sample.
(b) In simple random sampling all units of the population have equal chance of being selected in the sample.
(c) In any sampling method, the sample size does not depend on population.
(d) In systematic sampling, all units of the population have equal chance of being selected in the sample.
Answer:
(b) In simple random sampling all units, of the population have equal chance of being selected in the sample.

Question 4.
A parameter and statistic respectively are characteristics of which of the following?
(a) Population and Sample
(b) Sample and Population
(c) Sample and Sample
(d) Population and Population
Answer:
(a) Population and Sample

Question 5.
Which sampling is affected the most if there is hidden periodicity in population?
(a) Simple random sampling
(b) Stratified random sampling
(c) Systematic sampling
(d) Both (b) and (c)
Answer:
(c) Systematic sampling

Question 6.
Suppose we are using stratified sampling for a particular population and have divided it into strata of different sizes. How can we now make sample selection?
(a) Select an equal number of units from each stratum at random.
(b) Draw unequal number of units from each stratum and weigh the results.
(c) Draw number of units from each stratum proportional to their size in the population.
(d) None of the above
Answer:
(c) Draw number of units from each stratum proportional to their size in the population.

Question 7.
A security checkpoint that ‘checks every vehicle entering into the mall is an example of which of the following?
(a) Census inquiry.
(b) Stratified random sampling
(c) Systematic sampling
(d) Simple random sampling
Answer:
(a) Census inquiry

Section – B

State whether the following statements are ‘true’ or ‘false’:

Question 1.
A sampling plan that selects units from a population at uniform intervals in time value or position is called stratified sampling.
Answer:
False

Question 2.
A statistic is a characteristic of a population.
Answer:
False

Question 3.
Units in the sample should be selected within the same time duration.
Answer:
True

Question 4.
When properties of the units of the population have more dissimilarity, the use of stratified random sampling method is advantageous.
Answer:
True

Question 5.
In simple random sampling method, each unit of the population has an equal chance of beings included in the sample.
Answer:
True

Question 6.
A sampling method that divides the population into homogeneous groups from which random samples are drawn is known as systematic sampling.
Answer:
False

Question 7.
Each unit of the populations is examined in census inquiry.
Answer:
True

Answer the following questions in one sentence:

Question 1.
What is the process by which inference about a population is made from sample information?
Answer:
The process by which inference about a population is made from sample information is sampling.

Question 2.
Which sampling should be used when each group considered has small variation within itself but there is wide variation between different groups?
Answer:
Stratified random sampling should be used when each group considered has small variation within itself but there is wide variation between different group.

Question 3.
In which sampling method, units are selected from the population at uniform intervals?
Answer:
In systematic sampling method, units are selected from the population of uniform intervals.

Question 4.
Which table of random numbers is most commonly used?
Answer:
L.H.C. Tippet’s table of random numbers is most commonly used.

Question 5.
Which type of inquiry involves more errors?
Answer:
Population inquiry involves more errors.

Question 6.
What is meant by population inquiry?
Answer:
If information of all the units of the population is obtained, then it is called population inquiry.

Question 7.
What do you mean by a sample without replacement?
Answer:
A sample in which each unit is selected from the population without replacing the unit selected earlier in the population is called a sample without replacement.

Question 8.
When is the use of stratified random sampling considered to be favourable or suitable?
Answer:
When the population is heterogeneous the use of stratified random sampling considered to be favourable or suitable.

Question 9.
When can the systematic random sample be biased?
Answer:
When hidden periodicity in population coinsides with the selection of sample, the systematic random sample can be biased.

Question 10.
Define heterogeneous population.
Answer:
When there is considerable amount of variation among the units of population, then it is called heterogeneous population.

Question 11.
Give an example an inquiry where units are destroyed during inspection.
Answer:
Life of electric bulb is an example of an inquiry where units are destroyed during inspection.

Question 12.
If the three-digit random numbers are given and population size is of two digits, how will random numbers be used for selecting the sample?
Answer:
If the three-digit random numbers are given and the population size is of two digits, then for selecting the sample, the first two digits of random numbers is considered.

Question 13.
If the two-digit random numbers are given and population size is of three digits, how will random numbers be used for selecting the sample?
Answer:
If the two-digit random numbers are given and population size is of three digits, then considering two-digits of a column and first digit of the next column the sample is selected.

Question 14.
Define parameters of the population.
Answer:
The measures for population such as population mean, Population standard deviation, etc. are called parameters of the population.

Question 15.
Define sample statistics.
Answer:
The measures such as mean, standard deviation, etc. calculated from the numerical data obtained from sample units are called sample statistics.

Section – C

Answer the following questions as required:

Question 1.
When is sample inquiry undertaken?
Answer:
Sample inquiry is undertaken in following circumstances :

• The units in a population is very large.
• The population is spread over wide geographical area.
• The units under inquiry are to be destroyed during the inspection.
• The availability of time, money and expertise for conducting an inquiry is limited.

Question 2.
What is sampling?
Answer:
The procedure of selecting a sample from a population is called sampling.

Question 3.
State the methods of selecting a simple random sample.
Answer:
Two methods of selecting a simple random sample are :

• Method of lottery and
• Method of random number table.

Question 4.
Name various methods of sampling.
Answer:
The various methods of sampling are :

• Simple random sampling
• Stratified random sampling and
• Systematic sampling.

Question 5.
State the strategies used for deciding the number of units to be selected from each stratum in stratified random sampling.
Answer:
In stratified random sampling, a sample is drawn from each stratum by selecting the units at random in proportion to the size of each stratum.

Question 6.
Explain sample interval in systematic random sampling.
Answer:
The ratio of population size N and sample size n is called sample interval in systematicsampling. It is denoted by k which is a positive integer and defined as k = \(\frac{\mathrm{N}}{n}\).

Question 7.
Explain the process of stratification.
Answer:
A process of dividing heterogeneous, population into non-overlapping fairly homogeneous groups is called stratification and these groups are called strata.

Question 8.
Define stratum in stratified random sampling.
Answer:
In stratified random sampling the groups obtained due to stratification which differ from one another but internally they are almost same in terms of variation. Each of such strata is called stratum.

Section – D

Answer the following questions as required:

Question 1.
Explain the meaning of population inquiry and sample inquiry with an illustration.
Answer:
Population inquiry: The inquiry in which data are collected by inspecting all units of the population is called population inquiry.
Illustration: The population census conducted in our country for every ten years is a population inquiry. In population census information is collected from each family of the country as well as from the members of the family. Hence, it is population inquiry.

Sample inquiry: The inquiry in which data are collected by inspecting units of a sample selected from the population is called sample inquiry.
Illustration: An inquiry regarding the annual income of some families selected randomly from the population of families residing in Ahmadabad city is the sample inquiry.

Question 2.
Differentiate population inquiry and sample inquiry.
Answer:

Population Inquiry	Sample Inquiry
1. In population survey all units are examined. Hence it requires more time.	1. In sample survey few units are examined. Hence it requires less time.
2. The cost of survey is more.	2. The cost of survey is less.
3. As more units are to be examined proper accuracy cannot be maintained.	3. Accuracy can be maintained as few units are to be examined.
4. Population study is not possible when the units are to be destroyed during the study.	4. Sample study can be used when the units are to be destroyed during the study.
5. As more persons are to be employed in the work experts may not be available.	5. As few persons are to be employed, experts may be available.
6. As more units are to be studied the work becomes tedious.	6. As few units are to be studied the work is relatively easy.
7. As all units are examined complete information is available.	7. Limited number of units are examined hence complete information is not available.

Question 3.
State the characteristics of an ideal sample.
Answer:
An ideal sample must possess the following characteristics:

• It should be representative of the population.
• Its selection should be made without any kind of prejudice or bias. That is, its selection should be made at random.
• The selection of the units of a sample should be made during the same duration of time.
• Selection of sample units should be independent.
• It should be of appropriate size.

Question 4.
State the points to be considered while determining the sample size.
Answer:
The following points should be taken into account while determining the sample size:

• The purpose of the inquiry
• The size and scope of the population
• Availability of time, monetary resources and technical expertise,
• The variation among the values of a variable quantity of the units of population (heterogeneity of population) and
• Expected level of accuracy.

Question 5.
State the advantages of simple random sampling.
Answer:
Advantages of simple random sampling method are as follows:

• In simple random sampling each unit of the population has an equal chance of being selected in the sample.
• In this method the selection of sample being done without any bias or prejudice.
• For Homogeneous population, the sample obtained by this method shows good representation of population and hence reliable results can be obtained.
• In this method, from the sample data, reliable information about the characteristics of the population can be obtained with less cost and time.

Question 6.
Write a note on simple random sampling.
Answer:
A sampling in which every unit of the population has an equal chance of being selected in the sample, is called a simple random sampling.

This method is considered to be superior to all other methods of sampling, because,

• In this method there is no place for personal bias or prejudice for any unit of the population.
• The selection of each unit of sample is done independently, i.e., the selection of any unit does not depend upon the selection of any other unit.
• Which unit of the population will be selected in the sample is not decided in advance.

In this method the selection of sample units being done randomly, it is called simple random sample, which appropriately represents the population. The data of the sample units selected by this method gives reliable estimates of the population values. When the size of the population is large and the variation among the units of the population for any characteristics is more then the sample obtained by this method cannot give the reliable results.

Question 7.
Write a note on stratified random sampling.
Answer:
If the population is hetrogeneous, then simple random sample cannot be a representative sample of the population. In this situation, by using the stratified random sampling stratified random sample is be selected.
In this method, first of all units of a population are divided into different strata of homogeneous nature. All these strata are different among themselves but are internally homogeneous.

Now, a random sample is taken from each stratum and random samples so obtained from all strata are combined to get a sample which is called a stratified random sample and the method of selecting such a sample is called stratified random sampling method.

Illustration: Suppose out of 600 families in an area of a city, 200 families are having less Income, 300 families are having average Income and 100 families are having more Income. For socio-economic survey of these families sample of 60 families is to be selected. Taking the families of different income groups as strata families are randomly selected from each stratum in the proportion of 200:300:100. In this way, 20 families having less income, 30 families having average income and 10 families having more. Income are selected at random and the sample of 60 families so obtained is known as stratified random sample.

In stratified random sampling,

• The population is divided into different strata so each stratum gets representation. Hence the representative sample can be obtained for all the characteristics of the population.
• The work of selecting samples from different strata can be assigned to separate investigators or enumerators and hence administrative convenience can be improved.
• If proper care is not taken while dividing the population into different strata, then appropriate representative sample of the population cannot be obtained and therefore the reliability of results can be reduced.

Question 8.
State the disadvantages of stratified random sampling.
Answer:
The disadvantages of stratified random sampling are as follows :

• It is difficult to divide the population into homogeneous strata.
• If stratification is not proper, accuracy of the results obtained by this method decreases.
• The procedure of estimating population parameters in more complicated as compared to simple random sampling.

Question 9.
Write a note on systematic sampling.
Answer:
In systematic sampling, the first sample unit is rendomly selected and the remaining units of the sample are automatically selected in definite sequence at uniform interval from the list of population units.

• If the complete list of population units arranged in some systematic manner is available, then this method of sampling is advisable.
• Suppose, N units of the population are arranged in some systematic manner and numbered 1 to N. A sample of size n is to be drawn. Then sampling interval k = \(\frac{N}{n}\); where k is a positive integer is determined and a random number is selected from first k units of population and select every kth unit therefore.
• Set of such selected units is called systematic sample and the method of obtaining such a sample is called systematic sampling.

Illustration: If the random number selected from the first k unit is 4, then the units at order 4, 4 + k, 4 + 2k, 4 + 3k, …, etc. of the list of population units form the systematic sample. ,

Question 10.
State the advantages of systematic random sampling.
Answer:
The advantages of systematic random sampling are as follows :

• It is easier to draw a sample without mistakes as the order of the sample unit is automatically determined.
• Sample is evenly spread over the population.
• It requires less time and labour compared to simple random sampling and stratified random sampling.

Question 11.
Why is population inquiry usually not feasible in practice?
Answer:
Sometimes even in situations where population inquiry can be undertaken, preference is given to sample inquiry because a population inquiry involves more time, money and man power. Moreover a large numbers of errors creep into the population inquiry due to the extensive and complicated task of organising population inquiry.

Question 12.
State advantages of sampling.
Answer:
The advantages of sampling are as follows :

• In sample study as few units are to be examined, detailed study can be done.
• As few units are to be studied the survey work requires less time.
• The cost of study is also very less.
• In sample study few persons are required for the survey work, hence experts can be appointed. This will increase the reliability of the results.
• In certain survey special types of equipments are required. This is possible only if few units are to be studied.
• When the test is of destructive nature, sampling is only the way out. In such cases population study is not possible.
• A large area is covered in the available time and money.
• If proper method of sampling is employed, the results obtained will represent the population adequately.

Question 13.
Use the following random numbers to select a random sample of 5 ATMs without replacement from a total of 100 ATMs of a bank:
018, 502, 153, 096, 027, 007, 118, 245, 012, 054, 444, 211, 323, 428, 137.
Answer:
First of all, we will assign numbers 1 to 100 to the ATM of Banks.
Population size is 100, a three-digit number. Hence, the random numbers greater than 100 will be ignored from the given random numbers.

Random sample without replacement is to be obtained. Hence, repeated random numbers are ignored. Thus, the following random numbers are obtained :
018, 096, 027, 007, 012, 054.

As the random sample of 5 ATM is to be obtained, we select first five random numbers from the random numbers above.
Thus, the random sample of 5 ATM is obtained as follows :
018, 096, 027, 007, 012.

Question 14.
There are 70 students in a class¬room. A teacher wants to select 7 students for 7 activities. Obtain a random sample with replacement using the following random numbers:
274, 323, 923, 599, 667, 320, 910, 484, 786, 253, 009, 885, 115.
Answer:
First of all, we will assign numbers 1 to 70 to the students in a classroom.
Population size N = 70, a two-digit number. Hence, we consider only first two digits of random numbers and ignore the first two digits greater than 70.

• Random sample with replacement is to be obtained. Hence, we consider the repeated random numbers.
• The selected random numbers are : 27, 32, 59, 66, 32, 48, 25, 11.
• A random sample of 7 students is to be obtained. Therefore we select first seven random numbers from the above.
• Thus, random numbers of selected seven students are respectively 27, 32, 59, 66, 32, 48, 25, which is random sample with replacement.

Question 15.
Three-digit random numbers are given below:
170, 111, 352, 002, 563, 203, 405, 545, 111, 446, 776, 691, 816, 233, 616, 300, 250, 816, 010.
Using the random numbers, select a 2 % random sample with and without replacement from a population of 350 units.
Answer:
First of all we will assign numbers 1 to 350 to the units of population.

• Population size is N = 350, a three-digit number. Hence, the random numbers greater than 350 are ignored.
• The selected random numbers are : 170, 111, 002, 203, 111, 233, 300, 250, 010. 2
• 2 % means 350 × \(\frac{2}{100}\) = 7 units are to be selected in the sample.

Sample with replacement: Considering the repeated numbers, first seven random numbers are: 170, 111, 002, 203, 111, 233, 300.
Thus sample with replacement consists of the random numbers 170, 111, 002, 203, 111, 233, 300.

Sample without replacement: Ignoring the repeated numbers, the first seven random numbers are: 170, 111, 002, 203, 233, 300, 250.
Thus, sample without replacement consists of the random numbers 107, 111, 002, 203, 233, 300, 250.

Question 16.
Draw a random sample of 2 per cent students without replacement from 600 students of a particular college for giving their feedback on faculty members. There are 200 students in each of the three years (F.Y., S.Y. and T.Y.). Use the following three-digit random numbers:
For F.Y.: 158, 092, 411, 745, 009, 724, 674, 550, 716, 359, 419, 696, 200, 458.
For S.Y.: 384, 019, 679, 131, 390, 057, 299, 786, 006, 206, 729, 344, 543, 309.
For T.Y.: 227, 483, 741, 766, 027, 070, 648, 956, 198, 912, 200, 058, 696, 500.
Answer:
Here, N = 600
2 % sample without replacement is to be obtained. Therefore, n = 600 × \(\frac{2}{100}\) = 12.

First stratum :
EY. : N₁ = 200, n₁ = 200 × \(\frac{2}{100}\) = 4
Ignoring the random numbers greater than 200 and the repeated numbers, the selected random numbers are: 158, 092, 009, 200.
Hence, the number of the sample units of the sample of 4 students of First year are : 158, 092, 009, 200.

Second stratum:
S. Y. : N₂ = 200, n₂ = 200 × \(\frac{2}{100}\) =4
Ignoring the random numbers greater than 200 and the repeated numbers, the selected random numbers are: 019, 131, 057, 006.
Hence, the number of the sample units of the sample of 4 students of S.Y. are : 019, 131, 057, 006.

Third stratum:
T. Y. : N2 = 200, n₃ = 200 × \(\frac{2}{100}\) = 4
Ignoring the random numbers greater than 200 and the repeated numbers, the selected random numbers are: 027, 070, 198, 200, 058.
Hence, the random number of the sample units of the sample of 4 students of Third year are: 027, 070, 198, 200.

Question 17.
To study the usages of fertilizer, randomly select 10 farmers without replacement from 30 small farm owners and 20 large farm owners. There should be 6 small farm owners and 4 large farm owners in the randomly selected 10 farmers.
Random numbers for small farm owners:
2, 95, 18, 96, 20, 84, 56, 11, 52, 03, 10, 45.
Random numbers for large farm owners:
4, 40, 34, 11, 72, 11, 50, 55, 08, 13, 76, 18.
Answer:
Small farm owners : N = 30, n = 6

• Ignoring random numbers greater than 30 and the repeated numbers, the selected random numbers are :
  12, 18, 20, 11, 03, 10
• Hence, the numbers of sample units of the sample of 6 small farm owners are: 12, 18, 20, 11, 03, 10

Large farm owner : N = 20, n = 4

• Ignoring random numbers greater than 20 and the repeated numbers, the selected random numbers are: 04, 11, 08, 13, 18
• Hence, the number of sample units of the sample of 4 large farm owners are : 04, 11, 08, 13, 18.

Thus, randomly selected 10 farmers are:
Small farm owner: 12, 18, 20, 11, 03, 10
Large farm owner: 04, 11, 08, 13

Question 18.
There are 60 employees in the office of an I.T. company. 5 employees are to be selected using systematic random sampling for a trial of ‘work from home’ concept. Explain how can a sample be selected?
Answer:
Here, N = 60; n = 5
∴ Sample interval k = \(\frac{\mathrm{N}}{n}=\frac{60}{5}\) =12

• Assign numbers 1 to 60 to the employees of an IT company.
• Select a random number in sample interval 1 to 12.
• Suppose, the selected number is 7.
• Select every 12th number from 7th. Such four numbers are to be selected.
• The sample unit at 7, 19, 31, 43 and 55 will form a sample of employees.

Question 19.
Select all possible samples of size 4 using systematic sampling from a population of 20 units.
Answer:
Here, N = 20, n = 4. k = \(\frac{20}{4}\) =5
We have to select all possible systematic samples of size 4. k = 5. Therefore, selected possible random number may be 1, 2, 3, 4 or 5.

• If selected random number is 1 :
  Sample 1:1, 6, 11, 16
• If selected random number is 2 :
  Sample 2:2, 7, 12, 17
• If selected random number is 3 :
  Sample 3:3, 8, 13, 18
• If the selected random number is 4 :
  Sample 4:4, 9, 14, 19
• If the selected random number is 5 :
  Sample 5:5, 10, 15, 20

Question 20.
A Teacher wants to check home-work of 10 students out of 30 students of Standard XI of a school. How many random samples can be obtained using systematic sampling?
Answer:
Here, N = 30; n = 10
∴ k = \(\frac{30}{10}\) = 3
k = 3, therefore selected possible random number may be 1, 2 or 3.

• If selected random number is 1 ;
  Sample 1:1, 4, 7, 10, 13, 16, 19, 22, 25, 28
• If selected random number is 2 :
  Sample 2:2, 5, 8, 11, 14, 17, 20, 23, 26, 29
• If selected random number is 3 :
  Sample 3:3, 6, 9, 12, 15, 18, 21, 24, 27, 30