GSEB Class 11 Statistics Solutions Chapter 7 Sampling Methods

Section - A

 

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
In simple words: A sample represents a subset of the entire population, meaning it includes only a portion of the units, not all of them.

🎯 Exam Tip: Understanding the basic definition of a sample is crucial for foundational knowledge in statistics.

 

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.
(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.
In simple words: When the process of data collection itself destroys the units being studied, sampling is not just an option but a necessity.

🎯 Exam Tip: Recognize situations where sampling is indispensable, particularly in destructive testing scenarios, to demonstrate practical application of statistical methods.

 

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.
(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.
In simple words: Simple random sampling ensures that every unit in the population has an identical probability of being chosen for the sample.

🎯 Exam Tip: Grasping the principle of equal probability in simple random sampling is fundamental to understanding unbiased sample selection.

 

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
In simple words: A parameter describes a characteristic of the entire population, while a statistic describes a characteristic of a sample drawn from that population.

🎯 Exam Tip: Clearly differentiate between parameters (population) and statistics (sample) as this distinction is crucial for inferential statistics.

 

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
In simple words: If there's a recurring pattern in the population, systematic sampling can pick units that all share a similar characteristic, leading to a biased sample.

🎯 Exam Tip: Be aware of the limitations of each sampling method, especially how periodicity can compromise systematic sampling results.

 

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.
(d) None of the above
Answer: (c) Draw number of units from each stratum proportional to their size in the population.
In simple words: In stratified sampling with varying stratum sizes, it's best to select a proportional number of units from each stratum to ensure fair representation.

🎯 Exam Tip: Proportional allocation in stratified sampling is a key technique to ensure representativeness when strata sizes differ.

 

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
In simple words: When every single item or unit is examined, like every vehicle at a checkpoint, it constitutes a complete census.

🎯 Exam Tip: An inquiry that includes every single member or item in a group is defined as a census; recognize real-world examples of this method.

 

Section - B

 

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
In simple words: The statement describes systematic sampling, not stratified sampling, which involves dividing a population into distinct subgroups.

🎯 Exam Tip: Differentiating between various sampling methods based on their characteristics is a key concept for exams.

 

Question 2. A statistic is a characteristic of a population.
Answer: False
In simple words: A statistic describes a characteristic of a sample, whereas a parameter describes a characteristic of a population.

🎯 Exam Tip: It is important to correctly identify whether a descriptive measure pertains to a sample (statistic) or a population (parameter).

 

Question 3. If a unit is to be destroyed during an inquiry then sample inquiry is not only necessary but also inevitable.
Answer: True
In simple words: In destructive testing, sampling is the only feasible method because examining every unit would lead to its destruction.

🎯 Exam Tip: Understand that destructive inquiries necessitate sampling to preserve the population while still gaining insights.

 

Question 4. When properties of the units of the population have more dissimilarity, the use of stratified random sampling method is advantageous.
Answer: True
In simple words: Stratified random sampling is beneficial when the population is diverse because it allows for homogeneous subgroups to be sampled separately.

🎯 Exam Tip: Recognize that stratified random sampling is particularly effective for heterogeneous populations, ensuring better representation across different subgroups.

 

Question 5. In simple random sampling method, each unit of the population has an equal chance of beings included in the sample.
Answer: True
In simple words: Simple random sampling ensures that every single member of the population has an identical probability of being chosen for the sample.

🎯 Exam Tip: The core principle of simple random sampling is the equal probability of selection for all units, which is crucial for unbiased representation.

 

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
In simple words: Dividing a population into homogeneous groups and then drawing random samples from each is characteristic of stratified sampling, not systematic sampling.

🎯 Exam Tip: Accurately distinguish between stratified sampling (dividing into homogeneous groups) and systematic sampling (selecting units at regular intervals).

 

Question 7. Each unit of the populations is examined in census inquiry.
Answer: True
In simple words: A census inquiry involves the complete enumeration and examination of every individual unit within the entire population.

🎯 Exam Tip: The defining feature of a census is the inclusion of every unit, ensuring a comprehensive dataset for the entire population.

 

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 method through which conclusions about a larger population are drawn using data from a selected sample is known as sampling.
In simple words: Sampling is the technique of using data from a small group (sample) to understand a larger group (population).

🎯 Exam Tip: A clear definition of sampling and its purpose is often tested in introductory statistics.

 

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 employed when each group exhibits low internal variation but significant differences exist among the groups.
In simple words: When groups are similar internally but distinct from each other, stratified random sampling is best to ensure all group types are represented.

🎯 Exam Tip: Stratified random sampling is ideal for heterogeneous populations that can be divided into internally homogeneous subgroups.

 

Question 3. In which sampling method, units are selected from the population at uniform intervals?
Answer: In systematic sampling method, units are chosen from the population at regular, uniform intervals.
In simple words: Systematic sampling involves selecting items from a list or sequence at consistent, predetermined intervals.

🎯 Exam Tip: Remember that the hallmark of systematic sampling is the selection of units based on a fixed interval after an initial random start.

 

Question 4. Which table of random numbers is most commonly used?
Answer: L.H.C. Tippet's table of random numbers is the most commonly utilized.
In simple words: Tippet's table is a widely recognized and frequently used resource for generating random numbers in sampling.

🎯 Exam Tip: Knowing the common tools and tables used in statistical methods, like Tippet's table, can be helpful for both theoretical and practical questions.

 

Question 5. Which type of inquiry involves more errors?
Answer: Population inquiry typically involves a greater number of errors.
In simple words: A full population study, or census, often has more errors due to its large scale and complexity.

🎯 Exam Tip: Despite its comprehensiveness, population inquiries can be prone to more non-sampling errors due to logistical challenges.

 

Question 6. What is meant by population inquiry?
Answer: Population inquiry refers to the process of gathering information from all units within the entire population.
In simple words: A population inquiry is when information is collected from every single member of a group.

🎯 Exam Tip: Define population inquiry accurately as the examination of every unit in a target group to achieve full data coverage.

 

Question 7. What do you mean by a sample without replacement?
Answer: A sample without replacement is one where a selected unit is not returned to the population before the next unit is chosen.
In simple words: In sampling without replacement, once an item is picked, it cannot be picked again.

🎯 Exam Tip: Clearly distinguish between sampling with and without replacement, as it impacts the probability of selection for subsequent units.

 

Question 8. When is the use of stratified random sampling considered to be favourable or suitable?
Answer: The use of stratified random sampling is favorable or suitable when the population is heterogeneous.
In simple words: Stratified random sampling is best when the overall population has distinct, different subgroups.

🎯 Exam Tip: Stratified random sampling is particularly effective for diverse populations, enabling precise estimation for various subgroups.

 

Question 9. When can the systematic random sample be biased?
Answer: A systematic random sample can become biased if there is a hidden periodicity in the population that coincides with the sampling interval.
In simple words: Systematic sampling can be biased if a hidden pattern in the population lines up with the selection interval.

🎯 Exam Tip: Emphasize the risk of bias in systematic sampling when population elements exhibit a periodic trend aligning with the sampling interval.

 

Question 10. Define heterogeneous population.
Answer: A heterogeneous population is defined as one where there is a considerable amount of variation among its constituent units.
In simple words: A heterogeneous population consists of units that are very different from each other.

🎯 Exam Tip: Grasping the concept of a heterogeneous population is vital for understanding why certain sampling methods, like stratified sampling, are preferred.

 

Question 11. Give an example an inquiry where units are destroyed during inspection.
Answer: An example of an inquiry where units are destroyed during inspection is testing the life of an electric bulb.
In simple words: Testing how long light bulbs last is an example where the items are destroyed during the check.

🎯 Exam Tip: Real-world examples of destructive testing, such as product quality control where items are consumed or broken, illustrate the necessity of sampling.

 

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 three-digit random numbers are provided and the population size is a two-digit number, only the first two digits of the random numbers are considered for sample selection.
In simple words: When a population is two digits, but random numbers are three digits, use only the first two digits of the random number.

🎯 Exam Tip: When random numbers have more digits than the population size, truncate or adjust them to match the required range for selection.

 

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 two-digit random numbers are provided and the population size is a three-digit number, then two digits from a column and the first digit of the next column are considered to form a three-digit number for sample selection.
In simple words: If random numbers are two digits and the population is three digits, combine two digits from one column with the first digit from the next column to make a three-digit number.

🎯 Exam Tip: For population sizes larger than the random numbers' digit count, combine digits from adjacent numbers or columns to create appropriate random numbers.

 

Question 14. Define parameters of the population.
Answer: Measures such as population mean, population standard deviation, and others that describe characteristics of the entire population are called parameters.
In simple words: Population parameters are numerical descriptions, like the average or spread, for an entire group.

🎯 Exam Tip: A parameter is a fixed, numerical characteristic of a population, usually unknown and estimated by statistics.

 

Question 15. Define sample statistics.
Answer: Sample statistics are measures, such as mean, standard deviation, and so forth, calculated from the numerical data collected from units within a sample.
In simple words: Sample statistics are numbers calculated from a sample, like the sample mean, used to estimate population characteristics.

🎯 Exam Tip: A statistic is a variable, numerical characteristic of a sample, used to infer information about the population parameter.

 

Section - C

 

Answer the following questions as required:

 

Question 1. When is sample inquiry undertaken?
Answer: Sample inquiry is conducted under the following conditions:
- The population size is very large.
- The population is geographically dispersed over a wide area.
- The units being investigated are destroyed during the inspection process.
- There are limitations in terms of available time, financial resources, and specialized expertise for carrying out the inquiry.
In simple words: Sampling is done when the population is too big or spread out, when testing destroys the items, or when time and money are limited.

🎯 Exam Tip: Understand the practical reasons why sample inquiry is often preferred or necessary, especially concerning scale, cost, and destructive testing.

 

Question 2. What is sampling?
Answer: Sampling is the systematic procedure of selecting a representative subset, known as a sample, from a larger population.
In simple words: Sampling is the process of choosing a smaller group from a larger group to learn about the larger group.

🎯 Exam Tip: A precise definition of sampling is fundamental, emphasizing the selection of a representative subset to draw conclusions about the whole.

 

Question 3. State the methods of selecting a simple random sample.
Answer: The primary methods for selecting a simple random sample include:
- The lottery method.
- The random number table method.
In simple words: Simple random samples can be chosen using methods like drawing names from a hat (lottery) or using a table of random numbers.

🎯 Exam Tip: Knowing the practical techniques for implementing simple random sampling is important for both theoretical understanding and problem-solving.

 

Question 4. Name various methods of sampling.
Answer: The various sampling methods are:
- Simple random sampling.
- Stratified random sampling.
- Systematic sampling.
In simple words: Key sampling methods include simple random, stratified, and systematic sampling.

🎯 Exam Tip: Be able to list and briefly describe the most common types of probability sampling methods.

 

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, the strategy for determining the number of units from each stratum involves drawing a sample where units are selected randomly in proportion to the stratum's size within the population.
In simple words: In stratified sampling, the number of units chosen from each subgroup (stratum) is proportional to that subgroup's size in the total population.

🎯 Exam Tip: Proportional allocation is a common and effective strategy in stratified random sampling, ensuring fair representation from all strata.

 

Question 6. Explain sample interval in systematic random sampling.
Answer: In systematic random sampling, the sample interval is defined as the ratio of the population size (N) to the desired sample size (n). It is denoted by 'k', which is a positive integer calculated as \( k = \frac{N}{n} \).
In simple words: The sample interval in systematic sampling is calculated by dividing the total population size by the desired sample size, which tells you how often to select a unit.

🎯 Exam Tip: Understand how to calculate the sample interval \( k \) as \( N/n \) in systematic sampling, as this is a core formula for this method.

 

Question 7. Explain the process of stratification.
Answer: Stratification is the process of dividing a heterogeneous population into non-overlapping groups, known as strata, which are relatively homogeneous within themselves.
In simple words: Stratification is when you split a diverse population into smaller, more uniform groups to make sampling easier and more representative.

🎯 Exam Tip: Define stratification clearly as the division of a heterogeneous population into homogeneous subgroups (strata) to improve sample representativeness.

 

Question 8. Define stratum in stratified random sampling.
Answer: In stratified random sampling, a stratum refers to a subgroup obtained through stratification, where units within the group are almost identical in terms of variation, even though different strata vary from one another.
In simple words: A stratum is a distinct subgroup formed by dividing a diverse population, where members within the subgroup are very similar to each other.

🎯 Exam Tip: Emphasize that a stratum is an internally homogeneous subgroup created from a heterogeneous population, crucial for stratified sampling.

 

Question 9. Write a note on systematic sampling.
Answer: In systematic sampling, the initial sample unit is selected randomly, and subsequent units are then automatically chosen at a definite, uniform interval from the population list. This method is advisable if a complete list of population units is available in a systematic order. If the population size is N and the sample size is n, the sampling interval \( k \) is determined as \( k = \frac{N}{n} \), where \( k \) is a positive integer. A random number is chosen from the first \( k \) units of the population, and then every \( k^{th} \) unit thereafter is selected. The collection of these selected units forms a systematic sample, and this process is known as systematic sampling.
Illustration: If the random number chosen from the first \( k \) units is 4, then the units at order 4, \( 4 + k \), \( 4 + 2k \), \( 4 + 3k \), and so on, from the population list, will constitute the systematic sample.
In simple words: Systematic sampling starts with a random pick, then selects every Nth item from a list. It's easy but can be biased if there's a hidden pattern in the list.

🎯 Exam Tip: Explain the mechanism of systematic sampling, including the random start and fixed interval, and illustrate it with an example to show practical application.

 

Question 10. State the advantages of systematic random sampling.
Answer: The advantages of systematic random sampling are as follows:
- It is simpler to draw a sample without errors because the order of the sample units is automatically determined.
- The sample is evenly distributed throughout the population.
- It requires less time and effort compared to simple random sampling and stratified random sampling.
In simple words: Systematic random sampling is easy to perform, spreads the sample evenly across the population, and saves time and effort compared to other methods.

🎯 Exam Tip: Highlight the efficiency and uniform coverage as key benefits of systematic random sampling, making it a practical choice in many scenarios.

 

Question 11. Why is population inquiry usually not feasible in practice?
Answer: Population inquiry is often impractical because it demands significant time, financial resources, and human power. Furthermore, the extensive and complex nature of organizing a complete population inquiry can lead to a greater number of errors.
In simple words: A full population study is often not practical because it takes too much time, money, and people, and can lead to more mistakes due to its huge scale.

🎯 Exam Tip: Recognize that resource constraints (time, cost, manpower) and increased potential for errors are major reasons why population inquiries are often avoided in favor of sampling.

 

Question 12. State advantages of sampling.
Answer: The advantages of sampling are:
- Detailed studies can be conducted because only a few units are examined in a sample study.
- The survey work requires less time because fewer units are studied.
- The cost of the study is significantly lower.
- Fewer personnel are needed for the survey work, allowing for the appointment of experts, which enhances the reliability of the results.
- Specialized equipment for certain surveys is required only if a few units are to be studied.
- In cases of destructive testing, sampling is the only viable option, as population study is impossible.
- A large geographical area can be covered within available time and budget.
- If proper sampling methods are employed, the results obtained will adequately represent the population.
In simple words: Sampling saves time and money, allows for detailed study with experts, works for destructive tests, covers large areas efficiently, and provides reliable results if done correctly.

🎯 Exam Tip: List the key benefits of sampling, such as cost-effectiveness, time efficiency, depth of study, and applicability in destructive testing, for a comprehensive answer.

 

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, assign numbers from 1 to 100 to each ATM of the bank. The population size is 100, which is a three-digit number. Therefore, any random numbers greater than 100 from the provided list will be disregarded.
Since a random sample without replacement is required, any repeated random numbers will also be ignored. The selected random numbers, after filtering, are:
018, 096, 027, 007, 012.
These five random numbers constitute the sample of 5 ATMs.
In simple words: To pick 5 ATMs out of 100 using three-digit random numbers, first assign numbers 1-100 to ATMs. Then, from the given list, ignore any number over 100 and any duplicates. The first five valid unique numbers become the sample.

🎯 Exam Tip: When selecting a sample using random numbers, filter out numbers outside the population range and discard duplicates for sampling without replacement.

 

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, assign numbers from 1 to 70 to the students in the classroom. The population size N = 70, which is a two-digit number. Consequently, only the first two digits of the random numbers will be considered, and any first two digits exceeding 70 will be ignored.
Since a random sample with replacement is to be obtained, repeated random numbers will be considered. The selected random numbers are: 27, 32, 59, 66, 32, 48, 25, 11.
A random sample of 7 students is required. Therefore, we select the first seven valid random numbers from the list above.
Thus, the random numbers for the selected seven students are: 27, 32, 59, 66, 32, 48, 25, which forms the random sample with replacement.
In simple words: To select 7 students out of 70 with replacement using three-digit random numbers, first number students 1-70. Then, take the first two digits of each random number and ignore any over 70. The first 7 valid numbers (including repeats) form the sample.

🎯 Exam Tip: For sampling with replacement, always consider numbers within the population range, and remember that duplicates are allowed and included in the sample count.

 

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, assign numbers from 1 to 350 to the units of the population. The population size is N = 350, a three-digit number. Thus, any random numbers greater than 350 will be disregarded.
A 2% sample means \( 350 \times \frac{2}{100} = 7 \) units are to be selected in the sample.

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

**Sample without replacement:**
Ignoring repeated numbers, the first seven valid random numbers are: 170, 111, 002, 203, 233, 300, 250.
Thus, the sample without replacement consists of these random numbers.
In simple words: To take a 2% sample (7 units) from 350 units using three-digit random numbers, first label units 1-350. Ignore random numbers over 350. For 'with replacement', take the first 7 valid numbers, including repeats. For 'without replacement', take the first 7 unique valid numbers.

🎯 Exam Tip: Practice both "with replacement" and "without replacement" scenarios, paying close attention to whether duplicates are included or excluded based on the sampling method.

 

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: The total population N = 600 students. A 2% sample without replacement is required, so the sample size \( n = 600 \times \frac{2}{100} = 12 \) students. Since there are 200 students in each of the three years (F.Y., S.Y., T.Y.), each stratum will have \( n_i = 200 \times \frac{2}{100} = 4 \) students.

**First stratum (F.Y.):** \( N_1 = 200 \), \( n_1 = 4 \).
Ignoring random numbers greater than 200 and repeated numbers from the F.Y. list, the selected random numbers are: 158, 092, 009, 200.
Thus, the sample units for the 4 First-Year students are: 158, 092, 009, 200.

**Second stratum (S.Y.):** \( N_2 = 200 \), \( n_2 = 4 \).
Ignoring random numbers greater than 200 and repeated numbers from the S.Y. list, the selected random numbers are: 019, 131, 057, 006.
Thus, the sample units for the 4 Second-Year students are: 019, 131, 057, 006.

**Third stratum (T.Y.):** \( N_3 = 200 \), \( n_3 = 4 \).
Ignoring random numbers greater than 200 and repeated numbers from the T.Y. list, the selected random numbers are: 027, 070, 198, 200, 058.
Thus, the sample units for the 4 Third-Year students are: 027, 070, 198, 200.
In simple words: To get a 2% sample (12 students total, 4 from each year) from 600 students using stratified sampling without replacement: for each year (F.Y., S.Y., T.Y.), from the provided random numbers, pick the first 4 unique numbers that are 200 or less. These will be the selected students from each year.

🎯 Exam Tip: For stratified random sampling without replacement, ensure that for each stratum, you only select unique random numbers within that stratum's range, and that the number of selections matches the calculated sample size for that stratum.

 

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:** Population \( N = 30 \), Sample \( n = 6 \).
Ignoring random numbers greater than 30 and repeated numbers, the selected random numbers are: 12, 18, 20, 11, 03, 10.
Hence, the sample units for the 6 small farm owners are: 12, 18, 20, 11, 03, 10.

**Large farm owners:** Population \( N = 20 \), Sample \( n = 4 \).
Ignoring random numbers greater than 20 and repeated numbers, the selected random numbers are: 04, 11, 08, 13, 18.
Hence, the sample units for the 4 large farm owners are: 04, 11, 08, 13, 18.

Thus, the randomly selected 10 farmers (6 small, 4 large) are:
Small farm owners: 12, 18, 20, 11, 03, 10
Large farm owners: 04, 11, 08, 13
In simple words: To choose 6 small farm owners from 30 and 4 large farm owners from 20 without replacement, use the provided random numbers. For small farmers, pick the first 6 unique numbers less than or equal to 30. For large farmers, pick the first 4 unique numbers less than or equal to 20.

🎯 Exam Tip: Carefully apply the rules for sampling without replacement (ignoring numbers out of range and duplicates) for each stratum to ensure accurate selection.

 

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, the total population \( N = 60 \) employees, and the sample size \( n = 5 \) employees. The sample interval \( k \) is calculated as \( k = \frac{N}{n} = \frac{60}{5} = 12 \).
To select the sample:
- Assign numbers from 1 to 60 to all employees of the IT company.
- Select a random number within the first sample interval, which is from 1 to 12.
- Suppose the selected random number is 7.
- Then, select every 12th number starting from the 7th. This process will yield four additional numbers.
- The sample units at positions 7, 19, 31, 43, and 55 will form the systematic sample of employees.
In simple words: To pick 5 employees out of 60 for a trial using systematic sampling: number all employees 1-60. Calculate the interval (60/5 = 12). Pick a random start number between 1 and 12 (e.g., 7). Then, select every 12th employee from that start (7, 19, 31, 43, 55).

🎯 Exam Tip: Demonstrate understanding of systematic sampling by correctly calculating the interval, choosing a random start, and then applying the interval sequentially to select the sample units.

 

Question 19. Select all possible samples of size 4 using systematic sampling from a population of 20 units.
Answer: Here, the total population \( N = 20 \) units, and the sample size \( n = 4 \) units. The sample interval \( k \) is calculated as \( k = \frac{20}{4} = 5 \).
To select all possible systematic samples of size 4, the possible random starting numbers within the interval \( k \) (1 to 5) can be 1, 2, 3, 4, or 5.
- If the selected random number is 1, the sample is: 1, 6, 11, 16.
- If the selected random number is 2, the sample is: 2, 7, 12, 17.
- If the selected random number is 3, the sample is: 3, 8, 13, 18.
- If the selected random number is 4, the sample is: 4, 9, 14, 19.
- If the selected random number is 5, the sample is: 5, 10, 15, 20.
In simple words: To find all possible systematic samples of 4 from 20 units, the interval is 5. So, you can start with any number from 1 to 5, and then keep adding 5 to get the rest of the sample units (e.g., start at 1: 1, 6, 11, 16; start at 2: 2, 7, 12, 17, etc.).

🎯 Exam Tip: For systematic sampling, remember that the number of possible samples equals the sampling interval (k), and each sample is generated by a unique starting point within the first k units.

 

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, the total population \( N = 30 \) students, and the sample size \( n = 10 \) students. The sample interval \( k \) is calculated as \( k = \frac{30}{10} = 3 \).
Since \( k = 3 \), the possible selected random starting numbers can be 1, 2, or 3. Therefore, 3 random samples can be obtained using systematic sampling.
- If the selected random number is 1, the sample is: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28.
- If the selected random number is 2, the sample is: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29.
- If the selected random number is 3, the sample is: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
In simple words: To find how many systematic samples can be taken to check 10 homeworks from 30 students, divide the total students by the sample size (30/10 = 3). This means 3 different samples are possible, depending on the random starting point (1, 2, or 3).

🎯 Exam Tip: The number of possible systematic samples from a population is equal to the sampling interval \( k \), as each unique starting point within \( k \) generates a distinct sample.

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: To begin, we assign numbers from 1 to 70 to each student in the classroom. Since the population size \(N = 70\) is a two-digit number, we only consider the first two digits of the provided random numbers, discarding any that are greater than 70.

Given that a random sample with replacement is required, repeated random numbers are included.
The selected random numbers, after filtering for those not exceeding 70, are: 27, 32, 59, 66, 32, 48, 25, 11.
We need to obtain a random sample of 7 students. Therefore, selecting the first seven applicable numbers from the list yields the following: 27, 32, 59, 66, 32, 48, 25. This constitutes the random sample with replacement.
In simple words: To pick 7 students from 70 with replacement, assign numbers 1-70. Use the first two digits of given random numbers, ignoring those over 70. The first 7 valid numbers (allowing repeats) form the sample.

🎯 Exam Tip: For sampling with replacement, ensure that duplicate random numbers are not removed, as each selection is independent and the unit is returned to the population for potential re-selection. Always clarify the population size and the number of digits to consider from random numbers.

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, numbers 1 to 350 are assigned to the units within the population.
The population size, \(N\), is 350, which is a three-digit number. Therefore, any random numbers exceeding 350 must be disregarded.
The initial filtered random numbers are: 170, 111, 002, 203, 111, 233, 300, 250, 010.
A 2% sample implies selecting \(350 \times \frac{2}{100} = 7\) units for the sample.

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

For **Sample without replacement**: Ignoring the repeated numbers, the first seven unique random numbers are: 170, 111, 002, 203, 233, 300, 250.
Thus, the sample without replacement consists of the random numbers 170, 111, 002, 203, 233, 300, 250.
In simple words: From a population of 350, we need a 2% sample (7 units). Assign numbers 1-350. Filter out random numbers above 350. For "with replacement", take the first 7 valid numbers, including repeats. For "without replacement", take the first 7 unique valid numbers.

🎯 Exam Tip: Always calculate the required sample size first. Differentiate clearly between "with replacement" (where duplicates are allowed) and "without replacement" (where duplicates are ignored) when selecting units based on random numbers.

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: The total population size, \(N\), is 600.
We need to obtain a 2% sample without replacement. The sample size, \(n\), is therefore \(600 \times \frac{2}{100} = 12\).

**First stratum (F.Y. students):**
Population size \(N_1 = 200\). Sample size \(n_1 = 200 \times \frac{2}{100} = 4\).
Ignoring random numbers greater than 200 and any repeated numbers, the selected random numbers for F.Y. are: 158, 092, 009, 200.
Hence, the units for the sample of 4 F.Y. students are: 158, 092, 009, 200.

**Second stratum (S.Y. students):**
Population size \(N_2 = 200\). Sample size \(n_2 = 200 \times \frac{2}{100} = 4\).
Ignoring random numbers greater than 200 and any repeated numbers, the selected random numbers for S.Y. are: 019, 131, 057, 006.
Hence, the units for the sample of 4 S.Y. students are: 019, 131, 057, 006.

**Third stratum (T.Y. students):**
Population size \(N_3 = 200\). Sample size \(n_3 = 200 \times \frac{2}{100} = 4\).
Ignoring random numbers greater than 200 and any repeated numbers, the selected random numbers for T.Y. are: 027, 070, 198, 200, 058.
Hence, the units for the sample of 4 T.Y. students are: 027, 070, 198, 200.
In simple words: To get a 2% sample (12 students total) from 600 students across three years (200 each), we take 4 students from each year. For each year, we use the given random numbers, ignore those above 200, and remove duplicates to pick the 4 students for that year's sample.

🎯 Exam Tip: When dealing with stratified sampling and random numbers, remember to apply the population size filter and the "without replacement" rule to *each stratum independently* before combining the results.

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:** The population size \(N = 30\), and the required sample size \(n = 6\).
Ignoring random numbers greater than 30 and any repeated numbers from the given list, the selected random numbers are: 18, 20, 11, 03, 10. The number 12 is missing from the provided list of selected numbers in the original text, but it's important to note the process. Let's re-evaluate based on the *given* numbers: 2, 95, 18, 96, 20, 84, 56, 11, 52, 03, 10, 45.
Valid numbers (<=30): 2, 18, 20, 11, 03, 10. (6 unique numbers, exactly the sample size).
Hence, the sample units for the 6 small farm owners are: 2, 18, 20, 11, 03, 10. (Original text had 12, 18, 20, 11, 03, 10, which implies 2 was skipped or 12 was generated differently, but I will stick to processing the *given* random numbers.)

**Large farm owners:** The population size \(N = 20\), and the required sample size \(n = 4\).
Ignoring random numbers greater than 20 and any repeated numbers from the given list (4, 40, 34, 11, 72, 11, 50, 55, 08, 13, 76, 18), the selected random numbers are: 04, 11, 08, 13, 18.
Valid unique numbers (<=20): 4, 11, 8, 13, 18. We need 4, so we take the first 4.
Hence, the sample units for the 4 large farm owners are: 04, 11, 08, 13.

Thus, the randomly selected 10 farmers are:
Small farm owners: 2, 18, 20, 11, 03, 10
Large farm owners: 04, 11, 08, 13
In simple words: To select 6 small and 4 large farm owners without replacement, filter the provided random numbers. For small farmers, keep numbers 1-30 and take the first 6 unique ones. For large farmers, keep numbers 1-20 and take the first 4 unique ones.

🎯 Exam Tip: When performing sampling "without replacement," it is crucial to identify and discard any duplicate random numbers *before* finalizing the sample. Ensure the number of selected units matches the required sample size for each stratum.

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: In this scenario, the population size \(N = 60\), and the sample size \(n = 5\).
The sample interval \(k\) is calculated as \(k = \frac{N}{n} = \frac{60}{5} = 12\).

The selection process involves the following steps:

- Assign numbers from 1 to 60 to all employees of the IT company.
- Select a random starting number within the first sample interval, which is from 1 to 12.
- Let's assume the randomly selected starting number is 7.
- From this starting point (7), select every 12th employee. This means the subsequent selections will be \(7 + 12 = 19\), \(19 + 12 = 31\), \(31 + 12 = 43\), and \(43 + 12 = 55\).
- The sample will consist of the employees at positions 7, 19, 31, 43, and 55. These five employees will form the sample for the 'work from home' trial.
In simple words: To select 5 employees systematically from 60, first number them 1-60. Calculate the interval: 60/5 = 12. Pick a random starting number between 1 and 12 (e.g., 7). Then, select every 12th employee from that start (7, 19, 31, 43, 55).

🎯 Exam Tip: Systematic sampling requires calculating the sampling interval (k) and then selecting a random starting point within the first 'k' units. Subsequent units are chosen by adding 'k' to the previous selection. Ensure all steps are clearly laid out.

Question 19. Select all possible samples of size 4 using systematic sampling from a population of 20 units.
Answer: Here, the population size \(N = 20\), and the sample size \(n = 4\). The sample interval \(k = \frac{20}{4} = 5\).

To select all possible systematic samples of size 4, the possible random starting numbers within the first interval (1 to \(k\)) are 1, 2, 3, 4, or 5.

- If the selected random starting number is 1:
Sample 1: {1, 6, 11, 16}
- If the selected random starting number is 2:
Sample 2: {2, 7, 12, 17}
- If the selected random starting number is 3:
Sample 3: {3, 8, 13, 18}
- If the selected random starting number is 4:
Sample 4: {4, 9, 14, 19}
- If the selected random starting number is 5:
Sample 5: {5, 10, 15, 20}
In simple words: For 20 units and a sample of 4, the interval is 5. This means there are 5 possible starting points (1, 2, 3, 4, or 5). Each starting point generates a unique sample by adding 5 repeatedly until 4 units are selected.

🎯 Exam Tip: When asked for "all possible samples" in systematic sampling, generate a sample for each potential starting point within the first sampling interval. This demonstrates a complete understanding of the method's potential outcomes.

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: In this case, the total population of students, \(N = 30\), and the required sample size, \(n = 10\).
The sample interval \(k\) is calculated as \(k = \frac{30}{10} = 3\).

Since \(k = 3\), the possible random starting numbers for systematic sampling are 1, 2, or 3. Each of these starting numbers will generate a unique sample.

- If the selected random starting number is 1:
Sample 1: {1, 4, 7, 10, 13, 16, 19, 22, 25, 28}
- If the selected random starting number is 2:
Sample 2: {2, 5, 8, 11, 14, 17, 20, 23, 26, 29}
- If the selected random starting number is 3:
Sample 3: {3, 6, 9, 12, 15, 18, 21, 24, 27, 30}
Therefore, exactly 3 distinct random samples can be obtained using systematic sampling.
In simple words: With 30 students and needing to check 10, the interval is 30/10 = 3. This means there are 3 possible unique systematic samples, starting with either student 1, 2, or 3.

🎯 Exam Tip: The number of possible systematic samples is always equal to the sampling interval \(k\). Clearly demonstrating each possible sample set reinforces the answer and shows full comprehension.