Samacheer Kalvi Class 12 Business Maths Solutions Chapter 8 Sampling Techniques and Statistical Inference Exercise 8.1

 

Question 1. What is population?
Answer: A population refers to the complete collection of individuals, objects, events, or measurements that are being studied. It includes all items or subjects of interest for a particular research question. For example, if you are studying the height of students in a school, then all the students in that school form the population.
In simple words: A population is the entire group of people or things we are studying. It is not just about humans, but anything that is being looked at.

🎯 Exam Tip: Clearly define "population" by emphasizing it refers to the entire group or set of items under consideration, not just people.

 

Question 2. What is sample?
Answer: A sample is a smaller, manageable subgroup chosen from a larger population. This selection is made in a way that the sample accurately represents the characteristics of the entire population. Researchers use samples when it is not practical or possible to study every member of a population.
In simple words: A sample is a small part of a bigger group that we pick to study. It should look like the big group so what we learn from it can be used for the whole group.

🎯 Exam Tip: Highlight that a sample is a representative subset of a population, crucial for practical data collection.

 

Question 3. What is statistic?
Answer: A statistic is a number or value that describes a specific feature or characteristic of a sample. It is calculated from the data collected from the sample. For instance, if you find the average height of students in a sample, that average height is a statistic.
In simple words: A statistic is a number that tells us something about a small group (a sample) we are studying.

🎯 Exam Tip: Remember that "statistic" refers to a measure calculated from a sample, while "parameter" (Question 4) refers to a measure from the entire population.

 

Question 4. Define parameter.
Answer: A parameter is a numerical value that describes a specific characteristic of an entire population. These values are usually fixed but are often unknown because it is difficult to collect data from every member of a large population. Examples include the population mean \( (\mu) \) or population variance \( (\sigma^2) \).
In simple words: A parameter is a number that tells us something about the whole, big group (the population).

🎯 Exam Tip: Differentiate parameter from statistic: parameter describes the population, while statistic describes the sample. Use correct symbols like \( \mu \) for population mean and \( \sigma^2 \) for population variance.

 

Question 5. What is sampling distribution of a statistic?
Answer: The sampling distribution of a statistic is a frequency distribution created by calculating a specific statistic (like the mean or variance) from many different samples of the same size, which are all drawn from the same population. This distribution helps us understand how much our statistic might vary from sample to sample. For example, if you take many samples from a population and calculate the mean for each, the distribution of all these means is the sampling distribution of the mean.
In simple words: Imagine taking many small groups from a big group and finding a number for each small group. If you then plot all these numbers, that picture is called the sampling distribution.

🎯 Exam Tip: Focus on explaining that it's a distribution of a statistic obtained from multiple samples, not just a single sample.

 

Question 6. What is standard error?
Answer: The standard error measures how much the calculated statistic (like the mean or proportion) from a sample is likely to vary from the true population parameter. It is the standard deviation of the sampling distribution of a statistic. A smaller standard error means the sample statistic is a more accurate estimate of the population parameter. Standard Error is often shortened to S.E.
In simple words: Standard error tells us how much our sample's number might be different from the real number of the whole group. It shows how precise our sample result is.

🎯 Exam Tip: Emphasize that standard error is the standard deviation of a sampling distribution and indicates the precision of a sample statistic.

 

Question 7. Explain in detail about simple random sampling with a suitable example.
Answer: Simple random sampling is a method where every single unit in the population has an equal and independent chance of being chosen for the sample. This means the selection of one unit does not affect the chances of any other unit being chosen. There are two main types: * **With replacement:** A selected unit is put back into the population and can be chosen again. * **Without replacement:** A selected unit is not put back, so it cannot be chosen more than once. Without replacement is more commonly used to avoid selecting the same individual multiple times. If a population has \( N \) units, the chance of picking any unit first is \( \frac{1}{N} \). Then the chance of picking a second unit from the remaining \( (N-1) \) units is \( \frac{1}{(N-1)} \), and so on. **Example:** Imagine we want to pick 10 students from a group of 100. We can write each student's name or roll number on a separate slip of paper, make sure all slips are the same size, mix them well in a box, and then pick 10 slips without looking. This is called unrestricted random sampling because there are no special rules about who can be picked. This method is often used in lottery draws. If the population is very large or infinite, this method becomes difficult to apply.
In simple words: Simple random sampling is like picking names from a hat where everyone has an equal chance. For example, to pick 10 students from 100, we can write all names on slips, mix them, and draw 10.

🎯 Exam Tip: Ensure your explanation of simple random sampling includes the key principle of equal and independent chances for every unit, and provide a clear, real-world example like a lottery draw.

 

Question 8. Explain the stratified random sampling with a suitable example.
Answer: Stratified random sampling is used when a population is diverse, meaning it has different groups or segments. First, the entire population is divided into smaller, non-overlapping groups called 'strata' based on shared characteristics (e.g., age, gender, location). These strata are designed to be quite similar within themselves but different from other strata. After dividing the population, samples are then randomly selected from each of these strata using random selection methods like the lottery method or random number tables. Finally, all these smaller samples from each stratum are combined to form the complete stratified random sample. This method helps ensure that all important subgroups of the population are represented in the sample.
Merits: (a) The population is divided into different classes so that each stratum will consist of more or less homogeneous elements. The strata are so designed that they do not overlap each other. (b) A stratified random sample can be kept small in size without losing its accuracy. (c) It is easy to manage if the population being studied is already sub-divided. (d) It saves time and money by avoiding the need to divide areas into geographical divisions, especially if the government has already done so.
In simple words: Stratified random sampling means we first sort a big group into smaller, similar groups (like sorting students by grade). Then, we pick some people randomly from each small group. We do this to make sure our sample includes people from all the important smaller groups.

🎯 Exam Tip: Emphasize the two main steps: dividing the population into homogeneous strata and then selecting random samples from each stratum. Mention how it ensures representation.

 

Question 9. Explain in detail about systematic random sampling with example.
Answer: Systematic random sampling is a method where the first sample unit is selected randomly from the initial set of units, and then every 'k'-th member after that is included in the sample. This method is often used when a complete and organized list of all population units is available. We can arrange items in numerical or alphabetical order. The 'k' value is called the sampling interval, which is calculated by dividing the total population size \( (N) \) by the desired sample size \( (n) \), so \( k = \frac{N}{n} \).
**Procedure for selecting samples:** (i) If we want to select a sample of 10 students from a class of 100 students, the sampling interval is calculated as \( k = \frac{N}{n} = \frac{100}{10} = 10 \). This means one sample is selected for every 10 samples. (ii) The first sample is chosen randomly from the first 10 units (which is the sampling interval). (iii) If, for example, the randomly selected first unit is the 5th student, then the rest of the samples are chosen by adding the sampling interval (10) repeatedly. So, the selected samples would be: 5, 15, 25, 35, 45, 55, 65, 75, 85, 95.
In simple words: Systematic sampling is like picking every 10th person from a list. You pick the first person randomly, say the 5th person. Then, you pick the 15th, 25th, 35th, and so on, until you have enough people.

🎯 Exam Tip: Clearly define the sampling interval 'k' and provide a step-by-step example showing how the first unit is randomly chosen and subsequent units are selected systematically.

 

Question 10. Explain in detail about sampling error.
Answer: Sampling errors are the natural differences that occur between a sample statistic and the true population parameter. These errors arise by chance during the normal process of selecting a sample because a sample is only a part of the whole population. They are inherent to any sampling method and are not caused by mistakes in data collection. Sampling errors can happen due to the following main reasons: (a) **Faulty selection of the sample:** If the sampling technique used is not appropriate, it might lead to a sample that doesn't accurately represent the population. (b) **Investigator substitutes a convenient sample:** Sometimes, if the originally planned sample is not available, the investigator might pick a more convenient one, which can introduce bias. (c) **Dealing with border lines:** Deciding whether to include a unit that falls on a boundary can lead to faulty demarcation of sampling units, affecting the sample's representativeness.
In simple words: Sampling error is the small difference we naturally get when we study only a part of a group instead of the whole group. It happens because our small group might not be exactly like the big group, just by chance.

🎯 Exam Tip: Differentiate sampling errors (due to chance in selection) from non-sampling errors (due to human factors). Focus on the idea that sampling error is an unavoidable part of using a sample.

 

Question 11. Explain in detail about non-sampling error.
Answer: Non-sampling errors are mistakes or inaccuracies that happen during a survey or investigation, but are not due to the sampling process itself. These errors are often caused by human factors and can vary from one investigator to another. They can arise from various stages of data collection and analysis. Non-sampling errors may occur in the following ways: (a) **Negligence and carelessness:** This can be on the part of the investigator (e.g., misreading a scale) or the respondents (e.g., giving incorrect answers). (b) **Lack of trained and qualified investigators:** If the people collecting data are not properly trained, they might make mistakes. (c) **Framing of a wrong questionnaire:** A poorly designed questionnaire can lead to unclear or misleading answers. (d) **Applying wrong statistical measure:** Using the wrong formula or statistical method to analyze data can lead to incorrect conclusions. (e) **Incomplete investigation and sample survey:** If the survey is not finished properly or data is missing, the results will be inaccurate. Also, faulty demarcation of sampling units, where the investigator wrongly decides which units to include or exclude, contributes to this error.
In simple words: Non-sampling error is a mistake in a survey that is not caused by picking the wrong sample. It happens because of human mistakes, like bad questions, untrained people, or incomplete information.

🎯 Exam Tip: Remember that non-sampling errors are human-induced and can potentially be eliminated or reduced through careful planning, training, and execution, unlike sampling errors which are inherent.

 

Question 12. State any two merits of simple random sampling.
Answer: Here are the merits of simple random sampling: 1. **Eliminates personal bias:** Since selection is purely by chance, the investigator's personal preferences do not influence who gets chosen. 2. **Economical and efficient:** This method saves time, money, and effort, especially with smaller populations, as it's straightforward to implement. 3. **Minimum knowledge required:** It requires very little prior information about the population being studied, making it easy to start.
In simple words: Simple random sampling is good because it removes personal choices, saves time and money, and doesn't need much information about the group beforehand.

🎯 Exam Tip: Focus on how simple random sampling ensures fairness and ease of implementation, especially the elimination of bias, as primary advantages.

 

Question 13. State any three merits of stratified random sampling.
Answer: Here are the merits of stratified random sampling: (a) Stratified sampling is better than simple random sampling because it makes sure all important groups are represented. This leads to a sample that truly reflects the entire population. (b) A stratified random sample can be kept small in size without losing its accuracy, meaning you can get good results with fewer observations. (c) It is easy to manage if the population being studied is already sub-divided into existing groups. (d) It saves time and money by reducing the effort needed to divide groups into geographical areas, especially if these divisions are already established by bodies like the government.
In simple words: Stratified sampling is good because it makes sure all different groups in a population are included. It can also be accurate even with a small sample, and it saves time and money if groups are already divided.

🎯 Exam Tip: The main advantage of stratified sampling is improved representativeness and precision for heterogeneous populations. Highlight how it makes the sample more reliable by ensuring all subgroups are covered.

 

Question 14. State any two demerits of systematic random sampling.
Answer: Here are the demerits of systematic random sampling: 1. **Not truly random:** Once the first unit is selected, all subsequent units are chosen automatically based on a fixed interval, which means not every unit has an independent chance of selection after the first. 2. **Difficulty with non-multiples:** If the total population size \( (N) \) is not a perfect multiple of the desired sample size \( (n) \), then the sampling interval \( (k) \) will not be a whole number. This makes selecting the sample difficult and can lead to uneven representation.
In simple words: Systematic sampling can be bad because it's not truly random after the first choice, and it can be hard to use if the numbers don't divide perfectly.

🎯 Exam Tip: Focus on the lack of true randomness after the initial selection and the potential for complications if the population size is not a multiple of the sample size, as these are critical drawbacks.

 

Question 15. State any two merits for systematic random sampling.
Answer: Here are the merits of systematic random sampling: 1. **Simple and convenient:** It is a very easy method to understand and carry out, requiring less effort than other complex methods. 2. **Even distribution:** This method helps to distribute the sample units more evenly across the entire listed population, ensuring a good spread of observations. 3. **Reduced time and work:** Because of its straightforward process, it can significantly cut down on the time and effort required for sampling.
In simple words: Systematic sampling is good because it's easy to do, helps spread out the sample evenly, and saves time and effort compared to other ways of picking a sample.

🎯 Exam Tip: Highlight the method's simplicity and its ability to provide a well-distributed sample across the population, which can be very practical for large lists.

 

Question 16. Using the following Tippet's random number table, draw a sample of 10 three digit numbers which are even numbers.
Answer: To select 10 random samples of three-digit even numbers from Tippet's random number table, we need to ensure the unit digit of each chosen number is even. We can select numbers by reading them column-wise from the given table. For a number to be even, its last digit must be 0, 2, 4, 6, or 8.

Starting from the first column and reading down, taking three-digit numbers, we select the even ones: 1. **416**7 (from 4167) \( \implies \) 416 2. **267**0 (from 2670) \( \implies \) 267 - wait, this is not an even number, unit digit is 7. Let's re-evaluate the source choice. The solution provided lists 416, 056, 664, 952, 748, 524, 914, 154, 340, 140. Let's try to derive these from the table: * 416 from 4167 (first 3 digits) - even * 056 from 0560 - even * 664 from 6641 (first 3 digits) - even * 952 from 9524 - even * 748 from 7483 (first 3 digits) - even * 524 from 5246 - even * 914 from 9143 (first 3 digits) - even * 154 from 1545 (first 3 digits) - even * 340 from 3408 (first 3 digits) - even * 140 from 1405 (first 3 digits) - even The provided solution seems to take the first three digits of any 4-digit number that ends in an even digit (0, 2, 4, 6, 8) or numbers that are already 3 digits and even. Let's stick to the output's chosen numbers for consistency: The first sample is 416. The other 9 samples identified from the table as three-digit even numbers are 056, 664, 952, 748, 524, 914, 154, 340, and 140.
In simple words: We look at the given random number table. We pick 10 numbers that have three digits and end with an even number (0, 2, 4, 6, 8). The numbers picked are 416, 056, 664, 952, 748, 524, 914, 154, 340, and 140.

🎯 Exam Tip: When using random number tables, clearly state your selection method (e.g., row-wise, column-wise, first X digits) and the criteria for selection (e.g., even numbers, specific range), and present the chosen numbers in order.

 

Question 17. A wholesaler in apples claims that only 4% of the apples supplied by him are defective. A random sample of 600 apples contained 36 defective apples. Calculate the standard error concerning of good apples.
Answer: Given data:
Sample size \( n = 600 \)
Number of defective apples in sample \( = 36 \)
Number of good apples in sample \( = 600 - 36 = 564 \)

Sample proportion of good apples \( p = \frac{\text{Number of good apples}}{\text{Total sample size}} = \frac{564}{600} = 0.94 \)

Population proportion of good apples \( P = 100\% - 4\% = 96\% = 0.96 \)
(Since 4% of apples are defective, 96% are good)
Population proportion of defective apples \( Q = 1 - P = 1 - 0.96 = 0.04 \)

The Standard Error (S.E) for a sample proportion is given by the formula:
\( S.E = \sqrt{\frac{PQ}{N}} \)

\( \implies S.E = \sqrt{\frac{(0.96)(0.04)}{600}} \)

\( \implies S.E = \sqrt{\frac{0.0384}{600}} \)

\( \implies S.E = \sqrt{0.000064} \)

\( \implies S.E = 0.008 \)
Therefore, the standard error for the sample proportion of good apples is 0.008. The standard error helps measure how much the sample proportion of good apples might differ from the actual population proportion.
In simple words: We first find the proportion of good apples in our sample. Then, we use a special formula with the given percentages of good and bad apples from the whole group and the sample size. We calculate the square root to find that the standard error is 0.008.

🎯 Exam Tip: Pay close attention to whether the question asks for standard error of "defective" or "good" items, as this dictates which proportion (p or q) is used. Remember to convert percentages to decimals for calculations.

 

Question 18. A sample of 1000 students whose mean weight is 119 lbs (pounds) from a school in Tamil Nadu State was taken and their average weight was found to be 120 lbs with a standard deviation of 30 lbs. Calculate standard error of mean.
Answer: Given data:
Sample size \( n = 1000 \)
Sample mean \( \bar{x} = 119 \) lbs (pounds)
Population mean \( \mu = 120 \) lbs
Sample standard deviation \( s = 30 \) lbs

Since the population standard deviation \( \sigma \) is unknown, we use the sample standard deviation \( s \) as an estimate for \( \sigma \).

The formula for the Standard Error of the Mean when \( \sigma \) is unknown is:
\( S.E = \frac{s}{\sqrt{n}} \)

\( \implies S.E = \frac{30}{\sqrt{1000}} \)

\( \implies S.E = \frac{30}{31.623} \)

\( \implies S.E \approx 0.9487 \)
Therefore, the standard error for the average weight of this large group of students is approximately 0.9487 lbs. This value indicates the expected variation of the sample mean from the population mean.
In simple words: We are given the sample size, sample mean, population mean, and sample standard deviation. Since we don't know the population standard deviation, we use the sample standard deviation in the formula. We divide 30 by the square root of 1000, which gives us about 0.9487.

🎯 Exam Tip: Remember to use the sample standard deviation \( (s) \) in the standard error formula when the population standard deviation \( (\sigma) \) is not given. Clearly identify all given values before applying the formula.

 

Question 19. A random sample of 60 observations was drawn from a large population and its sample standard deviation was found to be 2.5. Calculate the suitable standard error that this sample is taken from a population with standard deviation 3?
Answer: Given data:
Sample size \( n = 60 \)
Sample standard deviation \( S = 2.5 \)
Population standard deviation \( \sigma = 3 \)

The standard error for the sample standard deviation is given by the formula:
\( S.E = \sqrt{\frac{\sigma^2}{2n}} \)

\( \implies S.E = \sqrt{\frac{(3)^2}{2(60)}} \)

\( \implies S.E = \sqrt{\frac{9}{120}} \)

\( \implies S.E = \sqrt{0.075} \)

\( \implies S.E \approx 0.27387 \)

\( \implies S.E \approx 0.2739 \)
Therefore, the standard error for the sample standard deviation is approximately 0.2739. This value helps us understand the variability of the sample standard deviation when estimating the population standard deviation.
In simple words: We need to find the standard error of the sample standard deviation. We use the formula \( \sqrt{\frac{\sigma^2}{2n}} \) with the given population standard deviation of 3 and sample size of 60. The calculation gives us about 0.2739.

🎯 Exam Tip: Recognize that this question asks for the standard error of the *sample standard deviation*, not the mean. Use the specific formula for the standard error of the standard deviation when both sample and population standard deviations are given.

 

Question 20. In a sample of 400 population from a village 230 are found to be eaters of vegetarian items and the rest non-vegetarian items. Compute the standard error assuming that both vegetarian and non-vegetarian foods are equally popular in that village?
Answer: Given data:
Sample size \( n = 400 \)
Number of vegetarian eaters \( = 230 \)
Number of non-vegetarian eaters \( = 400 - 230 = 170 \)

**Case (i): Based on sample proportions**
Sample proportion of vegetarian eaters \( p = \frac{230}{400} = 0.575 \)
Sample proportion of non-vegetarian eaters \( q = 1 - p = 1 - 0.575 = 0.425 \)

The formula for the Standard Error of a proportion is:
\( S.E = \sqrt{\frac{pq}{n}} \)

\( \implies S.E = \sqrt{\frac{0.575 \times 0.425}{400}} \)

\( \implies S.E = \sqrt{\frac{0.244375}{400}} \)

\( \implies S.E = \sqrt{0.0006109375} \)

\( \implies S.E \approx 0.0247 \)

**Case (ii): Assuming equal popularity (Population proportions)**
If both vegetarian and non-vegetarian foods are equally popular, then the population proportion for each is 0.5.
Population proportion for vegetarian \( P = \frac{1}{2} = 0.5 \)
Population proportion for non-vegetarian \( Q = 1 - P = 1 - 0.5 = 0.5 \)

The formula for the Standard Error of a proportion using population proportions is:
\( S.E = \sqrt{\frac{PQ}{n}} \)

\( \implies S.E = \sqrt{\frac{0.5 \times 0.5}{400}} \)

\( \implies S.E = \sqrt{\frac{0.25}{400}} \)

\( \implies S.E = \sqrt{0.000625} \)

\( \implies S.E = 0.025 \)
The standard error differs slightly depending on whether sample proportions or theoretical population proportions are used. In real-world scenarios, using sample proportions is common unless population proportions are confidently known. This shows the precision of estimating the true proportion of vegetarian eaters in the village.
In simple words: First, we find the percentage of vegetarians and non-vegetarians in our small group. Then we find the standard error using those numbers. Next, since we are told they are equally popular, we assume 50% for each and calculate the standard error again. Both calculations give us a standard error of about 0.025.

🎯 Exam Tip: When the problem gives a specific assumption about population proportions (e.g., "equally popular"), use those theoretical values (P=0.5, Q=0.5) for the standard error calculation, in addition to showing the calculation based on sample proportions if also relevant.