Sampling and Estimation – Masters 1st Sem.

Sampling and Estimation TU Solution

2018 June Make up Q.No. 11 – 3 Marks

Give a brief description about sampling distribution. [3]

A sampling distribution is a probability distribution of a statistic obtained from a large sample of a population. For a population statistic, the sampling distribution is the frequency distribution of possible outcomes. The variability of a sampling distribution is determined by the number of observations in a population, the number of observations in a sample, and the sample selection procedure. On calls the standard error of a sampling distribution.
When a large sample is taken from a population, the sampling distribution’s mean is the population’s mean, and vice Suppose your population mean (X bar) is 99. (as long as you have a sufficiently large sample size). In general, the standard error of a sampling distribution depends on the population’s mean, standard deviation, and sample size. A normal distribution is found in a population or a sample of numbers. A sampling distribution does not have to be bell-shaped because it includes multiple sets of data points.

नमूना वितरण भनेको जनसंख्याको ठूलो नमूनाबाट प्राप्त तथ्याङ्कको सम्भाव्यता वितरण हो। जनसंख्या तथ्याङ्कको लागि, नमूना वितरण भनेको सम्भावित परिणामहरूको आवृत्ति वितरण हो। नमूना वितरणको परिवर्तनशीलता जनसंख्यामा अवलोकनहरूको संख्या, नमूनामा अवलोकनहरूको संख्या र नमूना चयन प्रक्रियाद्वारा निर्धारण गरिन्छ। कलहरूमा नमूना वितरणको मानक त्रुटि।
जब जनसंख्याबाट ठूलो नमूना लिइन्छ, नमूना वितरणको मतलब जनसङ्ख्याको औसत हुन्छ, र मानौं तपाईंको जनसंख्याको मतलब (X बार) 99 हो। (जबसम्म तपाईंसँग पर्याप्त ठूलो नमूना आकार छ)। सामान्यतया, नमूना वितरणको मानक त्रुटि जनसंख्याको औसत, मानक विचलन, र नमूना आकारमा निर्भर गर्दछ। सामान्य वितरण जनसंख्या वा संख्याहरूको नमूनामा पाइन्छ। नमूना वितरण घण्टी आकारको हुनु पर्दैन किनभने यसले डेटा पोइन्टहरूको बहु सेटहरू समावेश गर्दछ।

2016 ONo. 11

Describe about probability and non-probability sampling.

Probability Sampling: Any sampling method that uses a random selection. An equal chance of being chosen for each unit in your population is required for a random selection method. Nowadays, we tend to use computers to generate random numbers for sampling. Types of probability sampling include:
1. Sampling at random
Sampling stratified
3. Systematic sampling.
4. Cluster sampling
5 .Multi-stage sampling
Non-Probability Sampling
Non-probability sampling is a sampling technique that does not give all individuals in a population the same chance of being chosen. The sample is chosen based on the sampler’s knowledge, opinion, and discretion. Non-random sampling is a type of non-random sampling. A non-probability sampling includes:
1. Purposive sampling
2. Constant sampling
Quota sampling
3. Self-selection
4. Snowball sampling

सम्भाव्यता नमूना: कुनै पनि नमूना विधि जसले अनियमित चयन प्रयोग गर्दछ। अनियमित चयन विधिको लागि तपाईको जनसंख्याको प्रत्येक एकाइको लागि छनोट हुने समान अवसर आवश्यक छ। आजकल, हामी नमूनाको लागि अनियमित संख्याहरू उत्पन्न गर्न कम्प्युटरहरू प्रयोग गर्ने झुकाव राख्छौं। सम्भाव्यता नमूनाका प्रकारहरू समावेश छन्:
1. अनियमित मा नमूना
नमूना स्तरीकृत
3. व्यवस्थित नमूना।
4. क्लस्टर नमूना
5. बहु-चरण नमूना
गैर-सम्भावना नमूना
गैर-संभाव्यता नमूना एक नमूना प्रविधि हो जसले जनसंख्यामा सबै व्यक्तिहरूलाई छनोट हुने समान मौका दिँदैन। नमूना नमूनाको ज्ञान, राय, र विवेकको आधारमा छनोट गरिन्छ। गैर-यादृच्छिक नमूना एक प्रकारको गैर-यादृच्छिक नमूना हो। गैर-सम्भाव्यता नमूना समावेश:
1. उद्देश्यपूर्ण नमूना
2. निरन्तर नमूना
कोटा नमूना
3. स्व-छनोट
4. स्नोबल नमूना

2014 Q.No. 11 – 3 Marks

 Give a brief overview of simple random sampling.

Simple random sampling is the most widely used sampling method. Each unit of the population has an equal chance of being sampled. It is also known as unrestricted random sampling.
If a unit is selected, noted, and returned to the population before the next drawing, give the required n sample units. This method is called simple random sampling with replacement (SRSWR). Simple random sampling without replacement occurs when a selected unit is not returned to the population before the next drawing (SRSWOR). In SRSWR, some units are duplicated, but in SRSWOR they are all unique.
The lottery method selects random number table samples.

सरल अनियमित नमूना सबैभन्दा व्यापक रूपमा प्रयोग गरिएको नमूना विधि हो। जनसङ्ख्याको प्रत्येक एकाइको नमूना बन्ने समान मौका छ। यसलाई अप्रतिबंधित अनियमित नमूनाको रूपमा पनि चिनिन्छ।
यदि एक एकाइ चयन गरिएको छ, नोट गरिएको छ, र अर्को रेखाचित्र अघि जनसंख्यामा फर्काइयो भने, आवश्यक n नमूना एकाइहरू दिनुहोस्। यो विधिलाई प्रतिस्थापन (SRSWR) संग सरल अनियमित नमूना भनिन्छ। प्रतिस्थापन बिना साधारण अनियमित नमूना तब हुन्छ जब अर्को रेखाचित्र (SRSWOR) अघि चयन गरिएको एकाइ जनसंख्यामा फिर्ता हुँदैन। SRSWR मा, केहि एकाइहरू नक्कल गरिएका छन्, तर SRSWOR मा तिनीहरू सबै अद्वितीय छन्।
लटरी विधिले अनियमित संख्या तालिका नमूनाहरू चयन गर्दछ।

2021 April Q. No. 11a

Write short note on Stratified random sampling

Stratified random sampling
This method is used to select samples from a mixed population. This method of sampling is suitable for representing different sections of the population such as male and female, educated and uneducated, employed and unemployed. A simple random sample is taken from each subgroup in stratified random sampling. The stratified sample is made up of units from each subgroup. Because samples are drawn from each stratum, we know that our study includes all segments of the population.

स्तरीकृत अनियमित नमूना
यो विधि मिश्रित जनसंख्याबाट नमूनाहरू चयन गर्न प्रयोग गरिन्छ। नमूनाको यो विधि जनसङ्ख्याका विभिन्न वर्गहरू जस्तै पुरुष र महिला, शिक्षित र अशिक्षित, रोजगारदाता र बेरोजगारको प्रतिनिधित्व गर्न उपयुक्त छ। एक साधारण अनियमित नमूना प्रत्येक उपसमूहबाट स्तरीकृत अनियमित नमूनामा लिइन्छ। स्तरीकृत नमूना प्रत्येक उपसमूहबाट एकाइहरू मिलेर बनेको छ। प्रत्येक तहबाट नमूनाहरू खिचिएको हुनाले, हामीलाई थाहा छ कि हाम्रो अध्ययनले जनसंख्याको सबै खण्डहरू समावेश गर्दछ।

2021 April (Back) Q. No. 11b

Write short note on Cluster random sampling

Cluster random sampling
Cluster sampling divides a population into non-overlapping clusters. Contrary to stratified random sampling, cluster sampling identifies clusters that are internally heterogeneous. In theory, each cluster is a microcosm of the population, containing a wide range of elements. A cluster is a grouping of people or things. Clusters are naturally occurring population groups that are randomly selected for inclusion in the overall sample. Although area sampling refers to population clusters such as geographical regions and cities, the terms cluster sampling and sampling are often used interchangeably.
Cluster or area sampling has many benefits. Convenience and cost are two major benefits. Because the study is limited to clusters, the cost of sampling from the entire population is reduced.

क्लस्टर अनियमित नमूना
क्लस्टर नमूनाले जनसंख्यालाई गैर-ओभरल्यापिङ क्लस्टरहरूमा विभाजन गर्दछ। स्तरीकृत अनियमित नमूनाको विपरित, क्लस्टर नमूनाले क्लस्टरहरू पहिचान गर्दछ जुन आन्तरिक रूपमा विषम छन्। सिद्धान्तमा, प्रत्येक क्लस्टर जनसंख्याको एक माइक्रोकोसम हो, तत्वहरूको विस्तृत दायरा समावेश गर्दछ। क्लस्टर भनेको व्यक्ति वा चीजहरूको समूह हो। क्लस्टरहरू प्राकृतिक रूपमा उत्पन्न हुने जनसङ्ख्या समूहहरू हुन् जुन समग्र नमूनामा समावेशको लागि अनियमित रूपमा चयन गरिएका छन्। यद्यपि क्षेत्र नमूनाले भौगोलिक क्षेत्रहरू र शहरहरू जस्ता जनसंख्या क्लस्टरहरूलाई जनाउँछ, क्लस्टर नमूना र नमूना शब्दहरू प्राय: एकान्तर रूपमा प्रयोग गरिन्छ।
क्लस्टर वा क्षेत्र नमूना धेरै फाइदाहरू छन्। सुविधा र लागत दुई प्रमुख फाइदाहरू हुन्। किनभने अध्ययन क्लस्टरहरूमा सीमित छ, सम्पूर्ण जनसंख्याबाट नमूनाको लागत कम हुन्छ।

2019 August Q. No. 11

Write short notes on:

A. Sampling Techniques

B. Importance of sample size

A. Sampling Techniques

There are several sampling techniques, which can be divided into two categories: Probability sampling vs. non-prob.

Probability sampling methods:

i. Simple random sampling

ii.Systematic sampling

iii. Started sampling

iv. Cluster sampling

v. Multistage sampling

vi. Probability proportion to size sampling

Non -Probability sampling methods

i. Convenience sampling

ii. Quota sampling

iii. Judgment (or purposive) sampling

iv. Snowball sampling

A. नमूना प्रविधिहरू

त्यहाँ धेरै नमूना प्रविधिहरू छन्, जसलाई दुई कोटिहरूमा विभाजन गर्न सकिन्छ: सम्भाव्यता नमूना बनाम गैर-समस्या।

सम्भाव्यता नमूना विधिहरू:

i. सरल अनियमित नमूना

ii.  व्यवस्थित नमूना

iii. नमूना लिन थाले

iv. क्लस्टर नमूना

v. बहु-चरण नमूना

vi. आकार नमूनाको सम्भावना अनुपात

गैर-सम्भावना नमूना विधिहरू

i.  सुविधा नमूना

ii. कोटा नमूना

iii. न्याय (वा उद्देश्यपूर्ण) नमूना

iv. स्नोबल नमूना

B. Importance of sample size
The sample size is the number of data points used in statistical analysis. The sample size can be made up of people, animals, food batches, machines, batteries, or whatever.
i. Sample size and effect sizes are the two major factors affecting study power.
ii. A study should only be conducted if there is a reasonable chance of obtaining useful information.
iii. A study with a small sample size may produce inconclusive results and may be unethical in that it puts human subjects or lab animals at risk.
iv. An overly large study wastes resources and exposes participants to unnecessary risk.
Determining the sample size for a study is thus an important step in its design.

B. नमूना आकार को महत्व
नमूना आकार सांख्यिकीय विश्लेषणमा प्रयोग गरिएको डेटा बिन्दुहरूको संख्या हो। नमूना आकार मानिसहरू, जनावरहरू, खाद्य ब्याचहरू, मेसिनहरू, ब्याट्रीहरू, वा जुनसुकैबाट बन्न सकिन्छ।
1. नमूना आकार र प्रभाव आकार अध्ययन शक्तिलाई असर गर्ने दुई प्रमुख कारकहरू हुन्।
2. उपयोगी जानकारी प्राप्त गर्ने उचित मौका छ भने मात्र अध्ययन सञ्चालन गर्नुपर्छ।
3. सानो नमूना आकारको अध्ययनले अनिर्णित परिणामहरू ल्याउन सक्छ र यसले मानव विषयहरू वा प्रयोगशाला जनावरहरूलाई जोखिममा पार्नेमा अनैतिक हुन सक्छ।
4. अत्यधिक ठूलो अध्ययनले स्रोतहरू बर्बाद गर्छ र सहभागीहरूलाई अनावश्यक जोखिममा पार्छ।
अध्ययनको लागि नमूना आकार निर्धारण गर्न यसको डिजाइन मा एक महत्वपूर्ण कदम हो।

2018 August Q.No. 11a [1.5 Marks]

Stratified sampling and cluster sampling

Stratified sampling
This method is used to select samples from a mixed population. This method of sampling is suitable for representing different sections of the population such as male and female, educated and uneducated, employed and unemployed. A simple random sample is taken from each subgroup in stratified random sampling. The stratified sample is made up of units from each subgroup. Because samples are drawn from each stratum, we know that our study includes all segments of the population.
Ideally, each stratum is homogeneous within and heterogeneous between. Stratification improves statistical efficiency in this case.

स्तरीकृत नमूना
यो विधि मिश्रित जनसंख्याबाट नमूनाहरू चयन गर्न प्रयोग गरिन्छ। नमूनाको यो विधि जनसङ्ख्याका विभिन्न वर्गहरू जस्तै पुरुष र महिला, शिक्षित र अशिक्षित, रोजगारदाता र बेरोजगारको प्रतिनिधित्व गर्न उपयुक्त छ। एक साधारण अनियमित नमूना प्रत्येक उपसमूहबाट स्तरीकृत अनियमित नमूनामा लिइन्छ। स्तरीकृत नमूना प्रत्येक उपसमूहबाट एकाइहरू मिलेर बनेको छ। प्रत्येक तहबाट नमूनाहरू खिचिएको हुनाले, हामीलाई थाहा छ कि हाम्रो अध्ययनले जनसंख्याको सबै खण्डहरू समावेश गर्दछ।
आदर्श रूपमा, प्रत्येक स्ट्र्याटम भित्र एकसमान र बीचमा विषम हुन्छ। स्तरीकरणले यस अवस्थामा सांख्यिकीय दक्षता सुधार गर्दछ।

Cluster sampling
Cluster sampling divides a population into non-overlapping clusters. Contrary to stratified random sampling, cluster sampling identifies clusters that are internally heterogeneous. In theory, each cluster is a microcosm of the population, containing a wide range of elements. A cluster is a grouping of people or things. Clusters are naturally occurring population groups that are randomly selected for inclusion in the overall sample. Although area sampling refers to population clusters such as geographical regions and cities, the terms cluster sampling and sampling are often used interchangeably.
Cluster or area sampling has many benefits. Convenience and cost are two major benefits. Because the study is limited to clusters, the cost of sampling from the entire population is reduced.

क्लस्टर नमूना
क्लस्टर नमूनाले जनसंख्यालाई गैर-ओभरल्यापिङ क्लस्टरहरूमा विभाजन गर्दछ। स्तरीकृत अनियमित नमूनाको विपरित, क्लस्टर नमूनाले क्लस्टरहरू पहिचान गर्दछ जुन आन्तरिक रूपमा विषम छन्। सिद्धान्तमा, प्रत्येक क्लस्टर जनसंख्याको एक माइक्रोकोसम हो, तत्वहरूको विस्तृत दायरा समावेश गर्दछ। क्लस्टर भनेको व्यक्ति वा चीजहरूको समूह हो। क्लस्टरहरू प्राकृतिक रूपमा उत्पन्न हुने जनसङ्ख्या समूहहरू हुन् जुन समग्र नमूनामा समावेशको लागि अनियमित रूपमा चयन गरिएका छन्। यद्यपि क्षेत्र नमूनाले भौगोलिक क्षेत्रहरू र शहरहरू जस्ता जनसंख्या क्लस्टरहरूलाई जनाउँछ, क्लस्टर नमूना र नमूना शब्दहरू प्राय: एकान्तर रूपमा प्रयोग गरिन्छ।
क्लस्टर वा क्षेत्र नमूना धेरै फाइदाहरू छन्। सुविधा र लागत दुई प्रमुख फाइदाहरू हुन्। किनभने अध्ययन क्लस्टरहरूमा सीमित छ, सम्पूर्ण जनसंख्याबाट नमूनाको लागत कम हुन्छ।

2021 April Q. No. 7 [5 Marks]

 A sample of 550 bulbs of a company showed an average life of1500 hours with a standard deviation of 35 hours. Obtain 95% and 99% confidence limits for the population mean assuming that the parent population from which samples are drawn in normally distributed.

2021 Apri Q. No. 10 [5 marks]

If the population proportion of success is 0.65, the sampling error is 0.093, and the level of significance is 0.05, compute the sample size.

2021 April (Back) Q. No. 7 [3 Marks]

A sample of 50 light bulbs from a manufacturing lot had an average life of 1400 hours with a standard deviation of 50 hours. Construct 99% confidence intervals for the true population mean.

2019 August Q. No.7 [5 Marks]

 A zookeeper took a random sample of 50 days and observed how much food an elephant ate on each of those days. The mean was 350 kg and the standard deviation was 25 kg. Construct 90% and 99% confidence intervals for the true population mean.

2019 Back Q. No. 10

The quality control manager at a light bulbs factory needs to estimate the mean life of a large shipment of life bulbs. The process standard deviation is known to be 100 hours. A random sample of 64 light indicated as the sample mean life of 350 hours.

  1. a. Obtain the standard error of the mean.
  2. b. Set up 95% confidence interval estimate if the true population mean of lightbulbs in shipment.

2018 August Q.No. 6 [3 Marks]

A random sample of size 64 has been drawn from a population with a standard deviation is 20. The mean of the sample is 80. Construct 95% and 99% confidence limits for the population mean.

2018 June Make up Q. No. 9 A researcher got involvement in knowing the responsible attitude of the people in a certain locality. He estimates the standard deviation is 0.05 sec. How large a sample must he take in order to be 95% confident that the error of his estimate of mean will not exceed 0.01 sec?

2018 June Make up Q. No. 10

A random sample of 600 apples was taken from large consignment and 75 of them were found to be bad. Find the 99 percent confidence limits for a percentage of bad apples.

2018 January Q.No. 8

A sample of 400 mobile sets was taken from a lot. 20 mobile sets were found to be damaged.

a. Find the standard error of the proportion of damaged mobile sets.

b. Estimate a 95% confidence interval of the percentage of damaged mobile set.

2017 Makeup Q.No. 9 | 2015 Make up Q. No. 9

A sample of 500 bulbs of a the company showed an average lifetime of 1400 hours with a standard deviation of 30 hours. Obtain 95% and 99% confidence limits for the population mean.

11.2016 Q.No. 8

A random sample of 700 units from a large consignment showed that 147 were damaged. Find 95% confidence limits for the proportion of damaged units in the consignment.

2015 Q No. & The quality control manager at a light bulbs factory needs to estimate the mean life of a large shipment of life bulbs. The process standard deviation is known to be 100 hours. A random sample of 64 light bulbs indicated a sample mean life of 350 hours.

a. Obtain the standard error of mean.

b. Set up 95% confidence interval estimate of the true population mean of lightbulbs in shipment.

c. Do you think that the manufacturer has the right to state that the light bulbs have

average lifetime of 400 hours?

2014 Q.No. 9

If the population proportion of success is 0.65 and n = 10, what will be the value of sampling error when the acceptance region is 0.95?

2014 No. 10

The quality control manager at a light bulbs factory needs to estimate the mean life of a large shipment of life bulbs. The process standard deviation is known to be 100 hours. A random sample of 64 light bulbs indicated a sample mean life of 350 hours. Set up 95% confidence interval estimate of the true population mean of light bulbs in shipment.

April 10, 2022

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