Introduction to Sampling and Sampling Distribution

1. Welcome To A Session On Sampling and Sampling Distribution www.AssignmentPoint.com

2. What is meant by sampling? Sampling is the process of selection of some part of an aggregate or totality on the basis of which a judgement or inference about the aggregate or totality is made. It is also viewed as the process of obtaining information about an entire population by examining only part of it. www.AssignmentPoint.com

3. What is the assumption behind sampling? The assumption behind sampling is that the sample data enable one to estimate the population parameters. www.AssignmentPoint.com

4. What is meant by a sample? The item selected from an aggregate or totality is technically known as sample. The selection process or technique is called sample design and the survey conducted on the basis of sample is described as sample survey. www.AssignmentPoint.com

5. Why do you use Sampling? Sampling is used in practice for a variety of reasons such as: Sampling can save time and money. A sample study is usually less expensive than a census study and produces results at a relatively faster speed. Sampling may enable more accurate measurements, for a sample study is generally conducted by trained and experienced investigators. www.AssignmentPoint.com

6. Why do you use Sampling? Sampling remains the only way when population contains infinitely many members. Sampling remains the only choice when a test involves the destruction of the item under study. Sampling usually enables to estimate the sampling errors and, thus, assists in obtaining information concerning some characteristic of the population. www.AssignmentPoint.com

7. Some issues to be considered in sampling Universe / Population: The term ‘Universe’ refers to the total of the items or units in any field of inquiry, whereas the term ‘population’ refers to the total of items about which information is desired. The attributes that are the object of study are referred to as characteristics and the units possessing them are called as elementary units. The population or universe can be finite or infinite: The population is said to be finite if it consists of a fixed number of elements so that it is possible to enumerate it in its totality. www.AssignmentPoint.com

8. Sampling frame: The elementary units or the group or cluster of such units may form the basis of sampling process in which case they are called as sampling units. Sampling design:A sample design is a definite plan for obtaining a sample from the sampling frame. It refers to the technique or the procedure the researcher would adopt in selecting some sampling units from which inferences about the population is drawn. Sampling design is determined before data are collected. www.AssignmentPoint.com

9. Statistic(s) and Parameter(s): A statistic is a characteristic of a sample. When certain measures such as mean, medium, mode or the like ones are estimated from samples, then they are called statistic(s). They describe the characteristics of a sample. Parameter(s): Parameters refer to the characteristics of a population. For instance, the population mean () is a parameter, whereas the sample mean (x) is a statistic. To obtain the estimate of a parameter from a statistic constitutes the prime objective of sampling analysis. www.AssignmentPoint.com

10. What is meant by sampling error? www.AssignmentPoint.com

11. What is meant by sampling error? Sampling error or error variance refers to inaccuracy in the information collected through a sample survey. In other words, sampling errors are those errors which arise on account of sampling and they generally happen to be random variations (in case of random sampling ) in the sample estimates around the true population values. www.AssignmentPoint.com

12. Sampling error = Frame error + chance + response error (If we add measurement error or the non-sampling error to sampling error, we get total error) www.AssignmentPoint.com

13. Sampling errors occur randomly and are equally likely to be in either direction. The magnitude of the sampling error depends upon the nature of the universe. The more homogeneous the universe, the smaller the sampling error. Sampling error is inversely related to the size of the sample i.e., sampling error decreases as the sample size increases and vice-versa. www.AssignmentPoint.com

14. The measure reflecting sampling error is often called the precision of the sampling plan. Sampling error is usually worked out as the product of the critical value at a certain level of significance and the standard error. As opposed to sampling errors, there may occur non-sampling errors which may creep in during the process of collecting actual information. Such errors occur in all surveys whether census or sample. There is no way to measure non-sampling errors. www.AssignmentPoint.com

15. Precision: Precision is the range within which the population average (or other parameter) will lie in accordance with the reliability specified in the confidence level as a percentage of the estimate ± or as numerical quantity. www.AssignmentPoint.com

16. Example If the estimate is Taka 4000 and the precision desired is ± 4%, then the true value will be no less than Taka 3840 and no more than Taka 4160. This is the range (Taka 3840 to Taka 4160) within which the true answer should lie. But if it is desired that the estimate should not deviate from the actual value by more than Taka 200 in either direction, in that case the range would be Taka. 3800 to Taka 4200. www.AssignmentPoint.com

17. Confidence level and significance level: The confidence level or reliability is the expected percentage of times that the actual value will fall within the stated precision limits. Thus, if a confidence level of 95% is desired , then there would be 95 chances in 100 (or.95 in 1). The sample results represent the true condition of the population within a specified precision range against 5 chances in 100 ( or .05 in 1) that it does not. Continued……. www.AssignmentPoint.com

18. Precision is the range within which the answer may vary and still be acceptable; • Confidence level indicates the likelihood that the answer will fall within that range; and • Significance level indicates the likelihood that the answer will fall outside that range. www.AssignmentPoint.com

19. Example: One must remember that if the confidence level is 95%, then the significance level is (100 - 95) i.e. 5%. If the confidence level is 99% the significance level is (100-99) i.e., 1% and so on. www.AssignmentPoint.com

20. Sampling distribution: If certain number of samples are taken and for each sample various statistical measures such as mean, standard deviation, etc., are computed, then each sample may give its own value for the statistic under consideration. All such values of a particular statistic, say mean, together with their relative frequencies constitute the sampling distribution of the particular statistic, say mean. www.AssignmentPoint.com

21. Accordingly, we can have sampling distribution of mean, or the sampling distribution of standard deviation or the sampling distribution of any other statistical measure. It may be noted that each item in a sampling distribution is a particular statistic of a sample. www.AssignmentPoint.com

22. What are inferences one may draw from sampling? • The sampling distribution tends quite closer to the normal distribution if the number of samples is large. • The significance of sampling distribution follows from the fact that the mean of a sampling distribution is the same as the mean of the universe. • The mean of the sampling distribution can be taken as the mean of the universe. www.AssignmentPoint.com

23. What are the important sampling distribution? Some important sampling distributions which are commonly used are: Sampling distribution of mean Sampling distribution of proportion Student’s distribution F distribution and Chi-square distribution. www.AssignmentPoint.com

24. A brief mention of each one of these sampling distributions will be helpful. • Sampling distribution of mean: Sampling distribution of mean refers to the probability distribution of all the possible means of random samples of a given size that we take from a population. Sampling distribution of mean is very useful in several decision situation for accepting or rejection of hypotheses. • 2.Sampling distribution of proportion: Like sampling • distribution of mean, we can as well have a sampling • distribution of proportion. This happens in case of statistics of attributes. www.AssignmentPoint.com

25. Example Assume that we have worked out the proportion of defective parts in large number of samples, each with say 100 items, that have been taken from an infinite population and plot a probability distribution of the said proportions, we obtain what is known as the sampling distribution of the said proportions. www.AssignmentPoint.com

26. 3.Student’s t-distribution: When population standard deviation() is not known and the sample is of a small size (i .e., n  30) , we use t distribution for the sampling distribution of mean and workout t variable • 4. F distribution: • Chi-square (2) distribution: www.AssignmentPoint.com

27. What are the central limit theorem? Central Limit Theorem When sampling is from a normal population, the means of samples drawn from such a population are themselves normally distributed. But when sampling is not from a normal population, the size of the sample plays a critical role. When n is small, the shape of the distribution will depend largely on the shape of the parent population, but as n gets large (n30), the shape of the sampling distribution will become more and more like a normal distribution, irrespective of the shape of the parent population. www.AssignmentPoint.com

28. Central Limit Theorem “The significance of the central limit theorem lies in the fact that it permits us to use sample statistics to make inferences about population parameters without knowing anything about the shape of the frequency distribution of that population other than what we can get from the sample” www.AssignmentPoint.com

29. Sampling Theory Sampling theory is a study of relationships existing between a population and samples drawn from the population. Sampling theory is applicable only to random samples. For this purpose the population or a universe may be defined as an aggregate of items possessing a common trait or traits. In other words, a universe is the complete group of items about which knowledge is sought. www.AssignmentPoint.com

30. Sampling Theory In the former case the universe in fact does not exist and we can only imagine the items constituting it. Tossing of a coin or throwing a dice are examples of hypothetical universe. Existent universe is a universe of concrete objects i.e., the universe where the items constituting it really exist. The universe may be finite or infinite. Finite universe is one which has a definite and certain number of items, but when the number of items is uncertain and infinite, the universe is said to be an infinite universe. Similarly, the universe may be hypothetical or existent. Continued……. www.AssignmentPoint.com

31. Sampling Theory On the other hand, the term sample refers to that part of the universe which is selected for the purpose of investigation. The theory of sampling studies the relationships that exist between the universe and the sample or samples drawn from it. www.AssignmentPoint.com

32. What is meant by statistical induction or statistical inference? The theory of sampling is concerned with estimating the properties of the population from those of the sample and also with gauging the precision of the estimate. This sort of movement from particular (sample) towards general (universe) is what is known as statistical induction or statistical inference www.AssignmentPoint.com

33. What are the objectives of the sampling theory? Sampling theory is designed to attain one or more of the following objectives: (i) Statistical estimation (ii) Testing of hypotheses (iii) Statistical inference www.AssignmentPoint.com

34. What are the problems in case of sampling? • We generally consider the following three types of problems in case of sampling of attributes: • The parameter value may be given and it is only to be tested if an observed ‘statistic’ is its estimate. • The parameter value is not known and we have to estimate it from the sample. • Examination of the reliability of the estimate i.e., the problem of finding out how far the estimate is expected to deviate from the true value for the population. www.AssignmentPoint.com

35. What is meant by standard error? The standard deviation of sampling distribution of a statistic is known as its standard error (S.E.) and is considered the key to sampling theory. www.AssignmentPoint.com

36. The utility of the concept of standard error in statistical induction arises on account of the following reasons: The standard error helps in testing whether the difference between observed and expected frequencies could arise due to chance. The criterion usually adopted is that if a difference is less than 3 times the S.E., the difference is supposed to exist as a matter of chance and if the difference is equal to or more than 3 times the S. E., chance fails to account for it, and we conclude the difference as significant difference. www.AssignmentPoint.com

37. 2.The standard error gives an idea about the reliability and precision of a sample. The smaller the S. E., the greater the uniformity of sampling distribution. and hence, greater is the reliability of sample. Conversely, the greater the S. E., the greater the difference between observed and expected frequencies. In such a situation the unreliability of the sample is greater. The size of S.E., depends upon the sample size to a great extent and it varies inversely with the size of the sample. If double reliability is required i.e., reducing S. E. to ½ of its existing magnitude, the sample size should be increased four-fold. www.AssignmentPoint.com

38. 3. The standard error enables us to specify the limits within which the parameters of the population are expected to lie with a specified degree of confidence. Such an interval is usually known as confidence interval. www.AssignmentPoint.com

39. Range Per cent values  1 S. E. 68.27%  2 S. E. 95.45%  3 S. E. 99.73%  1.96 S. E. 95.00%  2.5758 S. E. 99.00% www.AssignmentPoint.com

40. What are the statistical estimations? • In most statistical research studies, population parameters are usually unknown and have to be estimated from a sample. As such the methods for estimating the population parameters assume an important role in statistical analysis. • Statistical estimation is the procedure of using statistic to estimate a population parameter. • A statistic used to estimate a parameter is called an estimator and the value taken by the estimator is called an estimate. Continued……. www.AssignmentPoint.com

41. Estimation The estimate of a population parameter may be one single value or it could be a range of values. In the former case it is referred as point estimate, whereas in the latter case it is termed as interval estimate. The researcher usually makes these two types of estimates through sampling analysis. Continued…… www.AssignmentPoint.com

42. Estimation While making estimates of population parameters, the researcher can give only the best point estimate or else he shall have to speak in terms of intervals and probabilities for he can never estimate with certainty the exact values of population parameters. Hence it is necessary to know the various properties of a good estimator so that one can select appropriate estimators for study. www.AssignmentPoint.com

43. What are the properties of a good estimator? www.AssignmentPoint.com

44. Properties of a good estimator • One must know that a good estimator possesses the following properties: • An estimator should on the average be equal to the value of the parameter being estimated. This is popularly known as the property of being unbiased. • An estimator is said to be unbiased if the expected value of the estimator is equal to the parameter being estimated. The sample mean(X) is the most widely used estimator because of the fact that it provides an unbiased estimate of the population mean(). Continued……. www.AssignmentPoint.com

45. Properties of a good estimator • An estimator should have a relatively small variance. This means that the most efficient estimator, among a group of unbiased estimators, is one which has the smallest variance. This property is technically described as the property of efficiency. • An estimator should approach the value of population parameter as the sample size becomes larger and larger. This property is referred to as the property of consistency. www.AssignmentPoint.com

46. Nature of study: If items are to be intensively and continuously studied, the sample should be small. For a general survey the size of the sample should be large, but small sample is considered appropriate in technical surveys. Continued… www.AssignmentPoint.com

47. Type of sampling: Sampling technique plays an important part in determining the size of the sample. A small random sample is apt to be much superior to a larger but badly selected sample. Continued… www.AssignmentPoint.com

48. What are the various methods of Sampling? The various methods of sampling or sampling designs can be grouped under the heads as random sampling and non – random sampling. Random sampling methods are as follows: • Random sampling methods • Simple random sampling • Stratified sampling • Systematic sampling • Multi – stage sample www.AssignmentPoint.com

49. Non–Random sampling methods are as follows: • Non–random sampling methods • Judgment sampling • Quota sampling • Convenience sampling www.AssignmentPoint.com

50. Sampling methods Random Sampling Non - Random Sampling Multi stage Sampling Simple Random Sampling Systematic Sampling Stratified Sampling Judgment Sampling Convenience Sampling Quota Sampling www.AssignmentPoint.com