ECO 07 Solved Assignment 2022 – 23
IGNOU B.Com Free Solved Assignment 2022 – 23
Elements of Statistics ECO 07 Solved Assignment 2022 – 23
COURSE TITLE: ELEMENTS OF STATISTICS
ASSIGNMENT CODE: ECO-07/TMA/2022-23
COVERAGE: ALL BLOCKS
Maximum Marks: 100
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Q.1 What is sampling? Discuss the various methods of sampling? (20)
Ans: Meaning of Sampling: Sampling is simply the process of learning about the population on the basis of sample drawn from it. In sampling technique only, a part of the universe is studies and conclusions are drawn on that basis for entire universe. In most of the research work and surveys, the usual approach happens to be to make generalizations or to draw inferences based on samples about the parameters of population from which the samples are taken. The researcher quite often selects only a few items from the universe for his study purposes.
All this is done on the assumption that the sample data will enable him to estimate the population parameters. The items so selected constitute what is technically called a sample, their selection process or technique is called sample design and the survey conducted on the basis of sample is described as sample survey. Sample should be truly representative of population characteristics without any bias so that it may result in valid and reliable conclusions.
According to Goode and Hatt, “A sample as the name applies, is a smaller representative of a large whole”.
According to Pauline V Young, “A statistical sample is a miniature of cross selection of the entire group or aggregate from which the sample is taken”.
According to Bogrdus, “Sampling is the selection of certain percentage of a group of items according to a predetermined plan”.
Methods of Sampling
Sampling methods are classified into two categories: Probability sampling methods and non-probability sampling methods.
Probability sampling methods are those in which every item in the universe has a known chance or probability of being chosen for the sample. Non-probability sampling methods are those which do not provide every item in the universe with a chance of being selected. These two methods are further divided into:
Probability Sampling |
Non-Probability Sampling |
1. Simple random sampling 2. Stratified random sampling 3. Cluster sampling 4. Systematic Sampling 5. Multi-stage sampling |
1. Quota Sampling 2. Judgmental or Purposive Sampling 3. Convenience Sampling 4. Extensive Sampling |
(1) Simple Random Sampling: Off all the methods of selecting sample, random sampling technique is made maximum use of and it is considered as the best method of sample selection. Random sampling is made in following ways:
(i) Lottery Method: In this the numbers of data are written on sheet of paper and they are thrown into a box. Now a casual observer selects the number of item required in the sample. For this method it is necessary that sheet of paper should be of equal dimensions.
(ii) By Rotating the Drum: In this method, piece of wood, tin or cardboard of equal length and breadth, with number 0,1 or 2 printed on them, are used. The pieces are rotated in a drum and then requisite numbers are drawn by an impartial person.
(iii) Selecting from Sequential List: In this procedure units are broken up in numerical, alphabetical or geography sequence. Now we may decide to choose 1, 5, 10 and so on, if the division is alphabetical order we decide to choose every item starting from a, b, c and so on.
(iv) Tippet’s Number: On the basis of population statistics, Tippet has constructed a random list of four digits each of 10, 400 institutions. These numbers are the result of combining 41,600 population statistics reports.
Advantages;
1. Due to impartiality, there is possibility of selecting any unit as sample.
2. Units have the characteristic of universe; hence units are more representative.
3. Simplicity of method makes no possibility of error.
4. Error can be known easily
5. It saves money, time and labour.
Disadvantages:
1. The selector has no control over the selection of units. The researcher cannot contact the far situated units.
2. He cannot prepare the whole field when the universe is vast.
3. If units have no homogeneity, the method is not appropriate.
4. There is no question of alternatives. The selected units cannot be replaced or changed.
(2) Stratified Sampling: This method of selecting samples is a mixture of both purposive and random sampling techniques. In this all the data in a domain is spilt into various classes on the basis of their characteristics and immediately thereafter certain items are selected from these classes by the random sampling technique.
This technique is suitable in those cases in which the data has sub data and having special characteristics. For example, if we wish to collect information regarding income expenditure of the male population strata on the basis of shopkeeper, workers, etc. From these we shall select randomly some units for study of income-expenditure statistics.
Process of Stratifying: The stratification of domain or data should be with great care, because the success of the technique depends upon successful stratification. Following points should be born in mind:
1. We should process extensive information of all items including in a domain and should know which item make a coherent whole on the basis of similar traits and which others are different from them and why?
2. The size of each stratum should be large to enable use of random sampling technique.
3. In stratifying it must be kept in mind that various strata should have similar relation to the domain and should be themselves homogeneous.
4. The various strata should differ from each other should be the same as the proportion of stratum from the domain. Suppose a domain has four strata, accordingly the proportion of each stratum of domain is ¼. Now if the number of total items of the sample is 64, we shall select 16 items from each stratum and thus the proportion of selected items from each stratum will be ¼.
Advantage:
1. Neither group nor class of importance is totally neglected as units of each are represented in the sample.
2. If different classes are divided properly, selection of few units represents the whole group.
3. On the classification of regional basis, units are not in contact easily. This leads to economy of time and money.
4. There is a facility in substitution of units. If someone is not contacted easily, the other person of the same class can be substituted for him. Such inclusion result will not show any contradicting.
Disadvantages:
1. The sample does not become representative if selected sample has more or less units of a class.
2. If the sizes of different group are different, no equal proportional quality can be viewed.
3. Non-proportional selection leads to more emphasis in the end. During such time researcher can be biased, hence samples will not accurate.
4. If group is not expressed properly, the difficulty is seen about the unit to be kept under which group or class.
(3) Cluster Sampling: In this method of sampling, the population is divided into clusters or groups and then Random Sampling is done for each cluster. In some instances, the sampling unit consists of a group or cluster of smaller units that we call elements or sub-units. Cluster Sampling is different Stratified sampling. In the case of stratified sampling the elements of each stratum are homogeneous while in cluster sampling each cluster is heterogeneous within and a representative of the population.
(4) Systematic Sampling: This method of sampling is at first glance very different from random sampling. In practice, it is a variant of simple random sampling that involves some listing of elements. In systematic sampling each element has an equal chance of being selected, but each sample does not have the same chance of being selected. Here, the first element of the population is randomly selected to begin the sampling. But thereafter the elements are selected according to a systematic plan. Systematic sampling proceeds by picking up one element after a fixed interval.
(5) Multi-Stage sampling: This is not a favoured procedure of sampling. In this items are selected in different stages at random. For example, if we wish to know per acre yield of various crops in U.P., we shall begin by studying a single crop in one study. Here we shall begin by making at random selection of 5 districts in the first instance, and then of these 5 districts, 10 villages per districts will be chosen in the same manner. Now in the final stage, again by random selection 5 fields out of every village. Thus we shall examine per acre yield in 250 farms all over U.P. this number can increased or decreased depending upon the opinion of experts.
(6) Quota Sampling: This method of study is not much used. In this method entire data is spilt into as many as there are investigators and each investigator is asked to select certain items from his block and study. The success of this method depends upon the integrity and professional competence of investigators. If some investigators are competent and others are not so competent, serious discrepancies will appear in the study.
(7) Purposive or Judgmental or Selective sampling: In this method the investigator has complete freedom to choose his sample according to his wishes and desire. To choose or leave an item for the purpose of study depends entirely upon the wishes of investigator and he will chose items or units which in his judgment are representative of the whole data. This is a very simple technique of choosing the samples and is useful in cases where the whole data is homogeneous and the investigator has full knowledge of the various aspects of the problem.
Advantage:
1. More representation is possible in this method.
2. As sample is small in size, the method is less expensive and less time consuming.
3. The utility of this method increases when few units of universe have special importance.
4. When units are less in number, sample is profitable
Disadvantages:
1. Units are selected by researcher at his will. Hence sample is biased.
2. The error of the sample cannot be detected.
3. Researcher is unable to understand the whole group.
4. Those hypothesis on which inference of error of sample is attributed, are less used.
(8) Convenience Sampling: This is hit or miss procedure of study. The investigator selects certain item from the domain as per his convenience. No planned efforts are made to collect information. This is method by which a tourist studies generally the country of his visit. He comes across certain people and things, has transaction with them and then tries to generalize about the entire populace in his travelogue. This is essentially unscientific procedure and has no value as a research technique.
(9) Extensive sampling: This method is virtually same as census except that irrelevant or irascible items are left out. Every other item is examined. For instance, if we are to study the educational levels of Indians, we may leave foreigners living in India from our study. This method has all the merits and demerits of census survey and is very rarely used.
Q.2 If the mean of the given frequency distribution is 35, then find the missing frequency y. Also, calculate the median and mode for the distribution.
Class Distribution: |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |
Frequency: |
2 |
4 |
7 |
Y |
1 |
Ans: Missing Frequency:
Calculation of Median & Mode
Q.3 (a) Discuss the sources of secondary data. What precautions should be taken while using secondary data? (10)
Ans: Secondary Data: Data which are collected by someone else, used in investigation are knows as Secondary data. Data are primary to the collector, but secondary to the user. For example: Statistical abstract of the Indian Union, Monthly abstract of statistics, Monthly statistical digest, International Labour Bulletin (Monthly).
Merits and Demerits of Secondary Data:
Merits:
(a) While using secondary data, time and labour are saved.
(b) It may also be collected from unpublished form.
(c) If secondary Data are available, they are much quicker to obtain than primary data.
Demerits:
(a) Degree of accuracy may not be acceptable.
(b) Secondary Data may or may not fit the need of the project.
(c) Data may be influenced by personal bias of investigator.
Sources of Secondary Data:
(a) Official publication by the central and state governments, district Boards.
(b) Publication by research institutions, Universities etc.
(c) Economic Journals.
(d) Commercial Journals.
(e) Reports of Committees, commissions.
(f) Publications of trade associations, Chamber of Commerce etc.
Precautions in the use of Secondary Data:
The following aspects should be considered before use of secondary data:
(i) Suitability: The investigator must check before using secondary data that whether they are suitable for the present purpose or not.
(ii) Adequacy: The investigator has to determine whether they are adequate for the present purpose of investigators.
(iii) Dependability: Dependability of secondary data is determined by the following factors:
(a) The authority which collected the data.
(b) Procedure of Sampling followed.
(c) Status of Investigator.
(iv) Units in which data are available.
(b) Define Geometric and Harmonic mean? Differentiate between them? (10)
Ans: Geometric Mean (GM): Geometric mean (GM) is a type of average that is used to calculate the average return or growth rate of an investment over a specific period of time. It is calculated by taking the product of the values of a variable at different points in time and then taking the nth root, where n is the number of time periods.It is defined as nth root of the product of n items or values. i.e., G.M. = ^{n}√ (x1. x2. x3 ……xn)
Harmonic Mean (HM): Meaning, Uses, Merits and Demerits
Harmonic mean (HM) is another type of average that is used to calculate the average value of a set of numbers. It is calculated by taking the reciprocal of the arithmetic mean of the reciprocals of the numbers in the set.
H.M.=N/(1/x1 + 1/x2 + 1/x3 + …….. +1/xn)
Here are ten key differences between the geometric mean and the harmonic mean:
1. Definition: The geometric mean is the average of a set of numbers,
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Q.4 The wages of nine workers are given below:Rs. 310, Rs. 290, Rs. 320, Rs. 280, Rs. 300, Rs. 290, Rs. 320, Rs. 310, Rs. 280.
Find the following: (20)
(a) Standard Deviation.
(b) Variance.
Ans: Calculation of Standard Deviation and Variance
Q.5 Write short notes on the following: (5×4=20)
(a) Distrust of Statistics.
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(b) Statistical Derivatives.
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(c) Moving Averages.
Ans: Moving average: – Under this method the average value for a certain time span is secured and this average is taken as the trend value for the unit of time falling at the middle of the period covered in the calculation of the average. While using this method it is necessary to select a period for moving average.
Merits:
i) This method is simple to understand and apply.
ii) It is particularly effective if the trend of a series is very irregular.
iii) It is a flexible method of measuring trend because all figures are not changed if a few figures are added to the data.
Demerits:
i) Trend values cannot be computed for all years.
ii) No there is no hard and fast rule for selecting the period of moving average.
iii) This method is not appropriate if the trend situation is not linear.
(d) Properties of Normal Curve.
Ans: Properties of the Normal Distribution/Curve: The following are the important properties of the normal curve and the normal distribution:
1. The normal curve is “bell-shaped” and symmetrical in its appearance. If the curves were folded along its vertical axis, the two halves would coincide. The number of cases below the mean in a normal distribution is equal to the number of cases above the mean, which makes the mean and median coincide. The height of the curve for a positive deviation of 3 units is the same as the height of the curve for negative deviation of 3 units.
2. The height of the normal curve is at its maximum at the mean. Hence the mean and mode of the normal distribution coincide. Thus for a normal distribution mean, median and mode are all equal.
3. There is one maximum point of the normal curve which occurs at the mean. The height of the curve declines as we go in either direction from the mean. The curve approaches nearer and nearer to the base but it never touches it i.e., the curve is asymptotic to the base on either side. Hence its range is unlimited or infinite in both directions.
4. Since there is only one maximum point, the normal curve is un-modal, i.e., it has only one mode.
5. The points of inflexion, i.e., the points where the change in curvature occurs are
6. As Distinguished from Binomial and Poisson distributions when the variable is discrete, the variable distributed according to the normal curve is a continuous one.
7. The first and third quartiles are equidistant from the median.
8. The mean deviation is 4^{th} or more precisely 0.7979 of the standard deviation.
9. The area under the normal curve distributed as follows:
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