SLM_(1)

268 Business Mathematics and Statistics 5*83*8 8(53) Combined Mean X' = 53 53 =8 Variance of set A: 12 = 18 Variance of set B: 22 = 24 X'1 = 8, X'2 = 8, n1 = 5, n2 = 3, X' = 8. n 1 V 2 n 2 V 2 n1162 (x 1 x) 2 n 2 (x 2 x)2 n1 n2 8 s= 20.25 5*18*3*24*5(88)23(88)2 s= 53 9072 s= 8 s= s= s = 4.5

Problem 19: Find the standard deviation or the data given below, after finding the missing frequency. The mean of the data is 132. Class Frequency 100-110 2 110-120 4 120-130 7 130-140 – 140-150 5 150-160 2 160-170 1 CU IDOL SELF LEARNING MATERIAL (SLM)

Dispersion 269 Class MV fD d d 100-110 105 2 –30 –3 –6 110-120 115 4 –20 –2 –8 120-130 125 7 –10 –1 –7 130-140 135 — 0 0 0 140-150 145 5 + 10 +1 +5 150-160 155 2 +20 +2 +4 160-170 165 1 +330 +3 +3 21 + x –9 Solution: n = 21 + x where x is the missing frequency. = fd Mean = a + n *i (9) Mean = 135 + 21x * 10 90 132 = 135 + 21x 90

21x = 135 – 132 90 =3 21x Therefore 90 = 63 + 3x Therefore 27 = 3x Therefore x = 9 CU IDOL SELF LEARNING MATERIAL (SLM)

270 Business Mathematics and Statistics Calculation of S.D. Class MV fD d fd fd2 100-110 105 2 –30 –3 –6 18 110-120 115 120-130 125 4 –20 –2 –8 16 130-140 135 140-150 145 7 –10 –1 –7 7 150-160 155 160-170 165 900p0 5 + 10 +1 +5 5 2 +20 +2 +4 8 1 +30 +3 +3 9 30 -9 63 fd2 § fd ·2 SD = ¨ ¸ *i n ©n¹ 63 §9·2 SD = ¨ ¸ *10 30 © 30 ¹ SD = 2.100.09*10 SD = 2.01*10 SD = (1.418) * 10 SD = 14.18

§1· ¨¸ Note: 2.01 * 10 = Antilog © 2 ¹ log 2.01 * 10 §0.3032 · ¨ ¸ * 10 = Antilog © 2¹ = Antilog (0.1516) * 10 = (1.418) * 10 = 14.18 Problem 20: From a certain frequency distribution consisting of 18 observations, the mean and the standard deviation were found to be 7 and 4 respectively. But on comparing with the original data it was found that a figure 12 was wrongly written as 21 in the calculation. Calculate the correct mean and standard deviation. CU IDOL SELF LEARNING MATERIAL (SLM)

Dispersion 271 Solution: Sum of the observations Mean = 18 Sum of the observations 7= 18 Therefore sum of observations = 7 * 18 = 126 Correct sum of observations ? 126–21+12 ? 117 117 Correct Mean = 18 = 6.5 ¦ x 2 § ¦x ·2 SD=V= ¨¸ n ©n ¹ ?4 = ¦x 2 (7)2 18 ?16= ¦x2 -49

18 16+49= ` x2 18 ?x2 = 18 * 65 = 1170 This is incorrect 6x2 Correct 6x2 = 1170 – (21)2 + (12)2 n 1170 – 441 + 144 n 873 Correct SD = 873 (6.5)2 18 5 48.542.25 CU IDOL SELF LEARNING MATERIAL (SLM)

272 Business Mathematics and Statistics = 6.25 = 2.5 Problem 21: The arithmetic means of two samples of sizes 60 and 90 are 2 and 48 respectively. Their standard deviations are 9 and 12 respectively. Find the standard deviation of the combined sample of size 150. We have x'l = 52, x'2 = 48, n1 = 60, n2 = 90, s1 = 9 and s2 = 12. Let x' and 0 be the AM and standard deviation respectively of the combined sample size of 150. Then, n 1 V 2 n 2 V 2 n 1 (x ' x ') 2 n 2 (x '2 x '1 )2 1 2 1 V= n 1 n2 (608190*144)60(5249.6)2 90(4849.6)2 V= 150 17820 345.6230.4 V= 150

18396 V= 150 V = 11.07 11.2.5 Graphs: Histogram, Frequency Polygon and Curve, Ogive Curve Graphs Three types of graphs are used to represent Frequency distributions: histograms, frequency Polygons/Curves and the Ogive. Histogram This graph is used to draw a continuous frequency distribution. It consists of vertical bars adjacent to each other. The class boundaries are marked of on a suitable scale along the horizontal axis or the x-axis, while frequencies are marked along the, vertical axis or the y-axis. CU IDOL SELF LEARNING MATERIAL (SLM)

Dispersion 273 Rectangles are drawn on the class intervals marked of along the x-axis, with their heights proportional to the frequencies of the respective class intervals. The resulting figure would be a ‘histogram’ relating, to data of equal class intervals. Problem 1: Draw a histogram for the data given below: 16 20-30 30-40 40-50 50-60 60-70 70-80 14 12 10 8 6 4 2 0 10-20 In the above histogram one cm along the x-axis represents 10 marks and one cm along the y-axis represents two students. The total area of the histogram represents the total of the frequencies and the area of each rectangular bar is proportional to the frequency that it represents. Frequency Polygon and frequency Curve

A frequency polygon is constructed by joining the mid-points of the upper sides of adjacent rectangles in the histogram with small straight lines. The frequency polygon is shown in figure 3. In this case, it becomes a closed polygon when the mid-point of the * class 0-10 is joined with the mid-point of the upper side of the’ rectangle on the class interval 10-20. Similarly, the mid-point of the class interval 80-90 may be joined with the mid-point of the upper side of the last rectangle. The area of the closed frequency polygon is always equal to the area of the histogram. The frequency curve is formed by drawing a smooth curve through the mid-points of the upper sides of the adjacent rectangles of the histogram. CU IDOL SELF LEARNING MATERIAL (SLM)

274 Business Mathematics and Statistics The Ogive: This example explains the ogive. The cumulative frequency distribution or the data. Marks No. of Students Less than 20 4 Less than 30 10 Less than 40 18 Less than 50 33 Less than 60 43 Less than 70 48 Less than 80 50 Points are plotted on the x-axis and the y-axis representing the above frequency distribution on a suitable scale. These plotted points are (20, 4), (30, 10), (40, 18), (50, 33), (60, 43), (70, 48) and (80, 50). When these points are joined by a smooth freehand curve, the curve that is so formed is known as the ogive or the cumulative frequency curve. This curve enables us to find the median easily. or instance, or the given data, the procedure or finding the median is as follows: we draw a straight line parallel to the x-axis, through the point of the y-axis, This line cuts the curve at the point L. from L we draw a perpendicular on the x axis. The root of this perpendicular shows the value of the median (see Fig. 4). 11.3 Summary Measures that describe, characterize or represent a given data are known as ‘Statistical Averages’. There are two main objects while calculating a statistical average.

Firstly, to indicate precisely the sum and substance of an entire mass numerical data. Secondly, to serve as a means as well as a measure of comparison with other similar groups of numerical data. The different types of statistical averages are:  Arithmetic mean.   Median   Mode   Geometric mean CU IDOL SELF LEARNING MATERIAL (SLM)

Dispersion 275  Harmonic mean   Quadratic mean   Moving average   Progressive average The arithmetic mean (A.M.) is the quantity obtained by summing up the values of items in a variable and dividing the sum by the number of items. When the different items in a data are assigned weights according to their significance (relatively), the average then calculated is called: a weighted arithmetic average. The value of the middle item of a series that is ordered either in the ascending or descending order of magnitude is called Median. The value of the item in a variable that is repeated the greatest number of times is called the mode. In other words, it is the value of the item that occurs most frequently in the data. It is said to be the most prominent item as well as a typical measurement. If x1, x2, x3 … xn are n quantities, the geometric mean (G.M.) of these quantities is the nth root of the product of these quantities. If xl, x2, x3 … xn are n items, then the harmonic mean (H.M.) of these is the reciprocal of the average of the reciprocals of these n items. While calculating quadratic mean the average of the squares of the given quantities is to be obtained first and then its square-root calculated.

As its name indicates, it is an average that moves over a period of time. It’s possible to find out the trend of some given values over a period of time with this average. Progressive average is calculated to get a progressive outlook or trend in business, profits, sales, output etc. The values of those items that divide a given data into four, ten or a hundred parts are called quartiles, deciles and percentiles respectively. Hence a given data has three quartiles denoted by Q1, Q2, Q3; nine deciles D1, D2, D3 .... D9; and 99 percentiles Pl, P2, P3 ..... P99. CU IDOL SELF LEARNING MATERIAL (SLM)

276 Business Mathematics and Statistics 11.4 Key Words/Abbreviations MD =Mean Deviation QD = Quartile Deviation SD = Standard Deviation CV = Coefficient of variation 11.5 Learning Activity ? For the data given below, calculate: = Standard deviation and the coefficient of variation. = The wage limits or the central 50% of the wage earners Weekly Wages No. of Employees 208 21 9 22 11 23 15 24 18 25 26

26 17 27 14 28 5 29 2 2. From the prices of shares X and Y given below, state which is more stable. X 29 41 83 57 62 68 51 19 80 50 Y 50 53 47 61 49 63 70 57 60 40 3. From the prices of shares A and B given below, state which is more stable A 136 129 120 117 137 140 156 165 170 173 B 86 83 64 75 78 96 107 111 123 137 CU IDOL SELF LEARNING MATERIAL (SLM)

Dispersion 277 11.6 Unit End Questions (MCQ and Descriptive) A. Descriptive Type: Short Answer Type Questions = What do you understand by dispersion? Explain the different measures of dispersion pointing out their relative merits and demerits. = Explain the practical utility of the various measures of dispersion. = Distinguish between mean deviation and standard deviation. = What do you understand by coefficient of variation? Explain its uses. = Is standard deviation superior to the other measures of dispersion? Give reasons. = For the data given below calculate The coefficient of variation The wage limits or the central 50% of the employees The wage limit below which 15% of the employees have their earnings. Weekly Wages No. of Employees 20 3 21 9 22 23 11 15

24 18 25 26 26 17 27 14 28 6 29 2 7. Find in which factory payment of wages is more consistent. Daily Wages Factory AB 15-20 20-25 07 05 25-30 13 15 30-35 18 18 22 24 CU IDOL SELF LEARNING MATERIAL (SLM)

278 Business Mathematics and Statistics 35-40 15 13 40-45 10 08 45-50 05 07 8. Find the mean deviation from the median and mode in the following data: Age Number 16 30 17 32 18 35 19 46 20 51 21 60 22 43 23 32 24 19 9. Calculate the mean deviation from the mean and quartile deviation for the data given below Class f

0-9 4 10-19 6 20-29 8 30-39 15 40-49 10 50-59 5 60-69 2 [Ans: Mean deviation from mean = 11.94, quartile deviation = 10.91] 10. Find in which factory payment of wages is more consistent. Daily Wages Factory 15-20 AB 20-25 13 21 25-30 27 39 30-35 48 43 35-40 52 57 40-45 25 22 45-50 15 18 10 10 CU IDOL SELF LEARNING MATERIAL (SLM)

Dispersion 279 ` From the following observation prepare frequency distribution table in ascending order starting with 5-10 (inclusive method) and find the mean and standard deviation. Scores 19 18 15 35 32 30 8 12 17 8 18 19 18 30 36 42 35 37 30 39 25 24 26 28 35 8 17 19 22 24 10 16 15 18 17 12 21 19 30 31 19 26 24 24 29 27 25 28 22 22 21 28 = From the following observations prepare a frequency distribution table in ascending order starting with 100-110 (exclusive method) and find out the coefficient of variation. 125 108 112 126 110 132 136 130 149 155 120 130 136 138 125 111 119 125 140 148 147 137 145 150 142 135 137 132 165 154 B. Multiple Choice/Objective Type Questions 1. The simplest measure of dispersion is __________. (a) Range (b) Mean (c) Mode (d) Deviation 2. __________ deviation is also called as semi-inter quartile range. (a) Fractile (b) Quartile (c) Standard (d) None of these 3. __________ is the term used to denote the square of the standard deviation. (a) Mean (b) median

(c) Variance (d) None of these = __________ is used to draw a continuous frequency distribution. (a) Histogram (b) Graph (c) Frequency polygon (d) Ogive curve Answers: (1) (a); (2) (b); (3) (c); (4) (a) 11.7 References References of this unit have been given at the end of the book. ˆˆˆ CU IDOL SELF LEARNING MATERIAL (SLM)

280 Business Mathematics and Statistics UNIT 12 SAMPLING Structure 12.0 Learning Objectives 12.1 Introduction 12.2 Sample and Sampling 12.3 Main Object of Sampling 12.4 Basic Principles of sampling 12.5 Summary 12.6 Key Words/Abbreviations 12.7 Learning Activity

12.8 Unit End Questions (MCQ and Descriptive) 12.9 References 12.0 Leaning Objectives After studying this unit, you will be able to:  Explain about the sample, population, universe, census, survey.   Elaborate the main object of sample and the basic principles of sampling 12.1 Introduction An entire collection of all the individuals or things or units is called a Universe or Population. The entire count of all the units of the population is called a Census. A part of a storability of individual units (census) is a sample. Study of a census takes much time, money and energy. Sample studies are covered out. The basic principles of sampling deserve special study. CU IDOL SELF LEARNING MATERIAL (SLM)

Sampling 281 Census and Sample: Census being an exhaustive procedure necessitates much efforts and time. Hence, representative sample of a given, population is considered as sufficient. We study various sampling methods. 12.2 Sample and Sampling The word ‘sample’ means a part of any collection of things, individuals or results of operations that are quantitatively expressed. A totality or a collection of things or individuals is said to constitute a population. Hence, a sample simply means a part of a population. A finite population is one that has individuals or things that can be finitely expressed in numerical terms, whereas an infinite population is one that cannot be expressed finitely. For example, the heights of 100 persons form a finite population and is also called Universe. The following are a few examples of finite populations : population of marks of students at an examination, population of prices, weights, incomes etc. To draw certain conclusions about the characteristics of a population it is sufficient to select and study a part of it. In short for the purposes of statistical studies a sample would suffice. The entire work of selecting samples from a population is termed sampling. 12.3 Main Object of Sampling The principle aim of selecting a sample and studying it is to acquire the maximum information about the population with the least amount of time, money and energy. In brief, maximum information with minimum effort is the goal of sampling. 12.4 Basic Principles of Sampling

Before undertaking the work of sampling it is necessary to bear in mind the following points: ? No bias or prejudice should creep in the selection of a sample or samples. ? All the members of the sample should be governed by the same rules and conditions of sampling. ? Individual members of the sample should be entirely independent of one another. ? Special importance should never be attached to certain parts of the population while selecting members for the sample.

CU IDOL SELF LEARNING MATERIAL (SLM) 282 Business Mathematics and Statistics 12.5 Summary The word ‘sample’ means a part of any collection of things, individuals or results of operations that are quantitatively expressed. A totality or a collection of things or individuals is said to constitute a population. Hence a sample simply means a part of a population. A finite population is one that has individuals or things that can be finitely expressed in numerical terms, whereas an infinite population is one that cannot be expressed finitely. The entire work of selecting samples from a population is termed sampling. The principle aim of selecting a sample and studying is to acquire maximum information about the population with the least amount of time, money and energy. In brief, maximum information with minimum effort is the goal of sampling. The basic principles of sampling state the following: No bias or prejudice should creep into the selection of a sample or samples, All members of the sample should be governed by the same rules and conditions of sampling, Individual members of the sample should be entirely independent of one another, Special importance should never be attached to certain parts of the population while selecting members for the sample. 12.6 Key Words/Abbreviations

Bias, Unbaised sample, census, representative, sample survey, census survey or count. 12.7 Learning Activity ª To learn the procedure to be adapted for the purpose of conducting a sample survey. ........................................................................................................................................ ........................................................................................................................................ ª To understand the distinguishing features between a sample method of enquiry and the census method. ........................................................................................................................................ ........................................................................................................................................

CU IDOL SELF LEARNING MATERIAL (SLM)

Sampling 283 12.8 Unit End Questions (MCQ and Descriptive) A. Descriptive Type: Short Answer Type Questions Explain the following terms: Finite Population, Representative Sample, Biased Sampling. State the main objective of sampling and the basic principles of sampling. What are the requirements of a good sample? B. Multiple Choice/Objective Type Questions A sample discovered from a heterogeneous population is not-representative. (a) Correct (b) Incorrect (c) Relevant (d) None of these 2. Census survey is more accurate than a sample survey (a) Irrelevant (b) False (c) True (d) None of these 3. Sample does not represent all the features/characteristics of a given population. (a) Incorrect (b) True (c) False (d) None of these 4. Maximum information with minimum effort is the goal of sampling (a) Correct (b) Incorrect (c) Irrelevant *d) None of these Answers:

(1) (a); (2) (c); (3) (b); (4) (a) 12.19 References References of this unit have been given at the end of the book. ˆˆˆ CU IDOL SELF LEARNING MATERIAL (SLM)

284 Business Mathematics and Statistics UNIT 13 TYPES OF PROBABILITY Structure 13.0 Learning Objectives 13.1 Introduction 13.2 Methods of Sampling 13.3 Other Methods of Sampling 13.4 Statistical Laws 13.5 Sample Size 13.6 Summary 13.7 Key Words/Abbreviations 13.8 Learning Activity 13.9 Unit End Questions (MCQ and Descriptive) 13.10 References

13.0 Learning Objectives After studying this unit, you will be able to:  Describe the importance of the concept of productivity and its uses in the sampling process.   Illustrate the various methods of probability sampling and non-probability sampling.   Explain the relevance of statistical laws and the methods commonly used for the purpose of finding representative sample size. CU IDOL SELF LEARNING MATERIAL (SLM)

Types of Probability 285 13.1 Introduction When samples are drawn from a given population it is necessary to ensure their representativeness. For this purpose use of probability methods is considered as very much appropriate. Hence probability and non-probability sampling methods deserve special study. These methods are explained below. 13.2 Methods of Sampling Amongst the methods of sampling there are two categories: probability sampling and non probability sampling. In probability sampling, the process of selecting members is based on the chance factor. If the process of selection is not based on chance but on some arbitrary method then it is non-probability sampling. The following methods of sampling deserve special study: (xix) Random Sampling (probability sampling). (xx) Systematic Sampling (probability sampling). (xxi) Purposive Sampling (non probability sampling). (xxii)Stratified Sampling (probability sampling). (xxiii) Multi-stage Sampling (probability sampling). 13.2.1 Random Sampling

A sample that consists of members that are chosen at random from a population is called a random sample; the entire process of selecting members at random from a population is known as random sampling. This method ensures that the probability of each member of the population in the sample is equal. In other words, each member has the same chance of being included in the sample. Random sampling is a good method because members of the sample are chosen not only without bias or prejudice but purely according to the principles of random sampling. Further, to overcome bias or errors due to human weakness, certain methods of random sampling have been devised. For instance, the following are some of the methods: ? Method based on mechanical devices. ? Method based on random sampling numbers CU IDOL SELF LEARNING MATERIAL (SLM)

286 Business Mathematics and Statistics One of the simplest devices is the lottery drum method. Here, members of the population are numbered on tickets of uniform size and colour. These tickets are placed in a drum. Just like a pure lottery, tickets are drawn at random. In the case of a small size random sample a simple device is to number all the members of the population on small indentical cards — of uniform size and colour, like playing cards and draw them from the pack, one by one, with replacement after each shuffle and drawing. This procedure is continued till the cards drawn give the required size of the random sample. In the case of large scale random sampling these methods are not considered to be useful. Hence, tables of random sampling numbers such as Tippet’s random sampling numbers or Kendall and Babington- Smith’s random sampling numbers are used. 13.2.2 Systematic Sampling When complete population lists are available this method of sampling is considered to be the most appropriate one. Further it is a quick and easy method. The method consists selecting at random the first member and thereafter the other members in a systematic way. For example, if a population list comprises 48,000 members and a sample of 800 is required then we can select every sixtieth item of the population. But before doing that it is necessary to select the first member by random sampling, from the numbers 1 to 60. For instance if 22 happens to be the number selected at random then it will be the first number, 82 will be the second number, 142 the third and so on. Being quick and simple this method of sampling is considered to be very useful and hence has universal application. 13.2.3 Purposive Sampling (Judgement Sampling) The main object of this technique is to serve a particular purpose. Members of the sample are chosen strictly according to the criterion laid down. This sampling is also known as judgement

sampling. As the individual members of the sample are chosen in accordance with a certain principle, the method of such a selection is called purposive sampling. The nature of this sampling technique is such that members of the population have unequal probabilities of inclusion in the sample, which in turn means that some memebrs have a very high probability of inclusion over others. For instance, while undertaking an opinion survey, only the personal judgement of a particular group of persons may be taken into consideration. CU IDOL SELF LEARNING MATERIAL (SLM)

Types of Probability 287 13.2.4 Stratified Sampling (Probability Sampling) According to this method, the size of the sample the members of which are selected fromm a particular stratum by random sampling is proportional to the size of the stratum. All individual items selected by random sampling from each stratum constitute a sample called the stratified sample. Therefore the process of such a sample selection is termed stratified sampling. When compared with the systematic and random sampling methods, stratified sampling stands prominent as the better choice. This is because it overcomes the chief hurdle of random sampling, viz. giving unequal representation. Here unequal representation implies that certain parts of the population may be better represented than others in the sample. Further, the purposive sampling technique introduces a certain bias in the selection of members which is not so in the case of stratified sampling. The chief merit of this method, apart from stratification, is that the manner of picking up items from each stratum is purely according to random sampling principles. 13.2.5 Multi-stage Sampling This is the best method to use when large scale nationwide surveys are to be undertaken. The notable fact about this method is that the entire sampling is undertaken in a certain number of stages depending upon the nature of the survey. The only problem is that the larger the number of stages the greater is the likelihood of accuracy being sacrificed. The chief merit of this method of sampling is that it is less expensive and more practicable. Here’s an example of how multi-stage sampling draws random samples at each of the different stages. For a study of the smoking habits of people inhabiting a particular State a multi-stage sampling may be undertaken as follows : the process of selecting a few districts in the State by random sampling constitutes the first stage; selecting by random sampling a few towns and cities from these selected districts constiutes the second stage; selecting by random sampling a few blocks and localities from these selected towns and cities constitutes the third. Finally selecting, by random sampling, individuals from these selected blocks and localities constitutes the final stage.

13.3 Other Methods of Sampling The other methods of sampling are: « Cluster Sampling (probability sampling). « Quota Sampling (non-probability sampling). CU IDOL SELF LEARNING MATERIAL (SLM)

288 Business Mathematics and Statistics Convenience Sampling (non-probability sampling). Panel sampling (non-probability sampling). 13.4 Statistical Laws We now consider the following basic laws which provide the background to sampling techniques: ? The Law of Statistical Regularity. ? The Law of Inertia of Large Numbers. 13.4.1 The Law of Statistical Regularity This law asserts that a sample will always show the characteristics of the population to which it belongs. In other words when a few members from a population are chosen at random, then the members so selected undoubtedly explain the nature of the population. Further, whatever be the selected part of the population, it will reflect to a certain extent the regular characteristics of the entire population. In short the philosophy of this law is that a part represents the whole. Let us take an example. If we consider the marks secured by 40 candidates in a subject, amongst 200 candidates, then clearly the arithmetic average of the marks of these 40 candidates will not differ much from the average of the marks of all the 200 candidates. Of course, we might observe that the larger the sample size the more negligible would be the difference. The utility of this law can be at once seen when we deal with statistics of life insurance. Even statistics of agricultural experiments indicate the significance of this Law. 13.4.2 The Law of Inertia of Large Numbers

This law simply means that the larger the sample size the stronger would be its position in fully representing the entire population. In other words, large samples always tend to give more accurate results about the population to which they belong. For example, if we toss a coin only a few times then the proportion of tails to the total number of tosses is likely to differ significantly from the proportion of the number of heads to the total number of tosses. But if we toss a coin a large number of times then there is every possibility for the proportions of heads and tails to be each equal to half. This clearly points out that large samples are always more stable, in the sense that they represent the population more accurately. CU IDOL SELF LEARNING MATERIAL (SLM)

Types of Probability 289 13.5 Sample Size For the calculation of sample size, the followinig formula can be used., (i) Sample Size n = N 1Ne2 where N = total population, e = error (confidence level) such as 5% or 10% that is 0.05 or 0.10. Solved Problems Problem 1: Suppose N =- 10000, e = 0.05, then 10000 10000 n = 110000 (0.05)2 125 (ix) 384.615 (x) 385 approximately. A random sample of size 385 would represent a population of 10000 at 5% level (error). Problem 2: suppose N = 60000, e = 0.10, then 60000

n = 16000(0.10)2 98.36 98 approximately = Sample size in the case of proportions (xiv) 2 ·2 n = ¨ ¸ pq ©e¹ where Z = Standard limit depending upon confidence level. For instance, 1.96 limit corresponds to 5% level. e = Sampliing error such as 0.05. p = The given proportion CU IDOL SELF LEARNING MATERIAL (SLM)

290 Business Mathematics and Statistics Problem 3: Suppose Z = 1.96, e = 0.05V, p = 0.20 then, §1.96·2 §1.96·2 N= ¨ ¸ pq = ¨ ¸ (0.20) (0.80) © 0.05¹ © 0.05¹ 245.8624 246 approximately. = Method of deriving the formula for finding the sample size. We know that the sammpling error X' = P, X' = Sample Mean, P = Population mean. (confidence limits for mm are x ± Z (SE) where / is the confidence coefficient such as 1.96 at 5% level of significane and SE is the standard eerror. Now X – P = X(SE) V e = Z(SE) = Z n VV e = Z n , i.e., n = Z T § 2· 2 § Z· V ¨¸ n = z2¨ 2 ¸ = ¨ ¸ ©T ¹ ©T¹ V = standard deviation of the population

§Z·2 §Z·2 n = ¨ ¸ pq ¨ ¸ p(1p) ©e¹ ©e¹ = Use of tables for finding the sample size A researcher can use the tables prepared by Eckhardt, on the basis of the values of p, E and Z, E is the error and z is the confidence level. Refer Echardt (social research Methods, 1978:400) = Problems with reference to stratified sampling and cluster sampling. Problem 4: According to a survey on the basis of income levels, a researcher divides 25,000 families into five strata as mentioned below: Stratum 1: Income less than ` 5,000 p.m. = 8,000 Stratum 2: Income less than ` 15,000 and above ` 5,000 to 10,000 CU IDOL SELF LEARNING MATERIAL (SLM)

Types of Probability 291 Stratum 3: Income less than ` 5,001 p.m. = 4,000 Stratum 4: Income less tan ` 1 lakh p.m. = 2,500 Stratum 5: Income above ` 1 lakh p.m. = 500 Suppose the researcher has to study an appropriately representative sample of 100 families, then according to proportional memthod, he has to select at random from each stratum, the number of members mentoned below, as per the calculations. 4000 Stratum 3: Sample size = 25000 × 100 = 16 2500 Stratum 4: Sample size = 25000 × 100 = 10 500 Stratum 5: Sample size = 25000 × 100 = 2 The final sample size = 32 + 40 + 16 + 10 + 2 == 100 Strata with Different Variability We consider strata that not only differ in their sizes but also in their respective variability. Therefore, in regard to variability within a stratum, the standard deviation has to be taken into account.

Suppose we consider as the number of strata i.e., N1, N2, N3 … Nx with standard deviatons S1, S2, S3 … Sx = respectively and n = n1 + n2 + … nx where n1, n2, n3… nx denote the respective sample sizes of the x strata then the sample size of the pth strata is given by z x Np Sp np = NSNS5 NS 1122 xx where p = 1, 2, 3 … x 13.6 Summary The methods of sampling are: Random Sampling (Probability Sampling), Systematic Sampling (Probability Sampling), Stratified Sampling (Probability Sampling), Multistage Sampling (Probability Sampling) CU IDOL SELF LEARNING MATERIAL (SLM)

292 Business Mathematics and Statistics The other methods of Sampling are: Cluster sampling (Probability Sampling), Quota Sampling (Non-Probability Sampling), Convenience Sampling (Non-Probability Sampling), Panel Sampling (Non-Probability Sampling). Basic laws which provide the background techniques are: ‘The law of Statistical Regularity’, ‘The law of Inertia of large Numbers.’ For the calculation of Sample size the following formula can be used: Sample Size n = N/1 + Ne2 (Given by Taro Yemane) 13.7 Key Words/Abbreviations Random, Systematic, Stratified, Cluster, Multi-stage, Convenience Judgement, Statistical regularity, Inertia, Probability, Non-probability 13.8 Learning Activity 11. Give some examples of situations and circumstances when we can appropriately use probability sampling and non-sampling methods. ......................................................................................................................................... .........................................................................................................................................