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INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES BANGALORE BUSINESS STATISTICS MODULE I – MEASURES OF CENTRAL TENDENCY 1. Calculate the simple arithmetic average of the following items of sizes. 20, 50, 72, 28, 53, 74, 34, 54, 75, 39, 59, 78, 42, 64, 79 2. The following table gives the number of children born per family in 735 families. Calculate the average number of children born per family. Number of Number of Number of Number of Children born families children born families per family per family 0 96 7 20 1 108 8 11 2 154 9 6 3 126 10 5 4 95 11 5 5 62 12 1 6 45 13 1 3. The following data give the sizes of shoes sold by a store during a given week. Find the average size by the short cut method. Size of Shoes No. of pairs Size of Shoes No. of pairs 4.5 1 8 95 5 2 8.5 82 5.5 4 9 75 6 5 9.5 44 6.5 15 10 25 7 30 10.5 15 7.5 60 11 4 4. The following table gives the heights of 350 mean. Calculate the mean height of the group. Height (in cms) 159 161 163 165 167 169 171 17 (x) 3 No. of persons 1 2 9 48 131 102 40 17 (f) 5. Calculate the arithmetic average of the following by the Direct Method. Weekly wages (Rs.) 11-13 13-15 15-17 17-19 19-21 21-23 23-25 No. of labourers 3 4 5 6 5 4 3 6. The following table gives the marks obtained by a set of students in a certain examination. Calculate the average marks per student. Marks No. of students Marks No. of students 10-20 1 60-70 12 20-30 2 70-80 16 30-40 3 80-90 10 40-50 5 90-100 4 50-60 7 7. Find arithmetic mean for the following data. Class 50-59 40-49 30-39 20-29 10-9 0-9 interval Frequency 1 3 8 10 15 3 8. Given below is the distribution of 140 students obtaining marks or higher in a certain examination (all marks are given in whole numbers): X 10 20 30 40 50 60 70 80 90 100 C.F 140 133 118 100 75 45 25 9 2 0 Calculate the arithmetic mean marks obtained by the candidates. 9. The following table gives the life time in hours of 400 radio tubes of a certain make. Fine the mean lifetime of the radio tubes. Life time (in Number of Life time (in Number of hours) tubes hours) tubes Less than 300 0 Less than 800 265 Less than 400 20 Less than 900 324 Less than 500 60 Less than 1000 374 Less than 600 116 Less than 1100 392 Less than 700 194 Less than 1200 400 10. A market with 168 operating firms has the following distribution of average number of workers in various income groups. Income 150-300 300-500 500-800 800-1200 1200-1800 groups No. of 40 32 26 28 42 firms Average 8 12 7.5 8.5 4 No. of workers Find the average salary paid in the whole market. 11. Find the value of median of the following items: 5,7,9,12,10,8,7,15,21 12. Find the value of the median from the following data. 10, 18, 9, 17, 15, 24, 30, 11 13. Find the value of the median, Q1, Q3 from the following data. Daily wages (Rs.) 10 5 7 11 8 Number of workers 15 20 15 18 12 14. Find the median, Q1, Q3 from the following distribution. Class intervals Frequencies Class intervals Frequencies (Rs.) 1-3 6 11-13 16 3-5 53 13-15 4 5-7 85 15-17 4 7-9 56 TOTAL 245 9-11 21 15. Calculate median, Q1, Q3 from the following data. Value Frequency Value 0-4 328 30-39 5-9 350 40-49 10-19 720 50-59 20-29 664 60-69 Frequency 598 524 378 244 16. Calculate median, Q1, Q3 from the following data. Age Number of Age persons 55-60 7 35-40 50-55 13 30-35 45-50 15 25-30 40-45 20 20-25 Total Number of persons 30 33 28 14 160 17. Calculate median, Q1, Q3 from the following data. Value Frequency Value Less than 10 4 Less than 50 Less than 20 16 Less than 60 Less than 30 40 Less than 70 Less than 40 76 Less than 80 18. Calculate median, Q1, Q3 from the following data. Size Frequency Size More than 50 0 More than 20 More than 40 40 More than 10 More than 30 98 Frequency 96 112 120 125 Frequency 123 165 19. Compute median, Q1, Q3 from the following data. Mid values 115 125 135 145 155 165 175 Frequency 6 25 48 72 116 60 38 185 22 195 3 20. Find the mode of the following data relating to the weight of 10 students. 120, 130, 135, 130, 140, 130, 132, 132, 135, 141 21. Calculate the value of mode from the following data. Class intervals Frequency Class intervals 10-20 4 60-70 20-30 6 70-80 30-40 5 80-90 40-50 10 90-100 50-60 20 100-110 Frequency 22 24 6 2 1 22. Find the value of Mode from the following data. Size of the items Frequency Size of the items Frequency 100-110 4 140-150 33 110-120 6 150-160 17 120-130 20 160-170 8 130-140 32 170-180 2 23. Given Mean = 20; Mode = 15; Find the value of median 24. Given Mode = 25; Median = 20; Find the value of mean. 25. Calculate simple Geometric mean from the following data 133, 141, 125, 173, 182 26. The following data related to the distance traveled by 520 villagers to buy their weekly requirements. Miles 2 4 6 8 10 12 14 16 18 20 traveled No. of 38 104 140 78 48 42 28 24 16 2 villagers Calculate the arithmetic average. INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES BANGALORE BUSINESS STATISTICS MODULE I – MEASURES OF CENTRAL TENDENCY AND MEASURES OF DISPERSION 1. The profits of the company for the last 8 years are given below. Calcultate the Range and its co-efficient. Year Profits (in ‘000 Rs.) 1975 1976 40 30 1977 80 1978 100 1979 120 1980 90 1981 200 1982 230 2. Calculate co-efficient of Range from the following data. Weekly wages (Rs.) No. of labourers 50-60 60-70 70-80 80-90 90-100 50 45 45 40 35 100-110 110-120 30 30 3. Calculate Quartile Deviation and its co-efficient from the following data. Weight (in pounds) Frequency Cumulative Frequency 120 122 124 126 130 140 150 160 1 1 3 4 5 9 7 16 10 26 3 29 1 30 1 31 4. Calculate Semi-Inter Quartile Range and its co-efficient from the following data. Marks No. of Students 0-10 11 10-20 18 20-30 25 30-40 28 40-50 30 50-60 33 5. Calculate Quartile Deviation and its relative measure 60-70 22 70-80 15 80-90 22 Variable 20-29 30-39 40-49 Frequency 306 182 144 Variable 50-59 60-69 70-79 Frequency 96 42 34 6. Estimate an appropriate measure of dispersion for the following data: Income (Rs.) Less than 50 50-70 70- 90 90-110 No. of persons 54 100 140 300 Income (Rs) 110-130 130-150 Above 150 Total No. of persons 230 125 51 1000 7. The following are the marks obtained by a batch of 9 students in a certain test. S.No. Marks (out of 100) S.No. Marks (out of 100) 1 68 6 38 2 49 7 59 3 32 8 66 4 21 9 41 5 54 8. Calculate Mean Deviation (from arithmetic average) for the following values. Also calculate its co-efficient. 4800, 4600, 4400, 4200, 4000 9. Find mean deviation from the following data. No. of accidents Persons having said No. of number of accidents accidents 0 15 7 1 16 8 2 21 9 3 10 10 4 17 11 5 8 12 6 4 Total 10. Marks No. of students 11. Persons having said number of accidents 2 1 2 2 0 2 100 Calculate mean deviation from the following data. 0-10 10-20 20-30 30-40 40-50 50-60 6 5 8 15 7 6 60-70 3 Calculate the standard deviation from the following data. 160, 160, 161, 162, 163, 163, 163, 164, 164, 170 12. The mean of 200 items is 48 and then Standard deviation is 3. Find (i) the sum of all the items; (ii) the sum of squares of all the items. 13. For two groups of observations of the following results are available. Group I Total of (X-5) = 3; Total of squares of (X-5) = 43; N1 = 18 Group II Total of (X-8) = - 11; Total of squares of (X-8) = 76; N2 = 17 14. Calculate the standard deviation from the following data. Size of the items 6 7 8 9 Frequency 3 6 9 13 Size of the items 10 11 12 Frequency 8 5 4 15. Calculate standard deviation for the following distribution. Values 10 20 30 40 50 60 70 Frequency 1 5 12 22 17 9 4 16. The following data relate to the age of a group of Government employees. Calculate the arithmetic mean and standard deviation. Age 50-55 45-50 40-45 35-40 30-35 25-30 20-25 No. of 25 30 40 45 80 110 170 employees 17. Calculate the mean wages and standard deviation of all the workers taken together. Section No. of workers Mean wages Standard Deviation A 50 113 6 B 60 120 7 C 90 115 8 18. Calculate the combined mean and standard deviation of the two series. Series A Series B Mean 50 40 Standard Deviation 5 6 No. of items 100 150 19. An analysis of the monthly wages paid to the workers in two firms A and B, belonging to the same industry, gives the following results : Particulars Firm A Firm B Number of wage earners 586 648 Average monthly wages (Rs.) 52.5 47.5 Standard Deviation of the Square root of 100 Square root of 121 distribution of wages (Rs.) 20. The mean and standard deviation of the 20 items was found to be 10 and 2 respectively. Later, it was found the item 12 was misread as 8. Calculate correct mean and standard deviation. 21. The arithmetic mean and standard deviation of a series of 20 items were calculated by a student as 20 cms and 5 cms respectively. But, while calculating them, an item 13 was misread as 30. Find the correct arithmetic mean and standard deviation. 22. In a frequency distribution of 100 persons grouped in class intervals of Rs. 10-12, 12-14, 14-16 etc., revealed the mean wage to Rs. 32.02 and its standard deviation a Rs. 13.18. Later, it was found that the wage of a labourer who was getting Rs. 57 was misread as Rs. 27. Calculate the correct mean and standard deviation. SKEWNESS, MOMENTS AND KURTOSIS 1. Calculate the co-efficient of skewness from the following data. Value 10 20 30 40 50 60 70 Frequency 1 5 12 22 17 9 4 2. Assume that a firm has selected a random sample of 100 from its production line and has obtained the data shown in the table below. Class interval Frequency Class interval Frequency 130-134 3 150-154 19 135-139 12 155-159 12 140-144 21 160-164 5 145-149 28 Total 100 3. Calculate the co-efficient of skewness based upon Mean and Median from the following distribution. Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 interval Frequency 6 12 22 48 56 32 18 6 4. Calculate Karl Pearson’s Measure of Skewness on the basis of Mean, Mode and Standard Deviation. X 14.5 15.5 16.5 17.5 18.5 19.5 20.5 21.5 F 35 40 48 100 125 87 43 22 5. Calculate the Karl Pearson’s co-efficient of skewness from the following data. Marks 0 10 20 30 40 50 60 70 80 (above ) No. of 150 140 100 80 80 70 30 14 0 students 6. Consider the following distributions: Particulars Distribution A Distribution B Mean 100 90 Median 90 80 Standard Deviation 10 10 (i) The Distribution A has the same degree of the variation as the distribution B. (ii) Both the distributi8ons have the same degree of Skewness – State whether it is True / False. 7. From the data given below, calculate the co-efficient of variation. Pearson’s measure of skewness = 0.42 Arithmetic mean = 86 Median = 80 8. Given, Mean = 30; Standard Deviation = 8; Karl Pearson’s Coefficient of Skewness = +0.40; Find the Median and Mode. 9. Calculate Bowley’s co-efficient of skewness. Wages (Rs.) 30-40 40-50 50-60 60-70 70-80 No. of persons 1 3 11 21 43 80-90 32 90-100 9 10. Calculate the Bowley’s co-efficient of skewness from the following data. Class intervals 10-19 20-29 30-39 40-49 50-59 60-69 70-79 Frequency 5 9 14 20 25 15 8 11. The measure of skewness for a certain distribution is – 0.8. If the lower and upper quartiles 44.1 and 56.6 respectively. Find the median. 12. In a frequency distribution, the co-efficient os skewness based on quartiles is 0.6. If the sum of the upper and lower quartiles is 100 and median is 38, find the value of upper quartile. 13. From the data given below, calculate Karl Pearson’s, Bowley’s co-efficients of skewness. Mean = 150; Median = 142; Q1 = 62; Q3 = 195; Standard Deviation = 30. 80-89 4 INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES BANGALORE BUSINESS STATISTICS KARL PEARSON’S CORRELATION CO-EFFICIENT 1. Find Out the correlation co-efficient of the following 10 firms. Firms 1 2 3 4 5 Sales 50 50 55 60 65 Expenses 11 13 14 16 16 between the sales and expenses 6 65 15 7 65 15 8 60 14 9 60 13 10 50 13 2. Calculate the Karl pearson’s co-efficient of correlation from the following data. X 65 66 67 67 668 69 70 72 Y 67 68 65 68 72 72 69 71 3. Calculate the Karl Pearson’s co-efficient of correlation between the values of X and Y from the data given below. Values of X 15 16 17 18 19 20 Values of Y 80 75 60 40 30 20 4. With the help of following data in 6 cities, calculate the co-efficient of correlation by Karl Pearson method between the density of population and death rate. Cities Area in Sq. miles Population in ‘000 Number of deaths A 150 30 300 B 180 90 1440 C 100 40 56 D 60 42 840 E 120 72 1224 F 80 24 312 5. Calculate the co-efficient of correlation between the corresponding values of x and y in the following table. Values of X 2 4 5 6 8 11 Values of Y 18 12 10 8 7 5 6. Calculate the co-efficient of correlation between the variables X and Y. X 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Y 30,000 50,000 60,000 80,000 1,00,000 1,10,000 1,30,000 X Y 7. Calculate the co-efficient of correlation between X and Y. 2 4 6 8 5 9 13 17 8. Given the following information. Calculate the correlation coefficient by Karl Pearson’s method. Series Assumed Mean Total deviations from Sum of squares of assumed mean deviations from assumed mean X 41 - 170 8180 Y 32 - 20 2290 Sum of products of deviations of X and Y from their respective assumed mean = 2880. Number of pairs of observations = 10. 9. Calculate co-efficient of correlation from the following results. N = 10; Sum of X = 100; Sum of Y = 150; Sum of squares of (X – 10); Sum of squres of (Y – 15) = 215; Sum of product of (X – 10) (Y – 15) = 60. 10. Calculate the Karl Pearson’s co-efficient of correlation between X and Y from the following information. N = 12; Sum of X = 120; Sum of Y = 130; Sum of squares of (X – 8) = 150; Sum of Squares of (Y – 10) = 200; Sum of product of (X – 8) (Y – 10) = 50. 11. Test the significance of correlation for the following values based upon the number of observations (i) 10 and (ii) 100. The co-efficient of correlation + 0.4 and + 0.9. 12. Find the co-efficient of correlation from the following data by the method of Karl pearson. X 1 2 3 4 5 Y 166 184 142 180 338 13. Calculate the correlation co-efficient from the following 15 pairs of students in Projects A and Project B. (1,10) (2,7) (3,2) (4,6) (5,4) (6,8) (7,3) (8,1) (9,11) (10,15) (11,9) (12, 5) (13,14) (14,12) (15,13). Use Spearman’s formula to find the rank correlation co-efficient. 14. Ten students were given tests in English and Mathematics. Their marks are given below. Student No. 1 2 3 4 5 6 7 Marks in 78 40 50 55 52 49 60 English Marks in 70 60 60 75 69 55 70 Mathematics Determine Spearman’s rank correlation co-efficient. 8 54 9 59 10 58 65 65 60 14. Ten competitors in a beauty contest were ranked by three judges in the following orders. Use the Spearman’s rank correlation method. First 1 6 5 10 3 2 4 9 7 8 Judge Second 3 5 8 4 7 10 2 1 6 9 Judge Third 6 4 9 8 1 2 3 10 5 7 Judge 15. Calculate the rank correlation co-efficient from the following data. X 60 34 40 50 45 41 22 43 42 66 64 46 Y 75 32 34 40 45 33 12 30 36 72 41 57 16. Calculate the co-efficient of correlation from the following data by Spearman’s Rank Difference method. Prices of Tea (Rs) 75 88 95 70 60 80 81 Prices of Coffee (Rs.) 120 134 150 115 110 140 142 50 100 17. The co-efficient of rank correlation between the debenture prices and share prices of a company was + 0.8. If the sum of the squares of the difference in ranks was 33, find the value of n. 18. The co-efficient of rank correlation of marks obtained by 10 students in Statistics and Accountancy was found to be 0.2. It was later discovered that the difference in the ranks in the two subjects obtained by one of the students was wrongly taken as 9 instead of 7. Find the correct value of co-efficient of rank correlation. 19. Calculate the co-efficient of rank correlation from the following data. X 48 33 40 9 16 16 65 24 46 Y 13 13 24 6 15 4 20 9 6 57 19 20. Calculate concurrent deviations from the data given below. No. of workers 300 350 400 450 500 550 No. of bales consumed 30 32 33 35 40 50 21. If there is a lag of one year between price and supply (so that the supply increases after one year of price change), calculate the co-efficient of correlation between the price and supply from the following data. Year 1973 1974 1975 1976 1977 1978 1979 1980 1981 Price 78 89 97 69 59 79 68 61 60 Supply 115 125 137 156 112 107 136 123 108 22. From the following data, obtain two regression equations using the method of Least Squares. X Y 2 5 4 7 6 9 8 8 10 11 23. From the following data, calculate two regression equations using the method of Least Squares. X Y 10 100 15 90 35 110 40 80 50 120 24. From the following data, obtain two regression equations using the method of Least Squares. Sales 91 Purchasaes 71 97 75 108 69 121 97 67 70 124 91 51 39 73 61 111 80 57 47 INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES BANGALORE BUSINESS STATISTICS DECISION TREE / DECISION THEORY 1. A company has to choose one of the three types of Biscuits, Cream, Coconut and Glucose. Sales expected during the next year are highly uncertain. Marketing Department estimates the profits considering manufacturing cost, promotional efforts and distribution set up etc., as given in the table below. Types of Biscuits Cream (C) Coconut (Co) Glucose (G) Profits on estimated level of sales ( in lakhs) for quantities 5,000 15 20 25 10,000 25 55 40 20,000 45 65 70 2. A trading company of Delhi is considering expansion of its activities and planning to open a marketing office at Kanpur to boost up the sales in North- East UP. It is to be decided whether to operate from the existing office at Delhi and cover the area by frequent traveling or else establishing the office at Kanpur. The connected pay-offs and probabilities are as under: Alternatives A. Operate from Delhi B. Open office at Kanpur States of Nature (i) increase in demand by 30% (ii) no appreciable change (i) increase in demand by 30% (ii) no appreciable change Probability 0.60 Pay-off (Rs. In lakhs) 50 0.40 5 0.70 40 0.30 - 10 3. Under an employment promotion programme, it is proposed to allow sales of newspapers on the buses during off-peak hours. The vendor can purchase the newspaper at a special discounted rate of 25 paise per copy against the selling price of 40 paise. Anny unsold copies are, however a dead loss. A vendor has estimated the following probability distribution for the number of copies demanded. No. of copies to be demanded Probability 15 0.04 16 0.19 17 0.33 18 0.26 19 0.11 20 0.07 How many copies should he order so that his expected profir will be maximum ? 4. A manufacturer finds the opportunity to increase his business beyond his present on existing production capacity. In order to decide hether to increase the production capacity, he would need a reliable information about increase in demand of the product, based on which only, he can commit his resources. He has two choices open to him, firstly, the expansion of the existing capacity with a cost of Rs. 8 lakhs. Or the modernization of the plant at a cost of Rs. 5 laksh. The time required for implementation of both the options is expected to be the same. While considering the demand pattern, he estimates the high demand situation at a probability of 0.35 as compared to the modern rise in demand at 0.65 probability. He also estimates that he would be spending an additional amount of Rs. 12 lakhs for expansion against Rs. 6 lakhs for modernization, if the demand rise is high, whereas in the case of moderate demand increase, the expenditure involved would be Rs. 7 lakhs for the expansioin or Rs. 5 laksh for modernization process. (a) Calculate the conditional profits under various combibinations. (b) Establish expansion or modernization so as to maximize its expected monetary value (c) Work out EPPI, EVPI and EOL. 5. A manufacturer has faced with a problem of fast change in technology and hence, fast change in production line. At this point of time, the research and development wing of the organization has suggested an improved new product with easy acceptance. It will cost the manufacturer Rs. 60,000 for the pilot testing and development testing before establishing the product in the market. The organization has 100 customers and each customer, might purchase, at the most, one unit of the product, due to cost and newness. The selling price suggested is Rs. 6,000 for each unit. The probability distribution for proportion of customers buying the product is estimated as follows. Proportions of customers Probability 0.04 0.1 0.08 0.1 0.12 0.2 0.16 0.4 0.20 0.2 Work out the expected opportunity, losses and suggest whether the manufacturer should develop the product or not. 6. A food product company is contemplating the introduction of a revolutionary new product with new packaging to replace the existing product at the same price (S1) or a moderate change in the composition of the existing product with a new packaging at a small increase in price (S2) or a small change in the composition of the existing except the “New” with a negligible increases in price (S3). The three possible states of nature of events are (i) high increases in sales (N1), (ii) no change in sales (N2) and (iii) decreases in sales (N3). The marketing department of the company worked out the pay-offs in terms of yearly net profits for each course of action for these events (expected sales). This is represented in the following table. States of Nature N1 N2 N3 S1 7,00,000 3,00,000 1,50,000 Courses of action S2 5,00,000 4,50,000 0 S3 3,00,000 3,00,000 3,00,000 Which strategy should he choose on the basis of (a) Maximum criterion (b) Maximax criterion (c) Minimax Regret criterion (d) Laplace criterion. 7. An investor has 3 options to invest, but he can invest in only one option at a time. He can invest either in a departmental store, a cold storage or in a car maintenance shop. If he invests in a departmental stores and succeeds, he can invest in the car maintenance shop. If he invests in the cold storage and succeeds, he can invest in the departmental stores. If he invests in the car maintenance shop, and succeeds, he can invest in the cold storage and if he succeeds, he can invest in the departmental store. Based on the data given below, draw a decision tree and advice the investor on the best decision to take. Particulars Probability of Investment Loss on failure success Departmental 0.65 8,00,000 45,000 stores Cold storage 0.60 7,50,000 40,000 Car maintenance 0.70 7,40,000 3,50,000 shop 8. A business man has two options to sell his products. He can set up a show room in the city or can sell from his factory outlet. Setting up a show room will cost Rs. 3,00,000 with 60% probability of success. If the showroom succeeds, he can earn gross profits of Rs. 8,00,000 per year. If it fails, he can close the show room or rent it out for an annual rent of Rs. 2,40,000 ( for the rest of the year). The probability of getting rent is 80%. If he sells from the factory outlets, he has to incur Rs. 50,000 as renovating charges. The chances of successful selling here is 40%, with a net profit of Rs. 4,00,000 per year. (a) What would you advise the business man to do ? (b) Advise the business man on how a decision tree helps him to make decisions. 9. A business man from Chennai wishes to sell his products in Bangalore. He can set up a show room will entail costs of Rs. 6,00,000 with a 55% probability of success. If the showroom succeeds, he can gain a net profit of Rs. 10,00,000 per year. If it fails, he can either shutdown the show room or rent it out for an annual rent of Rs. 3,60,000 (for the rest of the year). The probability that he gets rent for the show room is 40%. If he sells through a wholesaler, he incurs Rs. 3,00,000 initial costs. The chances of selling successfully are 45% with a net profit of Rs. 5,50,000 per year. (a) Advise the business man on the best decisions. (b) How is the decision tree analysis useful in business decision ? INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES BANGALORE BUSINESS STATISTICS THEORY QUESTIONS (UNIVERSITY QUESTIONS) SECTION A MODULE I 1. 2. 3. 4. 5. Define Statistics. Define Matrix What is a function ? What do you mean by Quadratic function ? Give one example for each of the following : (a) Linear function (b) Geometric progression 6. What is meant by regret criterion ? 7. What is differentiatioin ? 8. Define Marginal Revenue. Express this concept as a derivative. 9. What is maxima and minima ? 10. Define marginal cost. Express this concept as a derivative. 11. Define Arithmetic Progression. 12. Find the sum of the first 20 terms of an Arithmetic Progression, 2,5,8,……….. ? 13. What is Integration ? 14. What is ratio ? 15. What is Proportion ? 16. Find the sum of the first ten terms of an Arithmetic Progressiion, 1,3,5,7………….. 17. Find the three numbers in Arithmetic Progression whose sum is 21 and product is 280 ? 18. the fourth term of a GP is ½ and the seventh term is 4/27. Find the 12th term. 19. Find the sum of the first 8 terms of a Geometric Progression, 4,8,16………. 20. Show that 1 + 3 + 5 + ……….. to n terms = n^2. 21. Find the sum of the series 3 + 7 + 11 ……. To 40 terms. 22. The third term of a Geometric Progression is 3/4 , and the seventh term is 4/27. Find the fourth term. 23. How many terms of Geometric Progression, 4,2,1 amount to 127/16 ? 24. The first term and the last term of a Geometric Progression are 4 and 324. The sum of the series is 384. Find C.R. 25. A firm produces a single product and it can market as many units as it is able to produce at a price of Rs. 1.75. Its plant and equipment can produce as many units as 5000 units a day. The total fixed cost is Rs. 2000 daily. Unit variable cost is Rs. 0.50. How many units per day must be produced in order that the firm breaks even ? MODULE II 1. What is the importance of statistics in managerial decision making ? 2. Define Business Statistics. 3. What do you mean by Primary and Secondary Data ? 4. What purpose does a measure of central tendency serve ? 5. What are averages ? How are they useful ? 6. What are the desirable properties of an average ? 7. Define Geometric mean and Harmonic zmean. 8. Define a data array. Obtain the median of the data set: 3,0,7,6, -10. 9. Define Arithmetic mean and median for the raw data. 10. Give any 4 merits of the Arithmetic mean. 11. What do you understand by Dispersion ? Define Dispersion and give the properties of a good measure of Dispersion. 12. What is co-efficient of variation ? How is this measure useful ? 13. What are Ogives ? MODULE III 1. 2. 3. 4. 5. What What What What What is is is is is Probability ? Poisson Distribution ? Binomial Distribution ? Normal Distribution ? standard normal variate ? MODULE IV 1. What is Decision theory ? 2. Briefly explain decision trees. 3. What is Expected Pay-off in decision tree ? 4. What is Risk and Certainty & uncertainty ? 5. What is Administrative Decision ? 6. What is Certainty Equivalent ? 7. What is conditional pay-off ? 8. What is Decision Table ? 9. What is Decision tree ? 10. What is Expected Monetary Value ? 11. What is Expected Opportunity Loss ? 12. What is Expected Value Perfect Information / 13. What is Laplace criterion ? 14. What is Maximax criterion ? 15. What is Minimax criterion ? 16. What is Maximin criterion ? 17. What is operating decision ? 18. What is Risk Premium ? MODULE V 1. 2. 3. 4. 5. 6. 7. What is sampling ? Define Stratified Random Sampling. Illustrate. Explain and illustrate Systematic Sampling. What is meant by Cluster sampling ? What is the significance of testing hypothesis ? What is meant by Type I & Type II error ? What are the three basic assumptions considered in the analysis of variance ? 8. What is the sum of the square ? What is the mean square ? 9. What is Chi-square test and for what it is used ? 10. What is Quota Sampling ? MODULE VI 1. What is correlation ? What is its significance ? 2. What is co-efficient of Determination ? How will you interpret its value ? 3. What do you mean by lag in correlation ? 4. Explain Regression and its uses. 5. What do you mean by regression analysis ? Give its uses. 6. What is Multiple Regression ? 7. Explain the difference between correlation and regression analysis. Correlation and Regression analyses are constructed under different assumptions. They furnish different types of information. 8. What are the components of Time series ? 9. List the different methods that can be used for determining trend. INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES BANGALORE BUSINESS STATISTICS THEORY QUESTIONS (UNIVERSITY QUESTIONS) SECTION B & C MODULE I 1. Define Arithmetic and Geometric Progression and give in each case (i) the expression for the sum of first n terms and (ii) a business application. 2. Define the concept of the derivative of a function. Give the business applications of this concept. 3. Define Marginal Cost, Average Cost and Total Cost. What is the relationship between the three ? MODULE II 1. “It is never safe to take the published statistics at their face value without knowing their meaning and limitations” - Elucidate this statement. 2. What are the various types of measurement of averages ? Write the relationship between them. 3. Write short notes on Skewness and Kurtosis. 4. What are the properties of a good measure of dispersion ? MODULE III 1. Define and illustrate the concept of probability. 2. Describe the features of the following probability models (i) Poisson model (ii) Normal model 3. How is a normal curve ? 4. Describe a binomial distribution. Illustrate the applications of this distribution in Business Management. 5. Briefly explain the use of the Baye’s formula 6. Explain and illustrate : (i) Relative frequency approach and (ii) Subjective approach for evaluating probabilities. 7. Write the features of the Exponential Distribution. MODULE IV 1. What is sampling ? Explain the various methods of sampling. 2. Graphs and Diagrams have an advantage over written reports. Comment briefly. 3. What is hypothesis ? Explain the different types of hypothesis. 4. Explain how to set up and test a hypothesis. 5. Explain and point out the difference between one tailed and two tailed tests. 6. What is (i) Chi-square test (ii) Sign test (iii) Median test ? MODULE V 1. What are the various decision criteria followed in Decision Theory ? MODULE VI 1. What is an Index Number ? How do Laspeyres method and Paasches method differ ? 2. Explain the method of construction used and interpretation of the Fisher’s Ideal price Index number. Why his Index number method is considered as the best method ? 3. What is correlation ? Distinguish between positive and negative correlation. What is the significance of the co-efficient of correlation ? How IS correlation ueful in dealing with business problems ? 4. Explain the method of construction used and interpretation of the Pearsonian correlation co-efficient. 5. “The value of the co-efficient of correlation lies beween + 1 to – 1. Elaborate. Give the example for the following diagram. 6. List the methods of determining correlation between two variables. 7. What is business forecasting ? Mention its methods. Distinguish between variance and its co-efficient of variance. 8. Distinguish between cyclical variation and seasonal variation. 9. Write the differences between correlation and regression. INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES BANGALORE BUSINESS STATISTICS PROBABILITY & SAMPLING DISTRIBUTION SECTION B & C 1. A print shop has three presses; P1, P2 and P3, each press is shut down 10% of the time. The probability of any press being shut down is not influenced by the operating conditions of the other two. Determine the probability that (a) all the three presses are shut down; (b) atleast one press is shut down; (c) P1 and P2, both are shut down. 2. In a Post Office, three clerks are assigned to process incoming mail. The first clerk processes 40%, second 35% and the third 25% of the mail. The first clerk has an error rate of 0.04, the second ha 0.06 and the third has 0.03. A mail selected at random from a day’s output is found to have an error. The Post Master wishes to know the probability that the mail was processed by the first, second or third clerk respectively. 3. A sample of 100 dry battery cells tested to find the length of the life produced the following results : Arithmetic mean = 12 hours, Standard Deviation = 3 hours. Assuming that the data are normally distributed, what percentage of battery cells are expected to have life (i) more than 15 hours; (ii) less than 6 hours; (iii) between 10 and 14 hours ? Z 2.5 2.0 1.0 0.67 Area 0.4938 0.4772 0.3413 0.2487 4. In an Intelligence test administered to 500 students, the average score was 42 and standard deviation was 24. Find (a) the number of students whose score exceeded 50 (b) the number of students who got between 30 and 40 (c) the number of students who got score above 60. 5. A survey conducted by a detergent soap manufacturer of Brand P indicates that when men go shopping, they are likely to buy brand P 3 out of 4 times. When women of shopping, they are likely to buy brand P 2 out of 3 times. Find the probability of Brand P being bought when men and women shop together. 6. 1000 light bulbs with a mean life of 120 days are installed in a firm. Their length of life is normally distributed with a standard deviation of 20 days. How many bulbs will expire in less than 90 days ? 7. The average monthly sales of 700 branch offices is 42. (Rupees in lakhs). The standard deviation of sales is 18 (Rupees in lakhs). Find the number of branches whose sales exceed 50 (Rupees in lakhs) assuming normal distribution. 8. An MBA graduate is selected for an interview in 3 different firms. For the first post, there are 5 candidates, for the second post, there are 10 candidates and for the third post, there are 9 candidates. What are his chances of getting atleast one post ? 9. Ten percent of the tools produced in a certain manufacturing process turn out to be defective. Find the probability that in a sample of 10 tools chosen at random, exactly two will be defective by using (a) Binomial Distribution (b) Poisson Distribution 10. The number of machine malfunctions per shift at a factory is recorded for 180 shifts and the following data are obtained : No. of 0 1 2 3 4 5 6 malfunctions No. of shifts 82 42 31 12 8 3 2 11. A Philanthropist wants to build a hospital in a rural area. He can build 100, 200 or 300 bed hospital, depending upon whether anticipated demand is low, medium or high. The expected net profits and the prior distribution regarding the states of nature is given below. State of Nature Action A1 (Build Action A2 (Build (Action A3 (Build 100 bed hospital) 200 bed hospital) 300 bed hospital) Low Demand 20,000 - 10,000 - 30,000 Medium Demand 25,000 30,000 - 5,000 High Demand 30,000 50,000 60,000 12. A company has to choose between two products, any one for manufacturing. A market research was conducted and the probabilities and the profits at different levels of market acceptability are as follows: Acts State of Nature Good Fair Bad Products “A” 0.75 0.15 0.10 Products “B” 0.60 0.30 0.10 Calculate the Expected Value of the choice of alternatives and advise the company. INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES BANGALORE BUSINESS STATISTICS SAMPLING DISTRIBUTION SECTION B & C 1. 1. A simple random sample size is 36, is drawn from a finite population consisting of 101 units. If the population standard deviation = 12.6, find the standard error of sample mean when the sample is drawn (a) with the replacement (b) without replacement. 2. The mean lifetime of a sample of 100 fluorescent light tubes produced by a company is computed to be 1570 hours, with a standard deviation of 120 hours. The company claims that the average life of the tubes produced by a company is 1600 hours. Using a level of significance at 5 % level, is the claim acceptable / 3. A sample of 400 men managers is found to have a mean height of 171.39 cms. Can it be reasonably regarded as a sample from a large population of men managers of mean height 171.17 cms and standard deviation of 3.30 cms ? Justify your answer statistically. 4. Intelligence test given to a group of boys and another of girls provided the following summarized information: Gender Mean Score Standard Deviation Numbers Girls 75 10 50 Boys 70 12 100 Is the difference in the mean scores of boys and girls statistically significant ? Show clearly the steps leading to your conclusion. 5. For a population with known variance 185, a sample of 64 individuals leads to 2217 as an estimate of mean. Compute the standard error of the mean and also construct an interval that would include the population mean 95% of the time. What will be the width of this interval if the sample size were to be only 16. 6. A finite population of 1000 units has been stratified into three groups having 200, 500 and 300 units respectively. Allocate a sample of size 100 units proportionately to the strata. What are the chief advantages of stratifying a population before drawing samples ? 7. The standard deviation of the wages of 400 textile workers is Rs. 12.8. Another sample of 600 textile workers gives the standard deviation at Rs. 15.7. Find out if the standard deviation of the first sample significantly differs from the combined deviation of the two samples which is 14. 8. A factory is producing 50,000 pairs of shoes daily. From a sample of 500 pairs, 2% were found to be sub-standard quality. Estimate the number of pairs that can be reasonably expected to be spoiled in the daily production and assign the limits at 95% level of confidence. 9. A psychologist claims lady students are smarter than men students. The IQ tests of 40 females had a mean of 131 and Standard deviation of 15. A sample of 36 males had a mean of 122 and Standard Deviation of 15. Do the data support psychologist’s claim ? Use the 5% level of significance. INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES BANGALORE BUSINESS STATISTICS CHI SQUARE TEST 1. One hundred students appeared for an examination and results were categorized as follows depending upon whether they have received special training: Special Training Result Pass Fail Yes 36 12 No 30 22 Test whether the special training was useful to the candidates. Clearly state the steps in your analysis. (You may use the information that the relevant table value at 5% level is 3.84). 2. A Manufacturing Limited launched a new car and then interviewed 500 people. Find out whether people consider the new model of the car as superior in style compared to the old model of car, using Chisquare test. Superior Not Superior Total New Model 280 60 340 Old Model 120 40 160 Total 400 100 500 3. Using Chi-square test, determine the incidence of accidents caused by technically skilled and unskilled personnel. Personnel Accidents No accidents Total Skilled 10 90 100 Unskilled 15 65 80 Total 25 155 180 4. 2000 families of a city were selected at random to test belief that families with higher income bought the sedan type of car and families with lower income bought the small car. Given the following results, use the chi-square test to find out if the belief is true. Income Sedan type of car Small car Total High 594 606 1200 Low 262 538 800 Total 856 1144 2000 5. Using chi-square analysis, determine the incidence of job related conflicts caused by trained and untrained personnel. Personnel Conflicts No conflicts Total Trained 20 180 200 Untrained 30 130 160 Total 50 310 360 6. The table below shows cross classification of 500 individuals by income level and preference for ice creams of Type A or Type B. Test whether the two attributes are associated. Draw suitable conclusion at 5% level. (Use table value of 9.4877). Income level Preference Total A Indifference B Low 170 30 80 280 Medium 50 25 60 135 High 20 10 55 85 Total 240 65 195 500 7. The following figures show the distribution of digits in numbers chosen at random from a telephone directory. Digit 0 1 2 3 4 5 6 7 8 9 Frequency 1026 1107 997 966 1075 933 1107 972 964 853 Test at 5% level whether the digits may be taken to occur equally frequently in the directory. (The table value for chi-square test for 9 degrees of freedom at 5% level of significance = 16.919). 8. A political party claims that men and women voters support it equally. In a sample survey, out of 360 men voters, 120 favoured the party, while 170 out of 490 women preferred it. Do the survey results support the claim ? Clearly display the main steps in your anlysis. Men Women Total Support 120 170 290 Do not Support 240 320 560 Total 360 490 850 INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES BANGALORE BUSINESS STATISTICS ANALYSIS OF VARIANCE TEST – F TEST (ONE WAY CLASSIFICATION AND TWO WAY CLASSIFICATION TESTS) 1. The following data represent the number of units of a product produced by 3 different workers using 3 different machines. Workers MACHINES X Y Z A 8 32 20 B 28 36 38 C 6 28 14 2. A new brand of soap was introduced in some targeted shops in 4 cities. The following table gives the sales. City Sales in thousands per month A 24 26 29 25 B 22 21 25 14 C 21 19 23 20 D 26 28 31 28 Use Analysis of Variance to test the significance of difference between the sales of soap in the four cities. 3. A new brand of tooth paste was introduced in some targeted sales points in 4 metros. The following table gives the sales. Metro cities Sales in thousands per month P 14 16 17 16 Q 12 11 13 9 R 11 9 11 11 S 16 18 20 15 Using ANOVA, test the significance of difference between the sales of the tooth paste in the four metro cities. 4. A company has 5 show rooms in 5 cities selling the same model of car. The number of cars sold over 4 months is given below. Using ANOVA, advice the company whether there is a significant difference in the sales among the different show rooms. Months Show Rooms A B C D E September 8 9 7 6 9 October 10 11 8 9 9 November 8 10 9 11 10 December 7 10 8 9 9 INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES BANGALORE BUSINESS STATISTICS INDEX NUMBERS 1. Calculate Fisher’s ideal index and test for the factor reversal test and the time reversal test for the following data: Commodity A B C D E Base year Price 30 32 30 31 32 Base year Quantity 95 115 120 125 125 Current year Price 22 24 25 27 28 Current year Quantity 215 220 219 222 224 2. Calculate Fisher’s ideal index number from the following data and prove that it satisfies the time reversal test and factor reversal test. Commodity A B C D E Base year Price 10 12 8 16 14 Base year Expenditure 55 65 40 80 70 Current year Price 14 16 10 18 12 Current year Expenditure 80 50 60 70 95 3. Given, the following data prices and average monthly quantities of four items purchased by children, compute price index for the year 2000 with the year 1995 as base by Fisher’s Method : ITEMS 1995 2000 Price Quantity Price Quantity Comic Books 8 1 10 2 Toffees 1 30 2 25 Ice Cream 5 5 6 10 Play Articles 10 1 15 1 4. Compute Fisher’s index number for prices, from the following information. Items Price Base year Current year Rice 400 850 Wheat 320 690 Sugar 720 1600 Dhaal 720 2100 Interpret the results. Quantity Base year Current year 100 120 20 60 10 10 10 20 5. Calculate Index number based on Bowley’s method and Fisher’s method for the following data: Commodity 2007 2008 Price Quantity Price Quantity A 2 8 4 6 B 5 10 6 5 C 4 14 5 10 D 2 19 2 13 6. Calculate Fisher’s ideal index from the following data and prove that it satisfies the time reversal test and factor reversal test: Commodities 2007 2008 Price Quantity Price Quantity P 14 70 18 95 Q 16 80 20 50 R 12 55 14 75 S 20 95 22 60 T 18 85 16 85 7. A ready made Garment Industry has worked out its seasonal index as given below : Quarter Seasonal Index January to March 98 April to June 90 July to September 82 October to December 130 If the total sales of garments in the first quarter is 1,00,000, determine how much worth of garments of this type should be kept in stock to meet the demand in each of the remaining quarters. 8. Calculate seasonal indices from the following data. Year QUARTER I II III 2000 68 62 61 2001 65 58 66 2002 68 63 63 IV 63 61 67 9. Construct Laspeyre’s, Paasche’s and Fisher’s ideal index for the following data. Commodity Base Year Current Year Price Quantity Price Quantity A 2 8 4 6 B 5 10 6 5 C 4 14 5 10 D 2 19 2 13 INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES BANGALORE BUSINESS STATISTICS CORRELATION AND REGRESSION ANALYSIS 1. Calculate whether there is any correlation between the salaries and the amount spent on car maintenance. Use Karl Pearson’s method for correlation, determine the probable error and the comment on the significance of correlation. Average salary (in Rs.) 10,000 12,000 15,000 18,000 20,000 Maintenance (in Rs.) 750 900 1,200 1,500 2,000 2. Obtain two regression equations for the following data. X 30 50 20 80 10 20 20 Y 50 80 30 110 20 20 40 40 50 3. For the data below on the related variables X and Y, (i) Draw the scatter diagram (ii) Compute the co-efficient of correlation (iii) Interpret the results. X 110 100 140 120 80 90 Y 70 60 80 60 10 20 4. Calculate the Pearson’s co-efficient of correlation between Advertisement Cost and the Sales as per the data given below : Advertisement (in ‘000 Rs.) 39 65 62 90 82 75 25 98 36 78 Sales (in Rs. Lakhs) 47 53 58 86 62 68 60 91 51 84 5. In a singing competition, 7 candidates were ranked by Judges A and B as follows. Candidate I II III IV V VI VII Rank by A 3 4 5 6 1 2 7 Rank by B 2 1 7 5 3 4 6 6. Four students of a class, the regression equation of marks in Statistics (X) on the marks in Accountancy (Y) was 3 y – 5 x + 180 = 0. The mean marks in Accountancy was 44 and the variance of marks in Statistics was 9/16th of the variance of marks in Accountancy. Find the mean marks of Statistics and the coefficient of correlation between marks in two subjects. 7. The data below show the experience (X years) of machine operators and their performance ratings as given by the number of good parts turned out (Y) per 100 pieces by the operator. Operator A B C D E F G H X 16 12 18 4 3 10 5 12 Y 87 88 89 68 78 80 75 83 8. A term lending institution has the following data relating to 9 units in Textile industry. Fixed Assets as a percentage of 65 66 67 67 68 69 71 72 73 total investment Long term debt as a percentage of 65 66 69 67 70 71 76 66 70 total investment 9. Obtain the regression equations for Operator A Experience in years (X) 17 Production in units daily (Y) 86 the following data : B C D E F G H 13 19 5 4 11 6 14 87 88 67 66 69 74 82 10. The following data gives the experience of lathe operators in a factory and the number of units of production turned out per day. Lathe Operator 1 2 3 4 5 6 7 Experience in years (X) 14 10 16 2 1 8 3 Units of production (Y) 84 80 87 60 54 78 56 Calculate the regression lines and (i) Estimate the probable units of production of a lathe operator with an experience of 15 years (ii) Estimate the years of experience of a lathe operator with a daily production of 96 units. 11. The manager of a company is interested in studying the effect of price (X) on the sales (Y) of a new product. He has selected four similar stores in the chain and measured demand in a two week period at each of the four prices, Rs. 20, Rs. 30, Rs. 40 and Rs. 50. The total demand at each of these prices is given in the following table. Price (Rs.) 20 30 40 50 Demand (Qtty) 138 111 84 47 (a) Find the least square regression line for this problem. (b) Find the co-efficient of determination. 12. The following data gives the work experience of machine operators in a factory and the number of units of production turned out per day : Machine Operator 1 2 3 4 5 6 7 8 9 Work Experience (X) 6 8 7 5 2 1 3 9 10 Work Experience (Y) 50 60 54 47 25 20 41 62 70 (a) Calculate the regression lines and estimate the probable units of production of a machine operator with an experience of 12 years. (b) Estimate the probable years of experience of a machine operator whose daily production is 85 units. 13. From the following data, obtain the regression equations of X and Y and predict X, when Y = 20. X 1 2 3 4 5 6 7 8 9 Y 9 8 10 12 11 13 14 16 15 INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES BANGALORE BUSINESS STATISTICS TREND ANALYSIS 1. Fit a straight line trend by the method of least squares to the data given below and project the probable sales for the next two years. A graph is not necessary. Year 2001 2002 2003 2004 2005 2006 Sales (in ‘000 Rs.) 164 180 186 187 190 192 2. Calculate the trend values by the method of least squares from the following data : Year 2001 2002 2003 2004 2005 Values 75 68 50 67 64 3. Fit a straight line trend by the method of least squares and plot on a graph for the following data about the sales of a trading firm. Year 2001 2002 2003 2004 2005 Sales (in lakhs of rupees) 75 90 91 95 98 4. Calculate the trend values by the method of least squares from the data given below : Year 2001 2002 2003 2004 2005 Value 78 72 54 70 67