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BAMS2923 STATISTICAL ANALYSIS FOR BUSINESS Tutorial 4 (CORRELATION AND REGRESSION) 1. The following shows the number of price quotations issued and the number of sales made by a random sample of 8 salesmen. The figures relate to the same period of time: No. of price quotations 105 213 96 157 114 103 237 185 Number of sales 78 104 63 83 54 59 137 96 (a) Construct a scatter diagram to illustrate these figures. (b) Calculate the product moment correlation coefficient. (c) Without performing any further calculations, interpret the results obtained in parts (a) and (b). 2.(i) An analysis of a number of companies showed the following relationship between expenditure on research & development and profits: Company Research & Development ($millions) Profits ($ millions) 35 68 46 303 15 70 80 290 35 108 61 140 52 90 (a) Calculate the product moment correlation coefficient. (b) Rank the data and calculate the Spearman's rank correlation coefficient. (c) Comment on the difference between the answers obtained in parts (a) and (b). 18 (ii) A bank has recently been analyzing its current account charges on quarterly average balance of its customers. The following data have collected from a random sample of its 7 current accounts: Account 2 3 4 5 6 number Average 83 26 54 104 43 65 38 balance ($) Account 5.42 14.27 7.83 2.62 8.64 4.20 10.01(ii) A bank has recently been analyzing its current account charges on quarterly average balance of its customers. The following data have collected from a random sample of its 7 current accounts: Account 1 2 3 4 5 6 number Average 83 26 54 104 43 65 38 balance ($) Account 5.42 14.27 7.83 2.62 8.64 4.20 10.01 charges($) Calculate the product moment correlation coefficient and coefficient of determination. Interpret your findings. 3. The petrol station A - J, which are situated in areas of similar traffic density are ranked firstly according to quality of service, secondly according to the size of the forecourt and thirdly according to the price of the petrol sold. Rank 1 indicates best service, largest forecourt and lowest price. The results including the average weekly petrol sales are given below: Station Quality of Forecourt Petrol Sales service size price ('000 of gallons) - - I Q T MOOD> 3 8 2 47 4.5 9 20 OOONOA 10 8 23 2 1 36 4 36 5 31 33 6 10 28 4.5 42 7 6 24 (a) Carry out calculations to decide whether price of petrol or quality of service is likely to be more important factor in determining the volume of petrol sales, or whether neither appears to important. 19 (b) Is there any evidence to suggest that those stations with large forecourts give better service? 4. The total monthly cost and the monthly output of electronic component in a factory for ten months are tabulated below: Output 21 3 5 24 19 15 11 9 14 9 ( '000s ) Total 65 30 31 71 54 52 40 33 45 38 cost6. The number of customer complaints received by a major retailer was worrying the management. 8 stores were selected. The area of selling space and the number of complaints against the particular store are given below: Store Selling area Number of (00m-) complaints A 12 42 B 95 124 C 8 12 D 45 102 E 34 53 F 72 145 G 60 88 H 19 26 (a) Draw a scatter diagram to represent the above data. (b ) Calculate the least squares regression line of number of complaints on selling area and draw the line on to the scatter diagram. Interpret the values of the constants 'a' and 'b' of the regression line Y =a + bX. (c) Hence using the least squares line, estimate the number of complaints received when the selling area of the store is: (i) 5,000 square meters; (ii) 10,000 square meters. (d) Which estimate obtained in part (c) would be more reliable? Give reasons. 21 7. The time taken to produce batches of an item is found to be related to the number of items in the batch. Records show the production times for 6 batches to be as follows: Batch sizes Production time(hours) 50 4.0 90 5.8 150 6.8 220 7.6 280 7.6 350 8.6 Two alternatives are suggested for estimating the production times, T, from the batch size, R. (i) T = 3.5 + R 50 (ii) T = 2 + R V8 (a) For each of the two relationships find the value of T corresponding to R = 32, 72, 128, 200, 288 and 338. (b) Use the values obtained in part (a) to construct a graph illustrating the two alternative relationships. Use the same graph, plot the scatter diagram. From observing your graph, state which best represents the data points given in the table. (c) Find the best fitting curve of the form T = A + B JR.4. The total monthly cost and the monthly output of electronic component in a factory for ten months are tabulated below: Output 21 3 5 24 19 15 11 9 14 9 ( '000s ) Total 65 30 31 71 54 52 40 33 45 38 cost ( $'000s ) (a) Calculate the product moment correlation coefficient and coefficient of determination and interpret your answers. (b) Find the least squares regression line of total cost on output. (c) State the fixed cost of the factory and the variable cost per unit of production. (d) Predict the total cost if the production level were: (i) 8,500 units (ii) 25,000 units Hence, comment on the accuracy of your estimates 5. The following data taken from a manufacturing company's budget relates to volume of sales and the corresponding expenses. Volume of sales (000 units) 5 6 7 8 9 10 Total expenses ($'000) 74 77 82 86 92 95 (a) Obtain the product moment correlation coefficient. (b) Comment, in the light of your results, on the relationship between volume of sales and the total expenses. 20 (c) Calculate the coefficient of determination and interpret the result. (d) Draw a scatter diagram and comment on the appropriateness of fitting a straight line relationship. By the method of least squares draw the regression line of best fit. (e) What will be the expected total expenses when the volume of sales is 7500 units? (f) If the selling price per unit is $11, at what volume of sales will the total income from sales equal the total expenses? 6. The number of customer complaints received by a major retailer was worrying the management. 8 stores were selected. The area of selling space and the number of.OOOO CELCOM... 4G 3:04 AM @ 10 37% X Chapter 2 C&R.pdf E 34 53 F 72 145 G 60 88 H 19 26 (a) Draw a scatter diagram to represent the above data. (b ) Calculate the least squares regression line of number of complaints on selling area and draw the line on to the scatter diagram. Interpret the values of the constants 'a' and 'b' of the regression line Y =a + bX. (c) Hence using the least squares line, estimate the number of complaints received when the selling area of the store is: (i) 5,000 square meters; (ii) 10,000 square meters. (d) Which estimate obtained in part (c) would be more reliable? Give reasons. 21 7. The time taken to produce batches of an item is found to be related to the number of items in the batch. Records show the production times for 6 batches to be as follows: Batch sizes Production time(hours) 50 4.0 90 5.8 150 6.8 220 7.6 280 7.6 350 8.6 Two alternatives are suggested for estimating the production times, T, from the batch size, R. (i) T = 3.5 + R 50 (ii) T = 2 + R V8 (a) For each of the two relationships find the value of T corresponding to R = 32, 72, 128, 200, 288 and 338. (b) Use the values obtained in part (a) to construct a graph illustrating the two alternative relationships. Use the same graph, plot the scatter diagram. From observing your graph, state which best represents the data points given in the table. (c) Find the best fitting curve of the form T = A + B JR