FORMULAE n! 1. Combination "C, =- (n-r)!r! n! 2. Permutations " p, = (n-r)! 3. Mean of the binomial distribution = np 4. Standard deviation = Vnpq 5 . Variance of the binomial distribution = np(1 - p) 6. Standard error of population proportion S, = Pq 7. Spearman's rank correlation coefficient r = 1- 6Ed2 n(n' - 1) Product moment coefficient of correlation 8. 9. Cost slope crash cost - normal cost normal time - crash time n 10. Harmonic mean (ungrouped data) hm = 11. Sample mean X = Ex n 12. Harmonic mean (grouped data) hm = n 13. Quartile coefficient of dispersion = 23 -21 14. Mean x = A+ E fd Ef or Mean x= MM N -Cfb 15. Median = Lb+ 2 C fm 16. Mode = Im+ d, + d 2 CFORMULAE 31. Conditional probability P(A B)= P(AnB) P(B) 32. Independence of A, B P(A )= P(A)orP(An B) = P(A) x P(B) 33. Continuous compounding A = P(1+r)" + b(1+r)" -b r 34. Quotient rule of differentiation f vu' -uv' 12 - ; where f = 35. Paasche'sModel : E(P, x 91), E(q, xpo) x 100 36. Poisson ModelP( X = x) = e-1 2" x !Question 6 (a) (b) Explain two importances of time series. (4 marks) Mutara town council is planning to undertake a project involving iive activities as indicated in the following table, all costs being in millions of shillings. Duration (days) Cost I Activity Preceding activity Normal j Crash Normal j Crash I Required: (i) Draw the project network diagram. (4 marks) (ii) Compute the project duration and total cost. (4 marks) If an activity on the critical path with the lowest cost slope is reduced by two days; Compute the: (ill) new project duration. (4marks) (iv) total cost. (4marks) (Total 20 marks) Question 3 (a) (b) (0) Graphically identify the three components of a quality control chart for the mean. (2 marks) The following table shows the moisture content in bread which has stayed overnight in a refrigerator for a period of five days. 1 2 i 3 i 4 i 5 | 33.8 34.0 34.1 33.9l34.2| i Dav l Moisture (millitres) Required: Determine the percentage of mean moisture centred in the range E123. (6 marks) A study was conducted among 100 professors from 3 different departments at a University College for their promotions. It was based on 3 categories of teaching, research and other university activities. The observed values of the number of professors promoted in each category are given in the following contingency table. Field of teaching I Basis of promotion Research I Others 25 | Total 100 Required: (i) construct a two-way contingency table With observed and expected values in each cell. (6 marks) (ii) Test. at 5% level (if significance. the hypothesis:Hu 'there is i'IIIJI relationship between the basis of promotion and the field of teaching'. (6 marks) (Total 20 marks Question 4 (a) (b) (0) Identify any two merits and any two demerits of the chain index method of computing index numbers. (4 marks) A family's monthly shopping list in 2015 and 2016 included the tollowing items: 2015 l 2010 I Item l Quantity Unit price (Shs) | Beans 10 kg l 2,500 l 3.000 I Rice 15 k 3 500 4.200 soap 4 bars 4,000 4.500 Required: (i) Compute the cost of living index using 2015 as a base year. (Smarks) (ii) Comment on the result obtained in (b) (i) above. (1 mark) A hardware store cleared their electricity bill of Shs 90,000 using Shs 5.000 and Shs 1,000 notes. In all a total of 50 notes were used. Required: Using the graphical method. determine the notes of each denomination that were used. (9 marks) (Total 20 marks) FORMULAE 17. Variance Var(x)= Efx 18. Standard deviation S = -X 2 If ( x-F) 2 VE Es 19. Sample standard deviation S = 1 n-1 20. Least squares regression equation of y on x is given by; y = a+ bx Where; b=- n [ xy - Ex Ey and a = [y b Ex n n 21. Least squares regression equation of x on y is given by; x = c + dy Where c= Ex dEy n[xy - Ex Ey and d = n n ny2 -( 22. Standardizing normal. x - H Z : 23. Confidence interval for sample mean =x1/ Jm 24. x2 = (0-E) E 25. Confidence interval of proportion = pz. \\ n pq 26. Pearson coefficient of skewness Sk = (x - mode) 3 x- median of SK = S 27. Expectation = ExP(X =x) 28. Laspeyres' price index = 2(Pixq.) [ (9. x Po) -x 100 29. Weighted aggregate price index = LWn x 100 : Two 30. Additive law of probability; P(AUB) = P(A) + P(B)-P(AnB)Question 5 (a) Distinguish between dependent and independent variables in regression analysis. (2 marks) (b ) The weekly bonus, in thousands of shillings, for 7 scientists of different ages (in years) in an organisation is shown in the following table. Age ( x ) 20 22 28 33 36 45 55 Bonus (y ) 69.8 78.5 108.8 134.0 144.0 182.5 198.6 Required: (i) Calculate the linear regression equation yon x using the least squares method. (11 marks) (ii) Explain what would happen to the gradient (slope) in (b) (i) above if all scientists had an increase of Shs 1,000 per week. (2 marks) (c) Bintu Company Lid manufactures pavers and concrete blocks. The following is a minimisation linear programming problem for the company and its final tableau: Minimise 3x +4y subject to: x-y 20 3x - 4y 20 x20, y 20 Where x, y represent pavers and concrete blocks respectively. Final simplex tableau of the minimisation problem. y u V r solution 1 6 0 0 11 0 5 3 1 0 16 0 2 4 0 -40 Required: Determine the: (i) dual to the minimisation linear programming problem. (3 marks) (ii) minimum value to the original problem. (2 marks) (Total 20 marks)ADD CUMULATIVE NORMAL DISTRIBUTION P(z) 3 5 6 8 1 0 1 2 0040 0080 0120 0160 0199 0239 0279 0319 0359 0.0 0.0000 0636 0675 0714 0753 0.1 0.0398 0438 0478 0517 0557 0596 0910 0948 0987 1026 1064 1103 114 0.2 0.0793 0832 0871 1293 1331 1368 1406 1443 1480 1517 0.3 0.1179 1217 1255 184 1879 0.4 0.1554. 1591 1628 1664 1700 1736 1772 1808 2190 2224 7 0.5 0.1915 1950 1985 2019 2054 2088 2123 2157 2549 0.2257 2291 2324 2357 2389 2422 2454 2486 2517 0.6 0.7 0.2580 2611 2647 2673 2704 2734 2764 2794 2823 2852 NWWWwwww 0.8 0.2881 2910 2939 2967 2995 3023 3051 3078 3106 3133 3289 0.9 0.3159 3186 3212 3238 3264 3315 3340 3365 3389 AUT 3508 1.0 0.3413 3438 3461 3485 3531 3554 3577 3599 3621 1.1 0.3643 3665 3686 3708 3810 3830 HNNNNNNN 3729 3749 3770 3790 1.2 0.3849 3869 3888 3907 3925 3944 3962 3980 3997 4015 4115 4131 4147 4162 417 wwa 1.3 0.4032 4049 4066 4082 4099 4236 4251 . 4265 4279 4292 4306 4319 1.4 0.4192 4207 4222 4406 4418 4429 1.5 0.4332 4345 4357 1370 4382 4394 4515 4525 4535 454 -NNN AAVIAN 1.6 0.4452 4463 4474 4484 4495 4505 NWWAUT NNWWA 4582 4591 4599 4608 4616 4625 463 1.7 0.4554 4564 4573 4671 4678 4686 4693 4699 4706 1.8 0.4641 4649 4656 4664 4756 4761 4767 NNWAA UIGOODE HENNN WAAVIO 1.9 0.4713 4719 4744 4750 $726 4737 4738 4817 4793 4798 4803 4808 481 4788 2.0 0.4772 4778 4783 4834 4838 4842 4846 4850 485 4957 HNNWW HNNNW Doooo 2.1 0.4821 4826 4830 4881 4884 4887 4890 2.2 0.4861 4864 4868 4871 4875 4878 OOH 4901 4904 1906 4909 4911 4913 916 2.3 0.4893 4896 4898 4925 4927 1929 4932 4936 24 0.4918 4922 4931 1934 4920 4946 4948 4949 4951 4952 4941 4943 4945 2.5 0.4938 4940 4956 4957 4959 4960 4961 4962 4963 4964 2.6 0.4953 4955 4968 4969 1970 4971 4972 4973 974 2.7 0.4965 4966 4967 4978 4979 4979 4980 4981 2.8 0.4974 4975 4976 4977 4977 4983 4984 4984 4985 4985 4986 4986 2.9 0.4981 4982 4982 4999 4999 5000 3.0 0.4987 4990 4993 4995 4997 4998 4998 The table gives P(z) = [ $(2)dz If the random variable Z is distributed as the standard normal distribution N(0, 1) then: $(Z) 1. P(o Zp) = 0 = /2-P 0.3 3. P(Z'> | Zpl ) = 1-2P= 20 0.2 01Question 1 (a) The Uganda Consumer Advocacy Forum suspected that the packaging of corntlakes cereals contains less than the advertised weight of 426 grams per packet. A sample of 40 packets was taken and the following weights. in grams, were obtained: 434.5 423.2 431.7 411.8 428.8 437.4 414.5 437.4 437.4 448.7 417.5 437.4 443.0 Required: (i) Starting with class intervals 395$x