VIKAS PRE UNIVERSITY COLLEGE, MANGALURU ANSWER KEY STATISTICS SECTION A SECTION B

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1 VIKAS PRE UNIVERSITY COLLEGE, MANGALURU ANSWER KEY STATISTICS SECTION A 1. Fertility refers to the births occurring to women of child bearing age 2. Marshall Edgeworth s index number does not satisfy Factor reversal test 3. Retail prices 4. Umbrellas are sold more in rainy season 5. Poisson distribution 6. Zero 7. If a single value is proposed as an estimate of the unknown parameter then it is point estimation 8. The probability of rejecting H 0, when it is not true is called power of a test. 9. If expected frequencies are less than 5, the pooling is done till it becomes more than or equal to A defect is a quality characteristic which does not confirm to specifications. 11. The item that has become inefficient with passage of time. 12. Inventory is a physical stock of goods kept for future use. SECTION B 13. Uses a. Life tables are used by life insurance companies to determine the rates of premium for policies of persons of different ages. b. It is used for the measurement of growth of population in the computation of net reproduction rate. 14. Fisher s index number is ideal because a. It is based on geometric mean, which is considered as an appropriate average for averaging ratios. b. It takes into account both current year as well as base year quantities. c. It satisfies both TRT and FRT d. It is free of bias = Prosperity, decline, depreciation (recession), recovery 17. Assumptions a. There are no sudden jumps in the values of dependent variable. b. There should be uniformity in the rise or fall of the values of the dependent variable. c. There will be no consecutive missing values in the series. 18. p(x) = (0.4) x (0.6) 1-x, x = 0, Chi-square distribution with two degrees of freedom and Mean = n = The error that occurs by rejecting null hypothesis when it is actually true is called type I error or First kind error.

2 The error that occurs by accepting null hypothesis when it is actually not true is called type II error or Second kind error 21. χ 8 σ 22. Controlling the quality of the finished products or manufactured products is called product control. Controlling the quality of the product during the manufacturing process itself is called process control ( ) * + ) # "! " # = 300 units $% STDR=, SECTION C Age Population Deaths Standard population(p) A=ASDR PA and above STDR = " # 26. Steps a. Defining the purpose of the index number b. Selection of base period c. Selection of commodities or items d. Obtaining price quotations e. Choice of an average f. Selection of weights g. Selection of suitable formula 27. Table Group Price Weights P WP Food Clothing Rent Fuel Others / 12.3/

3 28. Moving averages Year Sales Five yearly moving sum Five yearly moving average Let X denote year and Y denotes Production (tons) Here 4 values are known in y. Hence 4 y 0 =0 y 4-4y 3 +6y 2-4y 1 +y 0 =0 On substation, and simplification y 2 =Rs.1771 Another equation is obtained by raising suffix by 1 in the equation of 4 y 0 =0 y 5-4y 4 +6y 3-4y 2 +y 1 =0 On substation and simplification y 5 =Rs Given average mistakes λ=2 Let X denotes number of mistakes. Hence X~P(λ=2) Pmf is given by, 78 9:λ λ ; 9: ;,80,1,2,. <! P(X=2) = 9: ; 9: = <!! P[at most one mistake] = P[X 1]=p(0)+p(1)= = Let X denotes the number of red balls drawn of 5 balls. X~H(a=8, b=4, n=5) Pmf is given by, 78= a C b x Cn x a b Cn <! +, 80,1,.min C, C 8 4 x C 5 x 12 C5 P[X=3] = ## DE , 80,1, Given n 1 =64, xh = 50, s 1 =8 n 2 =48, xh = 54, s 2 =12 H 0 : µ 1 = µ 2 H 1 : µ 1 < µ 2 Test statistic is given by, Z JKL MH MH N O O " ) W XY YW ~N0,1 under H Z2 At 5% level of significance, the critical value k=1.64.

4 We reject H 0 if Z cal < -k. Otherwise we accept H 0. On comparison, we reject H 0 and we accept H 1. Hence the mean weight of boys is less than mean weight of the girls. 33. Given n=10, =39000, µ 0=40000, 0 s =1200 H 0 : µ = H 1 : µ Test statistic is given by, At 5% level of significance, for n-1 =9 degrees of freedom, the critical value (k) is 3.25 We reject H 0 if > k or. Otherwise we accept H 0. On comparison, we accept H 0. Hence the average life of tyre is 40000Km 34. From tables, A 2 = The control limits for chart is Central line, CL= = 50 Lower control limit, LCLL = A 2 = (5) = Upper control limit, UCL = A 2 = (5) = x+3y=36 x+2y=20 x=0, y=12 and y=0, x=18 x=0, y=10 and y=0, x=20 X Axis 1 cm = 5 units Y axis 1 cm = 2 units Corner points Z= 3x + 4y 0(0, 0) 0 A(0, 10) 1700 B(12, 4) 1880 C(18, 0) 1800 Z is maximum at B(12, 4) Hence x=12, y = 4 and maximum value of z is 1880

5 36. B 1 dominates B 2 and B 3. Hence B 2 and B 3 are deleted, B 1 A 1 6 A 2 7 A 3 5 A 4 3 A 2 dominates A 1, A 3 and A 4. Hence A 1, A 3 and A 4 are deleted B 1 A 2 7 Hence saddle point exists at [2, 1] Strategy of player A Strategy of player B is B 1 Value of the game is 7 SECTION D 37. Age Female population Female births Survival rates WSFR WSFR x S GRR = i Σ WSFR = =1100 Number of female born per woman = 1100/1000 = 1.1 Population tend to increase NRR = i Σ WSFR S = = 973 Number of female born per woman = 973/100 = Population tend to decrease 38. Table Commodities p 0 q 0 p 1 q 1 p 0 q 0 p 0 q 1 p 1 q 0 p 1 q 1 A B C D Laspeyre s price index number [ Paschee s price index number =110 Dorbish Bowley s price index number - \] ^ _ `100 =112

6 39. Table Year Production x x 2 xy Trend values Total Straight line trend equation is given by, Y=a+bx The normal equations are, Σy = n a + b Σx (1) Σxy=aΣx + b Σx (2) On substitution, From equation (1), a=90 From equation (2) b=3 First order trend line is given by,y = x Estimate for the year 2017 is given by substituting x=4 Y=102 Trend values are obtained by substituting values of x in the trend line. 40. a. Since 5 unbiased coins are tossed, p =½ and q=½. Expected frequencies = N p(x), Where p(x) = n C x p x q 1-x, x = 0, 1, 2, 3, 4, 5 No. of Heads Expected frequency Total 128 b. H 0 : accidents occur uniformly H 1 : accidents do not occur uniformly. Under H 0, the theoretical frequencies (E i ) are 70/7=10 a b Day Oi Ei b Sun Mon Tue Wed Thu Fri Sat Total Test statistic is given by, c def a b ~c!d ghi2 j b c def 6.8

7 Here n is number of expected frequencies calculated which are greater than 5. Hence at 5% level of significance critical value (right tail) for (7-1)=6 degrees of freedom is k 2 = We reject H 0 if c def kl otherwise we accept H 0. On comparison we accept H 0. Hence accidents occur uniformly. SECTION E 41. a. P(X 105)=P[Z ( )/5]=P(Z 1) = b. P(90mnm110=P(-2 Z 2) = = Given P 0 = 0.5, n = 400, x=220. p=x/n = 220/400 = 0.55 H 0 : P = 0.5 H 1 : P 0.5 Test statistic is given by, Z JKL op ~N0,1 under H ) q r.. ).s.s Y 2 At 5% level of significance, the critical value is k = We reject H 0 if Z cal > k or Z cal < -k. Otherwise we accept H 0. On comparison, we reject H 0 we accept H 1. Hence both coffee and tea drinkers are not equally popular. 43. H 0 : smoking and literacy are not associated. H 1 : smoking and literacy are associated.!etud Test statistic is given by, c def ~c eudtedut ghi2 j ## Hence at 1% level of significance critical value (right tail) for 1 degree of freedom is k 2 = We reject H 0 if c def kl otherwise we accept H 0. On comparison we accept H 0. Hence smoking and literacy are not associated 44. X 11 = 400, X 12 =100, X 22 = 100, X 23 = 100, X 33 = 100 Total cost is Rs Solution is non-degenerate. 2