BUSINESS MATHEMATICS AND STATISTICS

1. BUSINESS MATHEMATICS AND STATISTICS • SOLVED QUESTION PAPER OF SESSION : 2010-2011 SUBMITTED TO: SUBMITTED BY : Ms. BikramjeetKaurBavanpreetSingh Roll No. :171 ManveenKaur Roll no. : 172

2. SECTION-A

3. QUES.1(a):Difference between Determinant and Matrix Determinant Matrix • A determinant is always a square • Determinant has a definite value • Two vertical lines are used to denote determinants • A matrix may be rectangular or a square • A matrix has no definite value • We use square brackets to denote matrix

4. (b):Define Regression coefficients and Regression lines 1.Regression coefficients Just as there are two regression equations ,similarly there are two regression coefficients .regression coefficient measures the average change in the value of one variable for a unit change in value of another variable.

5. Regression coefficient of X on Y :this coefficient shows that with a unit change in the value of X ,what will be the average change in the value of Y. byx=r. deviation of x deviation of y (2)Regression coefficient of X on Y :this coefficient shows that with a unit change in the value of X, what will be the average change in the value of X.

6. Bxy=r.deviation of x deviation of y 2.Regression lines The regression line shows the average relationship between two variables . This is also known as the line of best fit on the basis of regression line , we can predict the value of a dependent variable . (1):Regression line of X on Y:the regression line of X and Y gives the best estimate for the value of X for any given value of Y .

7. (2):Regression line of Y on X:the regression line of Y on X gives the best estimate for the value of Y for any given value of X.

8. (c).Distinguish between imple random sampling and stratified random sampling. Simple random sampling Stratified random sampling It is that method in which each item of universe has an equal chance of being selected in the sample. Which item will put in sample and which not , such decision is not made by an investigator on his will but selection of units is left on chance. This method is used when units of universe are hetrogenous rather than homogenous . under this method , first of all units of the population are divided into different strata in accordance with their characterstics . then the samples are selected on random basis.

9. (d).Show that 1 a bc 1 b ca =(b-c)(c-a)(a-b) 1 c ab Sol. Operating r₂-r₂-r₁-r₃-r₃-r₁ 1 a bc 1 a bc 1 b ca = 0 b-c ca-bc 1 c ab 0c-a ab-bc Expanding along c₁, we get =(a-b)(c-a)(b-c)=(a-b)(b-c)(c-a)

10. (e).Simple interest on a certain sum of money for 1.5 years is ₹150 and compound interest is ₹153.33.find rate Simple interest for 1.5 years =₹150 Simple interest for 1 year = 150X²/₃ =100 Now , C.I. For 1st year =S.I. Of the 1st year=₹100 153.33-100=₹ 53.33 =C.I. For remaining ⅟₂years-S.I. For ⅟₂years =₹53.33-₹50=₹3.33 Rate =3.33 X 100 =6.66% 100X ⅟₂

11. (f).The arithmatic mean ,mode and the median of a group of 75 observations was calculated to be 27,34 and 29.ut was later discovered that 1 observation was wrongly read as 43 instead of 53.examine the error ƩX=NX N=75,X=27 ƩX=75X27=2025 CORRECTED ƩX=2025-43+53=2035 C0RRECTED X=2035/75=27.13

12. SECTION-B

13. QUES : Calculate mean,median and mode from the following data: X F • 0.5 • 1.0 • 3.0 • 5-7 • 7-10 • 10-15 • 15-20 • 20-25 • 2 • 3 • 2 • 4 • 4 • 12 • 6 • 2

14. Make equal intervals

15. X=ƩfX =377.5 =10.79 N 35 Median=17.5th item Median class=10-15 M=10+17.5-15X5=11.041 12 Modal class:10-15 Z=l₁ + f₁+f₀ X i 2f₁-f₀-f₂

16. X=10.79,M=11.04,Z=12 Ques : Calculate chain base index numbers chained to 2003 from the average price of following three commodities:

17. Chain base index numbers chained to 2003

18. Solution.

19. Ques : Below are given the figures of production of sugar factory: (a)Fit a straight line by the least squares method and tabulate the trend values. (b)Eliminate the trend . What components of the series are thus left over? (c)What is monthly increase in the production of sugar?

20. The equation of straight line curve is Y=a+bX a=ƩY=89 N b=ƩXY=2 ƩX²

21. Y = 89 = 2X origin:2004, X unit = 1 year • Calculation of trend values For 2001, X = -3 2002, X = -2 2003, X = -1 2004, X = 0 2005, X = +1 2006, X = +2 2007, X = +3

22. (b). Elimination of trend

23. (c).Monthly increase = b = 2 = 0.166 12 12 = 166.66 tonnes.

24. SECTION - C

25. Ques : A computer while calculating the correlation coefficient between two variables X and Y from 25 pairs of observation obtained from following results : N = 25 ƩX = 125 ƩX² = 650 Ʃy = 100 Ʃy² = 460 Ʃxy = 508 Incorrect correct X Y X Y 6 14 8 12 8 6 6 8

26. Correct ƩX = 125 - 6 – 8 + 6 = 125 Correct ƩY = 100 - 14 - 6 + 12 + 8= 100 Correct ƩX² = 436 Correct ƩY² = 520 Correct ƩXY = 520 r = 0.667

27. Ques: the following data relates to the scores obtained by a salesman of a company in an intelligence test and their weekly sales in thousand rupees :

28. (a). Obtain the regression equation of sales on intelligence test scores of the salesman. (b). If he intelligence test scores of a salesman is 65, what would be his expected weekly sales?

29. Solution :

30. X = 60 , Y = 50 Byx = 0.75 Regression equation of Y and X Y= 50 + 0.75X (b). For X = 65 , Y₆₅ = 50 + 0.75(65) =53.75

31. - Ques : 12 dice were thrown 4096 times . Each 4,5 or 6 spot appearing was considered to be a success . while 1,2 or 3 spot was a failure calculate the theoretical frequencies for 0,1,2,....,12 successes. Solution : N = 12 , P = probability of success N = =4096 The probability of 0,1,2,....,12 will be given by P(X)ᶰCᵪq = ᶰ-ᵡpᵡ for n = 1,2,3,...,12.

32. The expected frequencies are obtained by increasing the probability by N as follows :

33. THANK YOU