Mathematics And Stastistics For Managers

1. JOSEPH’S COLLEGE OF COMMERCE (AUTONOMOUS)

END SEMESTER EXAMINATION – OCTOBER 2013

M.I.B. – I SEMESTER

Mathematics and Statistics for Managers

 

Duration: 3 hours                                                                                        Max. Marks: 100

SECTION- A

 

1. Answer any SEVEN                             (7 x 5  = 35)

 

1. Assume that the life expectancy of women in a country follows linear relationship with respect to time. Given that in 1980 the life expectancy was 65 and in 2005 it was 75 years. Find the linear relationship between life expectancy and time. Estimate life expectancy of women in the year 2015.
2. A student has 3 places where he can take his breakfast. The college canteen charges Rs.20 for an egg roll, Rs.10 for a cake and Rs.5 for tea. A fast food place charges Rs.15 for an egg roll, Rs.15 for cake and Rs.6 for tea. A nearby restaurant charges Rs.40 for an egg roll, Rs25 for cake and Rs.10 for tea. A student wishes to buy 2 egg rolls one cake and a tea. Represent the following information in the form of a matrix. Through matrix operations find the cost of breakfast at different places.
3. “Each type of average has its own particular field of usefulness” Briefly discuss the characteristic features of different averages.
4. In a single throw of two dice, what is the probability of getting a total of 9?
5. A card is drawn from a regular pack of cards and a gambler bets that it is a spade or an ace. What are the odds in favour of his winning this bet.
6. Explain any ten properties of a normal distribution curve.
7. What is the difference between decision making under uncertainity and risk?

Explain the    various decision criterion under risk.

7. Find the coefficient of correlation between the values of x and y using concurrent deviation method.

x	78	89	97	69	59	79	68	61
y	125	137	156	112	107	136	123	108

 

8. Fit a straight line trend by the method of least squares for the following data about sales of a trading firm.

 

Year	2005	2006	2007	2008	2009
Sales ’00,000 Rs.	75	90	91	95	98

 

9. The following table shows the gain in weight by 25 children in a year. Find the mean, median and the mode of weight gained.

 

Gain in weight (kg)	1.5	2	2.4	3	3.2	3.4
No. of children	4	5	8	5	2	1

 

10. For the following data calculate Range, Inter Quartile Range, Quartile Deviation and its coefficient 10, 15, 18, 20, 20, 22, 23, 25, 27, 30

 

SECTION –B

 

II ) Answer any THREE questions. Each carries 15 Marks          .                                        (3×15=45)

 

11. a) Using matrices solve the following system of equations.

3x+4y+5z=18,   2x-y+8z=13,   5x-2y+7z=20

1. b) The total cost of production of a firm is given by the function C= 12x² – 8x + 4

Find i)  Total cost for an output of 15 units

1. ii)  The average cost for an output of 12 units.

iii) The marginal cost for an output of 16 units.

400. iv) The revenue function when the price is Rs.400.
401. v) Profit maximizing output with the above revenue function.
402. vi) Maximum profit

vii) Profit for an output of 20 units                                                             (8+7)

 

12. A purchasing agent obtains samples of 60 watt bulbs from two companies. He had the samples tested in his own laboratory for length of life with the following results:
13. a) Which company’s bulbs do you think are better in terms of average life?
14. b) If prices of both companies are same, which company’s bulbs would you buy?

 

Length of Life (in hours)	Samples from
	Company A	Company B
1700-1900	10	3
1900-2100	16	40
2100-2300	20	12
2300-2500	8	3
2500-2700	6	2

 

 

 

 

 

 

 

 

13. a) Given the prices and the average monthly quantities purchased by children calculate Index numbers based on Laspeyer’s, Paasche’s, Dorbish and Bowley’s and Fisher’s Method.

Item	2005		2010
	Price	Quantity	Price	Quantity
Comic Books	8	1	10	2
Toffees	1	30	2	25
Icecreams	5	5	6	10
Play-articles	10	1	15	1

 

 

 

 

 

 

 

 

1. b) Given the payoff table find the regret table and calculate EMV, EOL, EVPI, EPPI and identify the best decision.

 

 

Retailer’s demand	Probability	Conditional Payoffs (‘000 Rs) Stock per week
		1000 Pairs	3000 Pairs	5000 Pairs
1000 pairs	0.6	50	-10	-70
3000 pairs	0.3	50	150	90
5000 pairs	0.1	50	150	250

(8+7)

 

14. a) Let A and B be events with P(A)=3/8, P(B)=5/8, and P(A and B)=3/4 Find P(A or B), P(A/B), P(B/A), P(A^c), P(B^c), P( A^c and B^c).
15. b) In a railway reservation office, two clerks are engaged in checking reservation forms. On an average, the first clerk checks 55% of the forms, while the second does the remaining. The first clerk has an error rate 0 .03 and second has an error rate of 0.02. A reservation form is selected at random from the total number of forms checked during a day, and is found to have an error. Using Baye’s theorem Find the probability it was checked i) by the first  ii) by the second clerk.
16. c) An organization has two packaging machines: old and new. The new machine is more efficient if the materials are of good quality, on the other hand the old machine performs better if the materials are of poor quality. In the previous batches 80% materials have been of good quality and 20% of poor quality. The profit details are given below. Using decision tree decide which machine should be used under the condition that the quality of material is not known at this stage.

 

 

Quality Of Materials	Profit
	New Machine	Old Machine
Good	2400	2000
Poor	800	1600

 

15. a) Fit a poisson distribution for the following data and calculate the theoretical frequencies:

 

X	0	1	2	3	4
f	122	60	15	2	1

 

5. b) The marks of 1000 students in an examination follows normal distribution with mean 70 and standard deviation 5. Find the number of students whose marks will be i) less than 65 ii) more than 75  iii) between 65 and 75.
6. c) The number of  telephone lines busy at an instant of time is a binomial variate with probability 0.1 that a line is busy. If 10 lines are chosen at random, what is the probability that  i) no line is busy            ii) atmost 2 lines are busy.                                                                                                                                                         (5 + 5 + 5)

 

SECTION –C

 

III ) Compulsory Question                                                                                        (20marks)

 

16.a) A psychologist wanted to compare two methods A and B of teaching. He selected a random sample of 22 students. He grouped them into 11 pairs so that students in a pair have approximately equal scores on an intelligence test. In each pair one student was taught by  method A and the other by method B and examined after the course. The marks obtained by them are tabulated below. Find the spearman’s rank correlation coefficient. Interpret the result.

 

A	24	29	19	14	30	19	27	30	20	28	11
B	37	35	16	26	23	27	19	20	16	11	21

 

1. b)    The  general sales manager of Kiran- Enterprises, an enterprise dealing in the sale of readymade garments is trying with the idea of increasing his sales to Rs.80,000. On checking the records of sales during the last 10 years, it was found that the annual sale proceeds and advertisement expenditure were highly correlated to the extent of 0.8. The average and variance of sales and advertisement expenditure are given in the table.

Find the two regression equations. How much expenditure on advertisement would you suggest the general sales manager of the enterprise to incur to meet his target sales.

 

	Sales	Advertisement Expenditure
Average	45,000	30,000
Variance	1600	625
r=+0.8

(8+12)