Page 1 of 17

SOLVED EXAMPLES

UNIT-III AVERAGES- MEASURES OF CENTREL TENDENCY a. Calculate the median and mode for the sales from the data given below.

Sales(in Lacs): <10 10-15 15-20 20-25 25-30 30& above

No of Salesmen: 3 12 16 25 10 4

Working Table:

Median has 50% = 2

N observations on l.h.s

And

( )2

Nc h

Md lf

Where . 2

N

70

2= 35;

Hence class of Median is 20-25 & l=20,f=25,c=31, h=5

We put the values to get, (35 31)5

2025

Md

=20.8

Now, Mode= 1

1 2

( )

2

fm f hl

fm f f

Where, fm= 25,l = 20, f1 = 16, f2 =10, h =5

We put the values to get, (25 16)5

Mode= 202 25 16 10x

= 22.36

b. Calculation of Median and Quartiles Q1, Q2 &Q3

Daily wages (Rs.):<100 100-125 125-150 150-175 175-200 200& above

No of workers: 3 12 16 25 10 4

Working Table

1Q has 25% = 4

N observations on l.h.s

And, 1

( )4

Nc h

Q lf

Where 4

N=

125

431.25

We put the values to get,

1(31.25 12)25

10040

Q

=112.03

Similarly, 3Q has 75%=3

4

Nobservations on l.h.s.

3

3( )

4

Nc h

Q lf

where, 3

4

N=93.75, l=125,c=52,f=55& h=25

3

(93.75 52)25125

55Q

=143.97

c. Calculate the Quartile Deviation (QD) and it’s coefficient

Data:

Reduction in weight: 0-20 20-40 40-60 60-80 80-100

No of patients 12 23 35 20 10

Sales

f

c.f.(<)

<10 3 3

10-15 12 15

15-20 16 31

20-25 25 56

25-30 10 66

30& above 4 70=N

Total 70

Wages(Rs) ` f c.f.(<)

<100 12 12

100-125 40 52

125-150 55 107

150-175 10 117

175-200 5 122

200& above 3 125 =N

Total 125

Page 2 of 17

Calculation Table:

1Q has 25% = 4

N obs on l.h.s.

And, 1

( )4

Nc h

Q lf

Where4

N=

100

4 = 25, l=20, c=12,f=23,h=20

We put the values to get, 1(25 12)20

2023

Q

=31.30

IIIy, 3Q has 75%=3

4

Nobs on l.h.s. And, 3

3( )

4

Nc h

Q lf

Where, 3

4

N=75, l=60,c=70,f=20& h=20 3

(75 70)2060

20Q

=65

Now Q.D= 3 1

2

Q Q=

65 31.30

2

16.85& C.Q.D. =

3 1

3 1

Q Q

Q Q

65 31.30

65 31.30

0.35

d. CALCULATION OF S.D.

Ex: Compare the performance of SHIKHAR DHAWAN/VIRAT KOHLI

BASED ON LAST 5 ODI

Ans: TO COMPARE THE VARIATIONS WE OBTAIN CV FOR BOTH,

CLACULATION FOR DHAWAN CLACULATIONS FOR KOHLI

C.I. f c.f.(<)

0-20 12 12

20-40 23 32

40-60 35 70

60-80 20 90

80-100 10 100=N

Total 100

ODI NO DHAWAN KOHLI SD X2

VK X2

ODI #

3397

116 14 116xM+=13456 14xM+=196

ODI #

3395

17 115 289 13225

ODI #

3388

16 2 256 4

ODI #

3387

15 31 225 961

ODI #

3383

69 102 4761 10404

Total 233 264 18987 24790

ΣX 233 Mean= 46.6

ΣX2

18987 S.D.= 40.3217

ΣX 264 Mean= 52.8

ΣX2

24790 S.D.= 46.585

Page 3 of 17

Mean X =x

N

=

233

5=46.6 Mean X =

x

N

=

264

5=52.8

S.D.=

2

2Xx

n

=

21898746.6

5 =40.32 S.D.=

2

2Xx

n

=

22479052.8

5 =46.58

C.V.= 86.52 Less C.V.=88.22 More

Hence Dhawan was more consistent than Kohli.

S.D. FOR GROUPED DATA: WHEN FREQUENCIES ARE GIVEN,

S.D. =

2

2Xfx

N

Table of calculation:-

DATA:

Daily wages: 0-4 5-9 10-14 15-19 20-24 25-29

No of workers: 3 6 16 6 5 4

FOR CV WE FIND MEAN & SD.

WHERE,

Mean= X =fx

N

=560

40=14 S.D.=

2

2Xfx

N

= 2968014

40 =6.78

And, C.V.=. .

x100S D

Mean=48.44

C.I. f X Fx f 2x

e.g. 10-15 8 10 15

2

=12.5

8x12.5 =100 100x12.5 =

1250

Not 100x100

Total N = ∑fx = ∑f 2x =

DATA CALCULATION OF S.D. Daily

wagesin100Rs. No of

workers: f X Mid point fx fx2

0-4 3 2 3X2=6 6x2 =12

5-9 6 7 42 294

10-14 16 12 192 2304

15-19 6 17 102 1734

20-24 5 22 110 2420

25-29 4 27 4X27=108 2916

Total N=40

Σfx=560 Σfx2 =9680

Page 4 of 17

Ex:. The data given below are the score of students in a common examination test (CET).

Calculate the Standard Deviation (S.D.). Also find the coefficient of variation.

Score: 0-50 50-100 100-150 150-200 200-250 250-300

No of Student: 6 125 75 45 14 5

Ans: Table of s.d. calculation

Where, Mean X =fx

N

=

31300

270=115.92,

S.D =

2

2Xf x

N

=24343750

115.92270

51.48

C.V. = . .

x100S D

Mean=

51.48

115.92x100= 44.4

Ex: . Calculate the standard deviation (S.D.) from the data given below. Monthly rent paid (000Rs): 0-5 5-10 10-15 15-20 20-25

No of families: 4 23 28 15 2

Where, Mean X =fx

N

=

830

72=11.52,

S.D =

2

2Xf x

N

= 211300

11.6672

=4.56

Score f X Fx f2x

0-50

50-100

100-150

150-200

200-250

250-300

6

125

75

45

14

5

25

75

125

175

225

275

150

9375

9375

7875315

01375

3750

703125

1171875

1378125

708750

378125

Total N =

270

∑fx=

31300 ∑f

2x

4343750

C.I f x Fx fx2

0 to 5 4 2.5 10 25

5 to 10 23 7.5 172.5 1293.75

10 to 15

28 12.5 350 4375

15 to 20

15 17.5 262.5 4593.75

20 to 25

2 22.5 45 1012.5

72=N 840fx 11300fx2

Page 5 of 17

UNIT-IV PROBABILITY

a. Coin problem: A group of 3 coins is tossed as a time, find the probability of getting,

i. Exactly 2 Heads ii. At most 2 Heads

Solution: When three coins are tossed up at a time the sample is

S =HHH, HHT, HTH, THH,HTT,THT, TTH, TTT n(S) =8.

Event A: Exactly two Head turns up. A= { HHT, THH, HTH} n(A)= 3

... P(A) =

( )

( )

n A

n S =

3

8

Event B: At most two Heads: .B:HHT, HTH, THH,HTT,THT, TTH, TTT

n(B) =7. ... P(B) =

( )

( )

n B

n S=

7

8.

b. Dice Problem: A pair of fair dice is rolled. Write down the sample space and find the

probability that, the sum of dots on the uppermost face is i) 6 or 10. ii) Multiple of 4.

iii) < 6.

Solution: When a pair of dice is rolled, the sample space is

S=(1,1), (1,2), (1,3),( 1,4), (1,5), (1,6), (2,1)…(2,6).. (5,6), (6,6) n(S)= 36.

To find the probability we define the events,

i) Event A: the sum of dots on the uppermost face is 6 or 10.

A= {(1,5) (5,1), (2,4) (4,2) (3,3) (4,6) (6,4) (5,5)} n(A)= 8.

... P(A) =

( )

( )

n A

n S =

8

36

ii) event B: The sum of the dots on the uppermost faces is divisible by 4.

B: {(1,3),(3,1),(2,2),(2,6),(6,2),(3,5),(5,3),(4,4),(6,6)} n(B)=9.

... P(B) =

( )

( )

n B

n S=

9

36=0.25.

iii) event C: the sum of the dots is < 6.

C: { (1,1),(1,2),(1,3),(1,4),(4,1)(2,1),(2,2),(2,3),(3,1),(3,2)} n(C) = 10.

... P(C)=

( )

( )

n C

n S=

8

36.

UNIT-V: DECISION THEORY

a. Solve the Decision problem using MAXIMAX, MAXIMIN & LAPLACE criteria

Pay-off Table MAXIMAX MAXIMIN MAX(AVERAGE)

Stock/Demand 90 100 110 120 Col A Max Col B Min Col C Average

A1- 90 2250 2250 2250 2250 2250

MAX 2250 TOATL/4= 2250

A2- 100 1000 2500 2500 2500 2500 1000 2125

A3- 110 -250 1250 2750 2750 2750 -250 1625

A4- 120 -1500 0 1500 3000 MAX 3000 -1500 750

Total 3000 -1500 2250

Page 6 of 17

b. Solve the Decision problem using MAXIMAX, MAXIMIN & LAPLACE criteria

c. Solve the Decision problem using EMV criteria

EMV criteria:

We calculate the EMV values as follows

EMV (A!) = 20x0.3 +5x0.4 +(-1)x0.3= 7.7---Max

EMV (A2) = 8x0.3 +5x0.4 + 4x0.3 = 5.6

EMV (A3) = -10x0.3 +5x0.4 +10x0.3= 2

Hence the optimum decision is A1

d. Solve the decision problem using Minimax Regret criteria

Pay-off Table Regret Table Mark max for the States of nature

Action/

States of

nature A1 A2 A3 A1 A2 A3

S1 50 10 100 100-50=50 90 0

S2 700 500 60 700-700=0 200 630

S3 500 900 80 900-500=400 0 810

Max regret 400 200 Min(Max) 810

e. Solve the Decision problem using EOL criteria

Pay-off We prepare the Regret (OL) table as follows

EOL (A1) = 0x0.3+0x0.4+11x0.3 =3.3----Min EOL Optimal Decision: A1

EOL (A2) = 12x0.3+0x0.4+6x0.3 =5.4

EOL (A3) = 30x0.3+0x0.4+11x0.3 = 12.3

Action/

States of

nature A1 A2 A3

Optimal Decision

S1 50 10 100

S2 700 500 60

S3 500 900 80

Max 700 MAX 900 100 A2

Min 50 10 MAX 60 A3

Average 416.667 MAX 470 80 A2

Pay-off Table

Action State of nature

S1 S2 S3

A1 20 5 -1

A2 8 5 4

A3 -10 5 10

Probability 0.3 0.4 0.3

Action State of nature

S1 S2 S3

A1 20 0 11

A2 8 0 6

A3 -10 0 0

Action State of nature

S1 S2 S3

A1 20-20 = 0 0 11-11=0

A2 20-8= 12 0 11-6=5

A3 20-(-10)=30 0 11-0-11

Page 7 of 17

TUTORIAL ASSIGNMENT:-I

Unit-II: PERMUTATIONS & COMBINATIONS

Q1 Evaluate the following,

i) 5P3 +

7P2 ii)

8P3 +

6P4 iii)

10P8 +

10P7 iv)

8C4 +

7C3 v)

8C6 +

8C7

Q2. In how many possible the letters in the word FATHER be arranged so that,

a) All the vowels are always together b) they are not together

Q3. Five books on Mathematics, 4 books on English & 3 books on History are to be put

in a shelf in a row. In how many possible ways can this be done so that,

Books of same subjects are always together

Only English books are together

No two Mathematics are together

Q4. A box contains 6 Green & 5 Red balls, a pair of balls is drawn at random. Find the

no of possible selections so that,

Both the balls are of same colours.

They are of different colours

Only red balls are drawn

Q5. In how many possible ways 3 cards can be drawn from the pack of 52 cards so that,

i) all 3 are Ace cards;

ii) there are two kings and one queen

iii) cards are of same suit

TUTORIAL ASSIGNMENT:-II

Unit-IV: PROBABILITY & RANDOM VARIABLES

1. A cubic die is rolled down. What is the probability of getting,

a) No of dots <4 b) no of dots as multiple of 3

2. A group of 3 coins is tossed up at a time. Find the probability that,

a) Only 1H turns up b) there are more H than T

3. A pair of unbiased dice is rolled down. Find the probability that,

a) sum of the dots is <6 b) the sum of the dots is 7 or 11

4. 3 cards are drawn from the pack of 52 cards. Find the probability that,

a) all 3 are Ace cards b) all are of same suit c) there are 2 kings & 1 queen

5. Given P(A)= 0.5, P(B) = 0.6 & P (A∩B) = 0.4 Find, i) P(AB) ii) P(A/B) iii) (only A)

6. For 2 independent events A & B, P(A) = ½ , P(B) = ¾.

Find, i) P(AB) ii) P(only B) &iii) (only A) iv) P(Only One)

7. A problem on Maths is given to 2 students A, B who attempt it independently. What is

the probability that, i) the problem is solved ? ii) it is solved by only one?

Given that their chances of solving are 1/3, & 3/4 respectively.

Page 8 of 17

TUTORIAL NO:III

Unit-IV:EXPECTED VALUE & VARIANCE

1. Find the expected value & variance of the r.v. X defined as ‘The no of Heads’ in

the experiment of tossing a unbiased coin four times.

2. Find the Mean & Variance of the r.v. X from the following probability

distribution.

X 5 10 15 20 25

p(x) 0.1 0.23 0.35 0.2 0.12

3. Find the value of k so that the given p(x) represents a probability distribution.

Hence find the expected value of X.

X: 4 5 6 7 8

p(x): 0.15 0.20 k 0.15 0.10

4. Find the Expected value and Variance of the r.v. X for the given probability

distribution. Hence find the expected value of X.

X: -1 0 1 2 3 4

P(x): 0.10 0.05 0.15 0.15 0.30 0.25

5. A man tosses a cubic die in a fun & fair game. According to the terms of the

game. He earns,

Rs10/- if the of the dots is multiple of 3,

Rs15/- if the no of dots is less than 3

& earns nothing otherwise.

Find his expected gain from the game if he has to pay Rs10/- as the entry fee.

TUTORIAL NO:IV

Unit-V:DECISION THEORY-I

1. Suppose that a decision maker faced with three decision alternatives (Acts) and three

state of nature(events) with the following pay-off table:

Action State of nature

E1 E2 E3

A1 15 20 26

A2 18 10 15

A3 25 16 9

Solve the Decision problem using,

a) Maximin b) Minimax regret c) Minimax

Page 9 of 17

2. Solve decision problem using Minimax regret criterion:

Event→

Action↓

E1 E2 E3

A1 5 10 18

A2 8 22 8

A3 21 18 12

A4 30 7 19

3 Determine the best decision according to EMV criterion.

Action/Events E1 E2 E3

A1 10 12 25

A2 18 24 10

A3 25 30 20

Probability: 0.2 0.5 0.3

TUTORIAL NO:V

Unit-V:DECISION THEORY-II

1. Draw a decision tree diagram to show the solution to the decision problem using EMV

criterion:

2. Draw a decision tree for the decision problem below and state the best possible decision.

Use EMV criteria.

Event→

Action↓

A1 A2

P(E)

E1 5 10 25%

E2 8 22 20%

E3 21 18 15%

E4 30 7 40%

Product/Market

demand

Poor Average Good

P 100 350 100

Q 150 50 150

P(Demand) 30% 55% 15%

Page 10 of 17

TUTORIAL ASSIGNMENT:-VI

Unit-III: AVERAGES (MEASURES OF CENTRAL TENDENCY)

1. The score of students in a class test are given below. Find the values of Mean, Median &

Mode. Also count the no of students with score above the Mean score.

18 20 12 15 15 15 16 8 9 11

11 6 20 14 15 13 18 19 10 14

2. The closing price of shares on 5 trading days of the market are as follows,

Calculate the mean price.

Closing price(Rs): 30-34 35-39 40-44 45-49 50-54

No of shares: 18 35 45 32 25

3. The Height of students in a class is given below. Calculate the values of Mean & Median .

Height in cms: 110-120 120-130 130-140 140-150 150-160

No of students: 11 35 45 14 5

4. Calculate the combined mean from the data given below.

Sample I II

No of items: 100 150

Means weight(kgs): 30 32

5. The daily wages paid to the workers are given below.

Wages in Rs: <250 250-300 300-350 350-400 400-450 450& above

NO of Workers: 10 30 48 22 15 5

Calculate the Median & 3 Quartiles. Hence state the wage limit that covers middle 50%

of workers.

TUTORIAL ASSIGNMENT:-VII

Unit-III: DISPERSION(MEASURES OF VARIATION)

1. Calculate the QD &it’s coefficient.

Sales Range (in LacsRs.) : Below35 35-40 40-45 45-50 50+

No of salesmen: 28 43 55 32 22

1. The score of candidates in a CAT examination is shown below. Calculate S.D. & C.V.

Score: 75-100 100-125 125-150 150-175 175-200

No of Candidates: 9 21 35 18 12

2. The data given below read the price range of car sales over the period of six months.

Calculate the Coefficient of Variation.

Price (in Lacs Rs.): 2.5-7.5 7.5-12.5 12.5-17.5 17.5-22.5 22.5-27.5 No

of cars sold (in 100): 3 17 33 16 6

3. Calculate the combined S.D. for the data below

Sample I II

No of items: 100 150

Mean weight (kgs): 15 16

S.D. 3 4

Page 11 of 17

TUTORIAL NO:X

Unit-II:LINEAR PROGRAMMING PROBLEM

1. Solve the following LPP by graphical method.

Min Z= 150x+100y s.t. 6x+2y ≥6; 2x+4y ≥6; x & y ≥ 0

2. Solve the following LPP by graphical method.

Max Z= 6x+5y s.t. 4x+5y≤ 20; 2x+6y≤ 12; x& y ≥0.

3. Solve the following LPP by graphical method.

Min Z= 10x+15y; s.t. 3x+y ≥3; x+2y ≥3; 2x+y≥4; x& y ≥0

4. Solve the following LPP by graphical method.

Max Z= 16x+15y s.t. 4x+5y≤ 20; 2x+6y≤ 12; x& y ≥0

TUTORIAL NO: VII

Unit-I: SAHRES & MUTUAL FUNDS

1. Calculate the brokerage @ 1.5% on the purchase of 300 shares @ Rs.65/-.

2. Calculate the dividend earned @40% on 250 shares of FV 5/- purchased @ 90/-

3. If a dividend of Rs. 1250 is earned on 150 shares of FV 2/-. Find the rate of dividend earned.

4. Mr. Akash purchased 200 shares of HDFC @ 575/- & sold for 650/- each after receiving a

dividend of 40% on FV 5/-. Calculate the % profit to him if he paid 2% brokerage.

5. Miss. Babita sold 200 Rs 10/- shares @ Rs.90/-.She invested the amount in buying 150 other

shares @125/- Find the extra amount required if any, when the brokerage paid was 1.25%

6. Miss. Anju invested Rs.54000 in buying certain ten rupee shares @ Rs.90/-. He sold 1/3rd

of

them @ 125/- after 10 days and the rest @ 110/- at the end of year after receiving a dividend of

10%. Find the gain to her in the transaction.

TUTORIAL NO: IX

Unit-I: SAHRES & MUTUAL FUNDS

1. Calculate the amount of entry load applicable @ 2.25% on the purchase of 500

units at NAV Rs. 48.55/-.

2. Mr Akash sold 400 units of HDFC mutual fund at NAV Rs. 125/- & purchased

SBI mutual fund at NAV Rs.78.25/-. Find the no of units purchased when load of

2.5% was applied in both the transactions.

3. If the NAV of a MF unit is increased from 28 to 45 in one year, what is the %

growth?

4. Anand invested Rs.1Lac in the gift fund of HDFC Mutual fund at NAV of Rs.

28/-.He sold the units at NAV Rs.32/-after receiving a dividend @ 25%. Find the

net % gain to him, if the load was 2.25% on both the transaction

Page 12 of 17

QUESTION BANK – SEM-I

SECTION-I (MATHS)

Unit-I: SHARES & MUTUAL FUNDS

1. Mr. Ajay invested Rs.45000 in buying certain ten rupee shares @ Rs.90/-.He sold half of

them @ 100/- after 10 days and the rest @ 80/- at the end of year after receiving a

dividend of 10%. Find the gain or loss to Mr. Ajay in the transaction.

2. Calculate the sale value of 500 shares @ Rs. 45/- if brokerage 0.3%.is applied.

3. Calculate the amount of brokerage paid on the purchase of 250 shares of FV Rs.10/-@

Rs.75/- when brokerage charged is 0.35%. Also calculate the purchase value.

4. Calculate the amount of dividend @ 40% earned on 200 shares of F.V. Rs 5/- which were

purchased at Rs.55/- each.

5. Mr. Rajesh bought 500 shares of FV 5/- @ Rs.150/-. He sold 40% of them @ 180/- and

the rest @ 225/- at the end of year after receiving a dividend of 25%. Find the net % gain

to Mr. Rajesh in the transaction. Brokerage of 1.5 % is applied on both the transaction.

6. Mr. Shah invested Rs.43680/- in buying certain ten rupee shares @ Rs.91/-. He sold half

of them @ 100/- and the rest @ 80/- at the end of year after receiving a dividend of 10%.

Find the gain or loss to Mr. Shah in the transaction.

7. The NAV of a mutual fund increased from 48 to 64 within a year. Calculate the rate of

growth.

8. Calculate the amount entry load @1.5%.applied on the purchase of 200 units of a Mutual

Fund at NAV Rs.75/-

9. A person invest Rs.1,00,000 in the gift fund of HDFC Mutual fund on 11/2/2007. Find

the no of units purchased by him at NAV Rs. 15/- with entry load of 2.5%.

10. Suppose a scheme with 1,000 units ha the following items in its balance sheet: Unit

Capital Rs. 10,000; Investments at market value Rs. 25,000; Other assets Rs. 3,500;

Other liabilities Rs. 2,000; Issue expenses not written off Rs. 500; Reserves Rs. 17,000.

What would be its NAV?

Unit-II: PERMUTATIONS & COMBINATIONS

1. Evaluate the following,

5P3 +

7P2 ii)

8P3 +

6P4 iii)

10P8 +

10P7

2. In how many possible the letters in the word ATTITUDE be arranged so that,

All the vowels are always together

3. All the consonants are togetherFour books on PHYSICS 3 books on CHEMISTRY & 2

books on BIOLOGY are to be put in a shelf in a row. In how many possible ways can this

be done so that,

Books of same subjects are always together

Only BIOLOGY books are together

BIOLOGY books are at end position

Page 13 of 17

4. Six boys & 2 Girls are to stand in a row for a group photo. How possible ways they can

have a photo so that,

2 Girls always stand together

They do not stand together

They stand at the end position?

5. From the digits 1,2,5,6,8 & 9 a 3 digits number is to be formed. In how many possible

this can be done so that,

No digit is repeated

Digits are allowed to repeat

Only even number without repeated is formed.

6. In how many possible ways 2 balls can be drawn out of 15 balls?

7. Find the no of possible ways to draw a pair of cards from the pack of 52 playing cards.

8. A box contains 6 Green & 5 Red balls, a pair of balls is drawn at random. Find the no of

possible selections so that,

Both the balls are of same colours.

They are of different colours

9. Out of 6 Batsmen, 5 Bowlers & 3 Wicketkeeper a Team of 11 players is to be formed.

How many ways can this be done so as to include,

5 Batsmen, 4 Bowlers & 2 Wicketkeeper

6 Batsmen, & at most 1 Wicketkeeper

10. In how many possible ways 3 cards can be drawn from the pack of 52 cards so that, i) all

3 are Ace cards; ii) there are two kings and one queen. iii) cards are of same suit.

11. Solve the following LPP by graphical method.

i) Max Z= 6x+7y s.t. 4x+5y≤ 20; 2x+6y≤ 12; x & y ≥0.

ii) Min Z= 10x+15y s.t. 3x+y ≥3; x+2y ≥3; x & y ≥0

iii) Max Z= 20x+30y s.t. 3x+3y≤ 36; 5x+2y≤ 50; x+y ≥3.

iv) Max Z= 20x+30y s.t. 4x+6y≤ 24; 3x+7y≤ 21; x+y≤ 5; x & y ≥0.

v) Min Z= 12+20y s.t. x+y ≥3; 2x+y ≥3; x+2y ≥4; x & y ≥0

vi) Min Z= 10x+15y s.t. 3x+y ≥3; x+2y ≥3; x & y ≥0

Page 14 of 17

SECTION-II (STATS)

Unit-III: AVERAGE & DISPERSION

1. What are the positional averages. How can they be approximated?

2. Write short note on ‘Partition values’.

3. Calculate the values of Mean, Median and Mode.

Height in cms: 10-20 20-30 30-40 40-50 50-60

No of students: 18 35 45 12 5

4. The sum of deviations of the x values in a certain group of observations taken from 25 is

120 and that from 35 is -880. Find the mean and no of observations in the group.

5. Calculate Median and Mode for the following distribution.

Income in Rs.: 5-10 10-15 15-20 20-25 25-30

No of Person: 30 50 100 40 30

6. Calculate 3 Quartiles from and hence state the salary limits that include middle 50% of

the employees.

Salary(per day Rs): Below 100 100-200 200-300 300-400 400-500

500-600

No of Employees: 5 12 23 35 20 5

7. Calculate the Q.D. & it’s coefficient from the data given below.

Wages : <800 800-1000 1000-1200 1200-1400 1400-1600

No of Workers: 20 60 80 30 10

8. Calculate the Q.D. & it’s coefficient from the data given below.

Age : <25 25-29 30-34 35-39 40-44 45& above

No of Workers: 10 30 48 22 15 5

9. The coefficient of Q.D. for a certain group of observations is 0.2 and the sum of lower

and upper quartiles is 100. Fine the two quartiles and the Q.D.

10. Find the mean deviation and it’s coefficient from mode for the data given below.

Price of shares: 200-220 220-240 240-260 260-280 280-300

No of shares: 24 32 50 17 7

11. Calculate the standard deviation (S.D.) and C.V. from the data given below.

Increase in height (in cms): 0-2 2-4 4-6 6-8 8-10

No of children 12 23 35 20 10

12. Calculate the coefficient of variation for the data given below.

Earning per share(in 100 Rs): 0-5 5-10 10-15 15-20 20-25

No of Shares: 5 12 20 10 8

13. Find the coefficient of variation for the data given below.

Daily wages: 200-220 220-240 240-260 260-280 280-300

No of workers: 24 32 50 17 7

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14. Compare the two groups on consistency level & state which group is more consistent.

Group A B

Mean 75 40

S.D. 8 5

Also calculate the combined S.D.

15. Calculate the combined standard deviation (S.D.) from the data given below.

Sample I II

No of items: 100 150

Means: 15 16

S.D. 3 4

Unit-IV: PROBABILITY

16. Find the probability of getting a prime number when a cubic die is tossed up.

17. A box contains 20 tickets numbered 1-20. A ticket is drawn at random form the box. Find

the probability that the number on the ticket is,

a perfect square

a multiple of 3

18. A pair of unbiased dice is rolled at a time find the probability that the sum of the dots

appearing on the uppermost faces is,

i) 6 or 10; ii) multiple of 4; iii) 10 or more.

19. Three unbiased coins are tossed up at a time. Find the probability that,

i) exactly 2 Heads appear; ii) at most 2 Heads appear.

20. Three cards are drawn from the pack of 52 cards. Find the probability that,

i) all 3 are Ace cards; ii) there are two kings and one queen;

iii) all are face cards iv) cards are of same color

21. Find the probability of getting a Face card when a card is drawn at random from a pack of

52 playing cards.

22. Given P (A) = 2/5, P (B) = 3/4 & P (A∩B) = ½. Find,

i) P (AUB); ii) P (A/B); iii) P (B/A)

Are events A & B independent?

23. Given P(A)= 0.5, P(B)= 0.7 & P(A∩B) = 0.4

Find, i) P (AUB); ii) P (A/B) iii) P( only A) iv) P(only one)

24. Find the expected value & variance of X from the probability distribution given below

X: 1 2 3 4 5

p(x): 0.15 0.20 0.4 0.15 0.10

UNIT V:-DECISION THEORY

25. Define the terms for a Decision Problem

i) Course of Action ii) State of Nature iii) Pay-off

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26. Explain the terms:

Maximax criteria Laplace criteria Maximin criteria

27. What is the Regret or Opportunity loss? How is it used to find optimum decision?

28. Solve the given decision problem using,

a) Maximin b) Minimax c) Laplace criteria.

Course of Action States of nature(Events)

S1 S2 S3

A 50 102 25

B 180 40 100

C 125 30 200

29. Obtain the best decision using Minimax criteria

Action Events

E1 E2 E3

A1 10 12 25

A2 18 24 10

A3 25 30 20

30. Solve the Decision problem using

i) Minimax ii) Maximax & iii)Laplace criteria

31. Given the pay-off matrix, solve the decision problem using,

i) Laplace ii) Maximin iii) Maximax

32. Given the pay-off matrix, solve the decision problem using,

i) Laplace ii) Maximin iii) Maximax

EVENTS COURSE OF ACTION

A1 A2 A3

E1 20 5 15

E2 8 25 4

E3 10 5 10

States of nature/ Action A1 A2 A3

S1 50 700 500

S2 10 500 900

S3 100 60 80

Action/Event E1 E2 E3

A1 10 25 16

A2 25 12 18

A3 30 22 15

A4 15 25 20

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33. Determine the best decision according to EMV criterion.

States of nature Course of action

A1 A2 A3

E1 10 12 25

E2 18 24 10

E3 25 30 20

Given, P(E1)= 0.4 P(E2)= 0.5 P(E3)= 0.1

34. Draw a decision tree for the decision problem below and state the best possible decision.

Action Events

E1 E2 E3

A1 15 8 16

A2 12 20 15

Probability: 0.2 0.5 0.3

35. Draw decision tree for the following problem and suggest a best course of action (Use

EMV)

DO TRY ALL THE PROBLEMS ON YOUR OWN.

Action

Choice of product

States of Demand & profit

Fair Good Best

P 900 290 500

Q 100 500 700

Probability 0.2 0.5 0.3