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,A March 2 16 '"

..,(REVISED COURSE)

(3 Hours)

TV-8310[Total Marks: 100

Con.1957-06.

N.S. (1)(2)(3)(4)(5)

Ouestion No.1 is compulsory.Answer any four out of remaining six questionsFigures to the right indicate full marks.Use of statistical tables is permitted.Write the sub-questions of main question collectively together.

r--

1. (a) Fita parabola to the following data:x: -2 -1 0 12

y: 1.0 1.8 1.3 2.5 6.3

(b) A tyre company claims that the lives of the tyres have mean of 42000 kms. with S.D. of 4000 kms.A change in the production process is believed to result in a better product. A test sample of 81ney/tyres has a mean life of 42500 kms. Test at 5% level of significance that the new product is'significantly better than the current one.

(c) Prove the properties of variance(i) V (aX + b) = a2 V (X)(ii) V (aX + bV) = a2V (X) + b2V (V) + 2ab cov {X,V}

(d) A firm produces two products A and S which are processed on two machines. The relevant datais given below. ' . ..

Product Amin/unit

21

5

5

5

5

/ Machine 1Machine 2

The profit per unit products A,S is 6 and 10 Rs. respectively. Determine the daily No. of units tobe manufactured of each product to get maximum profit. Formulate the LP.P. and\ solve by usinggraphical method.

2. (a) An 1.0. test was administered to 5 persons and after they were trained. The results are given below. 6

()

Test whether there is any change in 1.0. after the training programme, use 1% level of sig'nificance.

'~ (b) Out of 800 families with 5 children each how m~ny would you expect to have -(i) 3 boys and 2 girls (ii) 5 girls (iii) 5 boys?

6..

(c) Use Kuhn Tucker condition to solve the following N.LP.P.

, Maximize Z = 8x1+ 10x2 - x~ - x~

8

,~

Subject to 3X1'+2x2::;;6

3.X1' x2 ~ 0

(a) State and prove Spearman's rank correlation coefficient.

(b) A continuous random variable has p.d.f.f (x) = 1 - x, 0 < x < 1.

= x- 1, 1 < x < 2= 0, otherwise

Find mean and variance.

(c) Using the Lagrangian Multipliers solve the following N.LP.P.

6

6

8

Optimize

Subject to

Z = 2x~ + 2x~ + 2x~ - 24x1"" 8x2 -12x3 + 196

X1 + X2 + X3= 11

x1, X2, X3 ~ 0

I II III IV Vf--

I I.a. before training 110 120 123 132 125

L.a. after training 120 1,18 125 136 121

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. 4.(a) Solve L.P.P.by Simplex method

Maximize Z = 100 X1+ 50X2 + 50X3

Subject to 4x1 + 3x2 + 2x3 $10

3x1 + 8X2 + x3 5. 8

4X1 + 2X2 + X35. 6xl' x2' X3~. 0

6

A March2 17

j. .

Con.1957 -TV-831 0-06, .2

6

,,

(b) Show that recurrence relation for the moments of Poisson distribution is -

dllrIlr+1 = r Ilr-1 + m dm

(c) A random sample of 220 students in a college were asked to give opinion in terms of yes or noabout the winning of their college cricket team in a tournament. The following data are collected.

Class in Colle e I,

1st ear IInd ear IIlrd ear43 20 3723 57 40

8

Yes

No'

Test whether there is any association between opinion and class in coll'ege (use 5% level of significance).6

5. (a) If u = x - y, 'U = X + y and if x, yare uncorrelated prove that,

0'2 -0'2~ - x. y'u -

\) 0'2 +cr2x y

(b) In a large institution 2.28% of employees receive income below Rs. 4500 and 15.87% of employees'receive income above 7500 p.m. Assuming the income follows normal distribution. ' Find the mean

and S.D. of the distribution.

(c) Using Big-M method solve the loP.P.

Minimize Z = 10X1 + 3X2

Subject to X1 + 2X2 ~ 3

X1+ 4X2 ~ 4

X1, X2, ~ 0

6

8 ~

6. (a) Findihe dual ofMinimize

SUbjectto

6

Z = 2x 1 + X2 + 5xa

X1 + X2 + xa = 10

4X1 - X2+ 2xa ~ 12

3X1 + 2X2 - 3xa S;6

X1. X2, xa ~ 0

J.

Then write the 1st Iteration table by using Simplex method.(b) Solve the lPP by Big-M method.

Minimize Z =2X1. + X2+ 3xaSubject to 3X1 + X2 - 2xa ~ 1

X1 - 2X2 - xa ~ 2

X1, X2, X3 ~ 0

6

...........

7.(c) Prove that mean and standard deviation of standard normal variate are zero and one respectively.(a) Define Binomial distribution. Prove that mean = np apd variance = npq.(b) The equation of the two lines of regression for a bivariate data are:

9x + 10y - 67 = 0 ana 5x + 2y - 23 = 0

. . Find (i) mean values of x and y(ii) regression coefficient(iii) correlation coefficient.

(c) Using the principle of duality solve the following loP.P.

Minimize Z = 4X1 + 14x2 + 3xaSubject to - X1+ 3X2+ xa~ 3'

2x1 + 2x2 .:. xa ~ 2

X1,X2,xa~ 0

8

6

6

8