Annamalai University 2008-3rd Year B.Sc Mathematics " 760MATHEMATICAL STATISTICS " ( - IX ) ( PART - III - A - MAIN ) ( ) 5241 - Question Paper

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Name of the Candidate :

5 2 4 1 B.Sc. DEGREE EXAMINATION, 2008

(MATHEMATICS)

(THIRD YEAR)

(PART - III - A - MAIN)

(PAPER - IX )

760. MATHEMATICAL STATISTICS

(Including Lateral Entry )

December ]    [ Time : 3 Hours

Maximum : 100 Marks

Answer any FIVE questions. Statistical Tables can be used.

All questions carry equal marks.

(5 x 20 = 100)

1. (a) State and prove Bayes Theorem.

(b) A random variable X has the following probability function.

Value of ]	: 0	1	2	3	4	5	6	7
p(x)	: 0	k	2k	2k	3k	k2	2k2	7k2+k

(i)    Find k.

(ii)    Evaluate

p ( x < 6 ), p ( x >6) and p(0< x < 5).

(iii)    If p ( x < a ) > 1/2 , find the minimum value of a.

(10 + 10)

2. (a) State and prove product theorem on expectation for two random variables.

(b) Find the expectation of the number of failures preceding the first success in an infinite series of independent trials with constant probability of success.

10. (a) The annual sales of a company are given below :

Estimate the sales for the year 1980.

Year : 1970 1975 1980 1985 1990 1995 Sales : In Lakhs of Rs. 125 163 - 238 282 380

(b) The values of x and y are given below :

x :	5	6	9	11
y :	12	10	14	16

Find the value of y when x = 10, by using Lagranges interpolation formula.

Examine at 5% level, whether the two populations have the same variance.

<^FCM>5 = ⁴'¹⁵)'

(b) Construct a four yearly centred moving average from the following data :

Year : 1920 1930 1940 1950 1960 1970 1980 Sales : 129 131 106 91 95 84 93

( 10 + 10 )

9. Calculate Fishers ideal index number from the following data and show that it satisfies time reversal test and factor reversal test.

Commodity	1979		1980
	Price	Quantity	Price	Quantity
A	10	49	12	50
B	12	25	15	20
C	18	10	20	12
D	20	5	40	2

3. (a) Fit a second degree parabola to the following data :

X :	1	2	3	4	5	6	7	8	9
Y :	2	6	7	8	10	11	11	10	9

(b) Calculate the co-efficient of correlation between X and Y for the values given below :

X :	2	5	7	9	19	16
Y :	25	27	26	29	34	39

4. (a) The following table gives the marks obtained by 11 students in Mathematics and Statistics. Find the rank correlation coefficient.

Mathematics:	40	46	54	60	70	80	82	85	85	90	95
Statistics :	45	45	50	43	40	75	55	72	65	42	70

(b) The correlation co-efficient between the variables X and Y is 0-6.

If

c = 1-5, a = 2, x = 10

x    y

and

y = 20,

find the equations of two regression lines.    ( 10 + 10 )

5.    (a) With usual notation, for a Poisson

distribution,

prove that

d

ll . = m (r ii . + ll)

^{r+1    r}-^! dm ^r

(b) State the properties of normal curve.

( 10 + 10 )

6.    (a) Explain the four types of sampling with

examples.

(b) The means of two large samples of sizes 1000 and 2000 are 67-5 inches and 68 inches respectively. Can the samples be regarded as drawn from the same population with standard deviation 2-5 inches ?    (10 + 10)

7.    (a) State and prove Neyman Pearson

lemma.

(b) Two random samples give the following results :

n_: = 10, E ( x - x )² = 90

n₂ = 12, E ( y - 7 )² = 108

Test whether the samples have come from the normal population with the same variances.    (10 + 10)

8.    (a) In a test, given to two groups of

students drawn from two normal populations, the marks obtained were as follows :

Group -1 :	18	20	36	50	49	36	34	49	47
Group - II :	29	28	26	35	30	44	46	-	-

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