LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 35

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2008

ST 1812 – STATISTICAL COMPUTING – I

Date : 06/05/2008 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer the following questions. Each question carries 33 marks

(a) Write the quadratic form associated with the matrix

A =

Verify whether it is positive definite.

(b) Obtain the characteristic roots and vectors of the following matrix:

A =

Obtain the matrix U such that U^TAU = L.

(OR)

Find the inverse of the following matrix by partitioning method or

sweep out process.

A =

Sale Price (in lakh Rs)	No. of Rooms	Age of building
25.9	7	42
27.9	6	40
44	6	44
28.9	7	32
31.5	5	30
30.9	6	32
36.9	8	50
40.5	5	17
37.5	5	40
44.5	7	45

The data on sale prices of houses are given below with information on the number of rooms and age of the building:

Build a model with an intercept. Test for overall significance and the significance of the individual regressors. Comment on the adequacy of the model.

(OR)

(a) A model with a maximum of four regressors is to be built using a sample of

size 30. Carry out ‘Stepwise Building Process’ to decide the significant

regressors given the following information:

SS_T = 1810.50, SS_Res(X₁) = 843.79, SS_Res(X₂) = 604.22, SS_Res(X₃) = 1292.93, SS_Res(X₄) = 589.24, SS_Res(X₁, X₂) = 38.60, SS_Res(X₁, X₃) = 818.05, SS_Res(X₁, X₄) = 49.84, SS_Res(X₂,X₃) = 276.96, SS_Res(X₂, X₄) = 579.25, SS_Res(X₃,X₄) = 117.16, SS_Res(X₁, X_2,X₃) = 32.07, SS_Res(X₁, X_2,X₄) = 31.98, SS_Res(X₁,X_3,X₄) =33.89, SS_Res(X₂, X_3,X₄) =49.21, SS_Res(X₁, X_2,X₃,X₄) = 31.91

(b) The following are observed and predicted values of the dependent variable for a model with an intercept and two regressors.

Y	Y^
16.68	21.7
12.03	12.07
13.75	12.19
8	7.55
17.83	16.67
21.5	21.6
21	18.84
19.75	21.6
29	29.67
19	16.65

Compute the standardized residuals and find if there are any outliers.

The number of accidents taking place in a high way is believed to have mixture

of two Poisson distributions with mixing proportion 2/7 and 5/7. Fit the

distribution for the following data corresponding to one such distribution.

Marks	Number of days
0	98
1	78
2	56
3	73
4	40
5	8
6	2
7	1
>8	0

(OR)

Generate Five observations from a Normal distribution with mean 20 and variance 36 truncated at zero
Generate a sample of size 2 from a mixture of two Cauchy variates one of them has scale parameter 1 and location Parameter 1 and the other has Cauchy distribution with scale parameter 1 and location parameter 0.

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 23

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – November 2008

ST 1812 – STATISTICAL COMPUTING – I

Date : 13-11-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer any THREE questions. Each carries THIRTY FOUR marks.

(a). Fit a distribution of the type

P(x) = (1/2) [P₁(x) + P₂(x)]

where P₁(x) = (e^–μ μ^x ) / x! ; x = 0,1,2, …, μ >0

and P₂(x) = (e^–λ λ^x ) / x! ; x = 0,1,2, …, λ >0

for the following data on the frequency of accidents during 106 weeks in

Chennai:

No. of accidents : 0 1 2 3 4 5

No. of weeks : 26 46 20 6 5 3 .

Also test the goodness of fit at 5% level of significance.

(18)

(b) In a population containing 539 live birds of same weight and age , the birds

were divided into 77 equal groups. They were then given a stimulus to increase

the growth rate . The following data gives the frequency distribution of those

with significant weight at the end or 6 weeks. Fit a truncated binomial

distribution and test the goodness of fit at 5% of level of significance.

No. of birds : 1 2 3 4 5 6 7

Frequency : 7 16 22 18 9 3 2

(16)

(a). Find a g-inverse of the following matrix.

2 2 3 3 1

2 3 3 2 7

5 3 7 9 2

3 2 4 7 3

(17)

(b) Find the characteristic roots and vectors of the following matrix:

1 1 1

1 -1 2

0 1 1

( 17)

(a) Generate a random sample of size 15 from Cauchy distribution with p.d.f.

f(w) = (1/π) [ λ / { λ ² + (w – θ) ² } ] , taking θ = 50 and λ =5. 14)

(b) Verify whether or not the following matrix is positive definite:

12 4 -4

4 12 4

-4 4 20

(10)

3 2 3 1

4 3 5 2

2 1 1 0 (10)

Consider the following data for a dependent variable representing repair time in hours and two independent variables representing months since last service and type of repair.

Customer Months since Type of Repair Repair time

last service in hours

1 2 1 2.9

2 6 0 3.0

3 8 1 4.8

4 3 0 1.8

5 2 1 2.9

6 7 1 4.9

7 9 0 4.2

8 8 0 4.8

9 4 1 4.4

10 6 1 4.5

Using these data, develop an estimated regression equation relating repair time in hours to months

since last service and type of repair. Estimate the repair time if months since last service = 12 and

type of repair = 1.

The following table gives the annual return, the safety rating (0=riskiest, 10 = safest) and the annual expense ratio for 10 foreign funds (Mutual funds, March 2006).

Foreign Funds Safety Rating Annual Expense Annual

Ratio (%) Return (%)

1 7.1 1.5 49

2 7.2 1.3 52

3 6.8 1.6 89

4 7.1 1.5 58

5 6.2 2.1 131

6 7.4 1.8 59

7 6.5 1.8 99

8 7.0 0.9 53

9 6.9 1.7 77

10 7.7 1.2 61

Use F-test to determine the overall significance of the relationship at 0.05 level of significance.
Use t-test to determine the significance of each independent variable at 5 % level of significance.

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Loyola College M.Sc. Statistics April 2008 Statistical Computing – I Question Paper PDF Download

ST 1812 – STATISTICAL COMPUTING – I

Loyola College M.Sc. Statistics Nov 2008 Statistical Computing – I Question Paper PDF Download

ST 1812 – STATISTICAL COMPUTING – I