LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 29

FIRST SEMESTER – APRIL 2006

                                                 ST 1812 – STATISTICAL COMPUTING – I

 

 

Date & Time : 22-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

 

Answer any THREE questions

 

1. a) Find a G- inverse of the matrix

A =

 

1. b) Check whether the following vectors are linearly dependent:
2. i) X¢  = (1, -1, 2),       Y¢ = (2, 0, -1),             Z¢ = (0, -2, -5).
3. ii) X¢ = (1, 0,0), Y¢ = (0, 1, 0), Z¢ = (0, 0, 1).                            (20+14)

 

2. a) The following data relates to the results of an experiment the relative frequencies for 4 different types of genes are expected to be and
   where 0 < < 1.

The frequencies observed were 508, 432, 397 and 518 respectively.  Estimate

the parameter q by the method of maximum likelihood and find the estimate of

the  standard error of the estimator.

 

1. b) The scores of 17 students are given by the following table. Assuming that this

is a sample from normal population whose variance is s², obtain

1. a 95% confidence interval for s
2. a 99% confidence interval for s

 

Scores:

(Out of 100)         45     65     68     77     95     69     56         72        75

38     68     72     65     42     66     55         62

(14+20)

 

3. a) Below are given two random samples drawn from different normal populations:

Sample 1:    10        6          16        17     13     12     8       14     15        9

Sample 2:    7          13        22        15     12     14     18     8       21        23 10

 

Obtain a 99% confidences limits for the difference of means of the 2 populations.

 

1. b) Fit a normal distribution to the following data

 

C.I:	60-65	65-70	70-75	75-80	80-85	85-90	90-95	95-100
Frequency	3	21	150	335	325	135	26	4

(20+14)

 

4. a) Fit a multiple regression model of Y on X₁ and X₂ for the following data. Estimate Y when X₁ = 1350 sq.ft and X₂ = 2 years                                      (20)

 

1. b) Also, test the significance of the population multiple correlation coefficient at

5% level of significance.                                                                                (14)

 

FLAT PRIZE IN LAKHS (Y)	FLAT SIZE IN SQ.FT (X1)	AGE OF THE FLAT IN YEARS        (X2)
12.3	1050	1
15	1200	1
14.8	1180	3
11	950	2
10.3	900	3
16.9	1300	3
18	1400	3
6	450	4
5.2	480	5
4.6	420	4
18	1450	6
9.3	850	3
12.2	1020	7

 

5. a) Use Step-wise regression analysis to identify the most significant independent

variable(s) and comment on your finding regarding the significance of

population regression coefficients for the data given in question-4                (20)

1. b) Compute the condition index for the data in question-4 and examine whether

the multi-co linearity problem is present in the data or not.                          (14)

 

 

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                  LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034        M.Sc. DEGREE EXAMINATION – STATISTICS

AB 21

FIRST SEMESTER – NOV 2006

ST 1812 – STATISTICAL COMPUTING – I

 

 

Date & Time : 04-11-2006/1.00-4.00           Dept. No.                                                       Max. : 100 Marks

 

 

 

Answer any  THREE questions.

1. a.) The following data relates to the family size(X) and average food expenditure per week (Y) of 8 persons randomly selected from a small urban population.

Y: 40   50   50   70   80   100  110  105

X: 1      1     2     3     4      2      5      6

Assuming there is a linear relationship between Y and X, perform a regression of Y on X and estimate the regression coefficients. Also find the standard error of the estimate.

b.) Consider the following ANOVA table based on OLS regression.

Source of Variation     df        Sum of Squares

Regression                   ?          800

Residual                      45        ?

Total                            49        1200

• How many observations are there in the sample?
• How many independent variables are used in the model ?
• Find an unbiased estimate of the variance of the disturbance term?
• Calculate the value of the coefficient of determination and interpret it.
• Test the overall significance of the model at 5% level.

(20+14)

 

1. a.) Consider the following information from a 4 variable regression equation:

Residual sum of squares = 94;

Y = 10, 12, 14, 9, 7, 8, 2, 22, 4, 12.

i.)   Find TSS and ESS.

ii.) Test the hypothesis that R² = 0 Vs R² # 0 at 5% level.

b.) Test whether there is structural change in the model Y = β₀ + β₁X + u

between the two groups where the observations under group I and group

II are as given below:

Group I      Y: 10        15        17        14        12

X: 3          5          4          6          7

Group II     Y: 12        14        13        15        18

X: 5          3          7          6          4

Use 5% level.

c.) Consider the following OLS regression results:

Y = 16.5 + 2.1X₁ + 50X₂

(10)     (0.5)      (20)          n = 28

where the numbers in the parenthesis are the standard error of the

regression coefficients.

i.) Construct a 95% confidence interval for β₁.

ii.) Test whether in intercept is significantly different from zero at 5%

level.

(7+20+7)

 

 

3. a.) Consider the following data on annual income (in 000’s $) categorized by

gender and age.

Income: 12        10       14       15        6       11       17

Gender:  0         1          1          0        0         1         1

Age:  1         1          0         1         0         0         1

where Gender = 1 if male; 0 if female

Age = 1 if less than or equal to 35; 0 if greater than 35.

Perform a linear regression of Income on Gender and age. Interpret the results.

What is the benchmark category for the above model ?

b.) Fit a Poisson distribution for the following data relating to the number of

printing mistakes per page in a book containing 200 pages:

Number of mistakes:   0          1          2          3          4          5

Frequency:  60        50        40        30        15        5

(17+17)

4. Fit a normal distribution for the following heights (in cms) 0f 200 men

randomly selected from a village.

Height:            144 – 150        150 – 156        156 – 160        160 – 164

frequency:       3                       10                   25                    50

Height:            164 – 168        168 – 172        172 – 176

Frequency:                 63                      30                    19

Also test the goodness of fit at 5% level.                                           (33)

 

5. a.) Fit a truncated binomial distribution to the following data and test the

goodness of fir at 5% level.

X:        1          2          3          4          5          6          7

f:        6          15        18        12        9          8          2

b.) Fit a negative binomial distribution to the following data and test the

goodness of fit at 5% level.

X:        0          1          2          3          4          5

f:        180      120      105      90        40        12

(20+14)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 29

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2007

ST 1812 – STATISTICAL COMPUTING – I

 

 

 

Date & Time: 03/05/2007 / 1:00 – 4:00      Dept. No.                                       Max. : 100 Marks

 

 

 

Answer any THREE questions.

1. a.) Fit a linear model of the form Y_i = β₁ + β₂X_i + u_i for the following data relating to Y and X:

Y:        10        12.5     13.7     15.3     17        18.5

X:        3           5          7          10      12          6

Estimate the regression coefficients using OLS procedure and find the standard error of the estimate. Also find a 95% confidence interval for the regression coefficients and interpret them.

 

b.) Consider the following computer printout, where a faulty printer failed to print some of the

regression information.

The regression equation is Y =  ? +  ?X₁ + ?X₂

`                       Coefficient      St. error. Of  coeff.     T-Ratio

Constant          -7.6682            ?                                  -0.584

X₁                    51.0918           ?                                  6.80

X₂                    41.4607           ?                                  1.12

where the T-Ratio is calculated under the zero null hypothesis of the

regression coefficients.

Analysis of Variance

Due to             df                    Sum of Squares

Regression       ?                      17023

Residual          17                    6262

Total                19                    23285

• How many variables are there in the model?
• Find the missing values.
• Find R² and interpret it.
• Test the hypothesis H₀: R² = 0 Vs H₁: R² # 0 at 5% level.
• Find an unbiased estimate for the variance of Y.      (20+14)

 

2 a.) The following data relates to the income, sex and education level of

8  individuals selected at random:

Income                                 Sex                        Education level

($/week)     (1-Male;0-Female)    (1-Graduate;0-Non-graduate)

22                            1                                  1

20                            0                                  1

18                            0                                  0

25                            1                                  0

23                            1                                  1

17                            0                                  0

20                            0                                  0

21                            1                                  1

Fit a linear model and obtain the regression coefficients. Interpret the results.

 

 

b.) Consider the following OLS regression results with standard errors in

parenthesis:

S = 12,000 – 3000X1 + 8000(X1 + X2)

(1000)              (3000)              n = 25

where S = annual salary of economists with B.A. or higher degree

X1 = 1 if M.A. is highest degree; 0 otherwise

X2 = 1 if Ph.D is highest degree; 0 otherwise

a.) What is S for economists with a M.A. degree?

b.) What is S for economists with a Ph.D degree?

c.) What is the difference in S between M.A.’s and Ph.D’s?

d.) At 5% level of significance, would you conclude that Ph.D’s earn more per

year than M.A.’s?

e.) What is the bench mark category? Why it is not included in the model?                         (14+20)

 

 

3. a.) Use the data in the following table to test for the structural change of the

model Y = β₁ + β₂ Age + u  where Y denotes the  average amount of water

in liters a machine can desalinate per day in any given year. Assume that

after 5 years the capability of the machine deteriorates.

Y:     10        12        8          6          5          3          3          2          1          0          Age:    1          2          3            4          5          6          7          8          9          10

Note that the values of Y have been rounded off to the nearest integer.

• A die is tossed 120 times and the number of 1’s, 2’s …,6’s appearing was

obtained as below:

Number:     1          2          3          4          5          6

Frequency:     40        20        30        15        10        5

Fit a binomial distribution to the above data and test the goodness of fit at

5% level.                                                                                                                 (20+14)

 

 

4. a.) Fit a truncated Poisson distribution, truncated at zero, for the following

data:

X:        1          2          3          4          5          6

f:          86        52        26        8          6          1

Also test the goodness of fit at 5% level.

b.) Fit a negative binomial distribution for the following data and test the

goodness of fit at 5% level.

X:        0          1          2          3          4          5

f:        210      118      42        19        4          2                                               (17+17)

 

 

5. Fit a distribution of the form P(x) = 1/2 { P₁(x) + P₂(x) } where P₁ is a

geometric distribution with support 1,2,3,… and P₂  is a Poisson distribution.

X:        0          1          2          3          4          5          6          7          8

f:        71        110      119      50        34        8          5          2          1

Also test the goodness of fit at 5% level.                                                                    (34)

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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

1. (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

2. 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.

 

3. 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.

 

1. (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)

2. (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)

3. (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)

(c)   Find the rank of the given matrix A by performing row operations:

 

3        2          3          1

4        3          5          2

2        1          1          0                                                                                                         (10)

 

1. 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.

 

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

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

 

 

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     LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 35

FIRST SEMESTER – April 2009

ST 1812 – STATISTICAL COMPUTING – I

 

 

 

Date & Time: 04/05/2009 / 1:00 – 4:00       Dept. No.                                                       Max. : 100 Marks

 

 

Answer ALL the questions.  Each carries THIRTY FOUR marks

                                                                                        

1. a). In a population containing 490 live birds of same weight and age, the birds were divided into 70 equal groups.  They were then given a stimulus to increase the growth rate.  The following data gives the frequency distribution of birds with significant weights at the end of 6 weeks.  Fit a truncated binomial distribution and test the goodness of fit.

No of birds	1	2	3	4	5	6	7
Frequency	6	15	21	17	8	2	1

 

b). The following are the marks of 135 students of B.Com in a city college.

Marks	No. of Students
0-10	5
10-20	15
20-30	30
30-40	40
40-50	25
50-60	10
60-70	6
70-80	4

Fit a normal distribution by ordinate method and test the goodness of fit at                                    5% level of significance.

(OR)

c). Fit a truncated Poisson distribution to the following data and test the                            goodness of fit at 1% level of significance.

x	1	2	3	4	5	6
f	86	52	26	8	6	1

d). Fit a distribution of the form, where

, x=0,1,2,…,  > 0 and

; x=1,2,…, 0<p<1,

to the following data:

 

x	0	1	2	3	4	5	6	7	8
f	72	113	118	58	28	12	4	2	2

 

 

1. a). Find the inverse of the following symmetric matrix using partition method.

 

b). Obtain the characteristic roots and vectors for the matrix

 

 

 

 

(OR)

c). Find  for the following matrices:

i).

ii).

d). Draw a random sample of size 10 from the exponential distribution having the    density function

.

Also find the mean and variance of the sample observations.

 

1. a) The following data were collected on a simple random sample of 10 patients with hypertension.

 

Serial No.	mean arterial blood pressure (mm/Hg)	weight (kg)	heart beats / min
1	105	85	63
2	115	94	70
3	116	95	72
4	117	94	73
5	112	89	72
6	121	99	71
7	121	99	69
8	110	90	66
9	110	89	69
10	114	92	64

i). Fit a regression model and estimate effect of all variables / unit of measurement,        taking blood pressure as the dependent variable.

ii). Find R² and comments on it

(OR)

1. b) The following table explains a company monthly income based on their advertisement

on V-Slicer product.

 

Serial No.	Monthly Income on sales ($’000)	Advertisement on TV ($ ‘000)	Advertisement on News Paper ($ ‘000)
1	5.5	0.2	0.1
2	6.7	0.5	0.2
3	8.0	1.2	0.8
4	10.1	2.0	0.9
5	15	3.0	1.4
6	18.0	4.0	2.0
7	23	5.0	2.5
8	28	6.2	3.8
9	32	8.0	4.1
10	35	10.0	5.2

 

1. Draw a scatter diagram for the above data.
2. Fit a regression mode taking TV advertisement and News paper advertisement as independent variables and estimate monthly income when TV advertisement is 15 and News paper advertisement is 7 in 1000 dollars.

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034    M.Sc. DEGREE EXAMINATION – STATISTICSFIRST SEMESTER – APRIL 2011ST 18167 – STATISTICAL COMPUTING  I
Date : 20-04-2011 Dept. No.   Max. : 100 Marks    Time :                                              Answer all the questions              (4 x 25 =100 Marks)
1 a)  The following frequency distribution gives the number of albino children in families of five children          having at least 1 albino child:No. of Albinos (x) No. of Families (f)1 222 263 94 25 1
Fit a truncated binomial distribution for the above frequency distribution and test the goodness of          fit  at 5% level.
b)  Fit a normal distribution to the following data by area method   and   test   the goodness   of  fit               at 5%  level of significance:x f40 – 60 860 – 80 1280 – 100 20100 – 120 25120 – 140 45140 – 160 22160 – 180 16180 – 200 16200 – 220 4
( 15 +10)    (OR)
c) The table below gives the frequency distribution of the number  of dust nuclei in a small volume         of air that fell on to a stage in a chamber containing moister and filtered air: No.  of dust nuclei (x) 1 2 3 4 5 6 7 8f 60 84 98 70 37 20 5 3
It is suspected that a number of zero counts were wrongly rejected on the ground that the apparatus   was not working and hence not recorded.  Fit a truncated Poisson distribution to the above frequency distribution and test the goodness of fit.
d)   For the following frequency distribution, fit a negative binomial distribution and test the            goodness    of fit  at 5% level: x 0 1 2 3 4 5f 212 128 40 15 3 2

2  a)  Generate a sample of size 5 from the  Bivariate  normal distribution given below:     (OR)       b) Given the three selected points U1,    U2 and    U3 corresponding to t1 = 2 , t2 = 30 and                         t3 = 58 as follows:           t1 = 2,                U1 = 55.8               t2 = 30,             U2 = 138.6                t3 = 58,             U3 = 251.8
Fit a logistic curve by the method of selected points. Also obtain the trend values         for t = 5, 18, 25, 35, 46, 50, 54, 60, 66, 70.
3. a)Find the inverse of the following matrix A using partitioning method:                     A =      (Or)         b) (i)  Obtain the Rank, Index and Signature of the following matrix A:                          A  =
(ii) Verify whether or not the following matrix is negative definite:
B  =                                                                                     (15 + 10)
4)     a) Determine Tolerance and Variance Inflation Factor(VIF) for each explanatory variable based               on the data and fitted auxiliary regression equations given below:Y 8 9 7 5 6 4 5 2 1 3X1 5.2 5.6 4.8 4 6 5 4.5 2.3 1.5 2.6X2 5.1 5.2 4.7 3.2 3.2 5.4 3.9 2.6 1.8 2.1X3 2.3 1.2 1.5 1.6 1.4 1.8 1.9 1.8 1.5 1.6

Fitted Auxiliary regression equations areX1 = 2.211 + 0.95X2 -0.961X3X2 = -0.805 + 0.704X1 + 0.966X3X3 = 1.568 – 0.102 X1 + 0.139X2
(OR)
Y 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1X1 2.45 1.2 2.5 2.14 1.6 2.19 2.1 2.8 1.5 2.8 2.18 1.1 2.22 2.23 1.5 2.11 2 1.9 1.4 2.7X2 0 1 0 0 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 1X3 1 0 1 1 0 1 0 0 0 0 1 0 1 1 0 1 0 0 0 0b) Consider the following data and the fitted Logistic regression model  Determine the following:(i) Optimal Cut point based on Gains table                                                        (ii)     Classification table based on the optimal cut point , Sensitivity and Specificity.(18+7)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2012

ST 1817 – STATISTICAL COMPUTING – I

 

 

Date : 03-05-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

1. a) For the following frequency distribution fit a poisson distribution and test the goodness of fit at 5 % level. (12 marks)

 

X	f
0	212
1	128
2	37
3	18
4	3
5	2

 

1. b) The following data gives the frequency of accidents in a city during 100 weeks.

 

No. of accidents	No.of weeks
0	25
1	45
2	19
3	5
4	4
5	2

 

Fit a distribution of the form  P (X = x ) =  for the given data and test goodness of fit at 5 %  level.                                                              (21 marks)

 

 

 

 

 

2. a) Five biased coin were tossed simultaneously 1000 times and at each toss the no. of heads was observed. The following table gives the no.of heads together with its frequency (17 marks)

 

 

X ( no. of heads)	0	1	2	3	4	5
f(x)	38	144	342	287	164	25

 

Fit a binomial distribution to the above data and test whether the fit is good at 5 % level.

 

b)

Travel time Y	9.3	4.8	8.9	6.5	4.2	6.2	7.4	6	7.6	6.1
No.of deliveries	4	3	4	2	2	2	3	4	3	2
Miles travelled	100	50	100	100	50	80	75	65	90	90

 

• Build a multiple linear regression model for the above data.
• Determine                                                                                                  (17 marks)

 

3. a) Find the inverse of the following matrix using partitioning method.

 

 

A =                                                                             (23 marks)

 

1. b) Find the rank of A, where A = (10 marks)

 

 

 

 

4.a)  Determine the characteristic roots and vectors of the matrix

 

(15 marks)

 

1. b) Write the quadratic forms of the matrix

 

 

A =                                                                  (18 marks)

 

5.Compute tolerance and variance inflation factor for each explanatory  variable based on auxiliary regression equation and the  given data .

 

 

 

 

 

 

 

 

 

 

Y
8	5.2	5.1	2.3
9	5.6	5.2	1.2
7	4.8	4.7	1.5
5	4	3.2	1.6
6	6	3.2	1.4
4	5	5.4	1.8
5	4.5	3.9	1.9
2	2.3	2.6	1.8
1	1.5	1.8	1.5
3	2.6	2.1	1.6

 

(33 marks)

 

 

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