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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