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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