LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
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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.
- a.) Fit a linear model of the form Yi = β1 + β2Xi + ui 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)
- 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)
- 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)
- 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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