LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
|
THIRD SEMESTER – APRIL 2006
ST 3807 – COMPUTATIONAL STATISTICS – II
Date & Time : 02-05-2006/1.00-4.00 P.M. Dept. No. Max. : 100 Marks
Answer any three questions and each question carries 33.5 marks.
- The following data relates to three body measurements taken on boys and girls of the same community. Test whether boys and girls differ from each other on the basis of their measurements assuming the variance – covariance matrices of these two data sets are equal.
| Boys (centimeter) | |||
| Serial Number | Height | Chest | Mid Upper Arm |
| 1
2 3 4 5 6 |
78
76 92 81 81 84 |
60.6
58.1 63.2 59.0 60.8 59.5 |
16.5
12.5 14.5 14.0 15.5 14.0 |
| Girls | |||
| Serial Number | Height | Chest | Mid Upper Arm |
| 1
2 3 4 5 6 7 8 9
|
80
75 78 75 79 78 75 64 80 |
58.4
59.2 60.3 57.4 59.5 58.1 58.0 55.5 59.2
|
14.0
15.0 15.0 13.0 14.0 14.5 12.5 11.0 12.5 |
- a) For the girls data given in question number 1 obtain
i). sample correlation matrix and test for the significance of the correlations.
ii). Partial correlation r12..3
iii). Multiple correlation R1.23 and test its significance.
- b) Draw a Q-Q plot for the girls data on chest measurement in question number 1 to find whether the data is from a normal distribution.
- a). The following frequencies with the corresponding probabilities observed in a genetical experiment are given below :
| Cell Number | 1 | 2 | 3 | 4 |
| Probabilities | (1+q) / 4 | (1 – q) / 4 | (1 – q) / 4 | (1 + q) / 4 |
| Frequencies | 1997 | 906 | 904 | 32 |
Obtain the Maximum Likelihood Estimator of q and the variance of this estimator.
b). Obtain the estimator by the method of modified minimum chi-square for the data in question 3. a).
c). Estimate the parameter q assuming a truncated Poisson distribution truncated at 0 for the given data
| X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Frequency | 60 | 50 | 40 | 25 | 10 | 8 | 4 | 3 |
- a) Classify the states of the Markov chain having the following transition probability matrix (tpm)
Find Pn and lim Pn
n→∞
- Consider the Markov chain with state space {0, 1, 2, 3} and one step tpm
where q = (1-p), 0<p<1. Comment on the nature of the states.
- a) Consider a Markov Chain having the state space {0, 1, 2} and transition matrix
- Show that the Markov chain is irreducible.
- Obtain the period for this Markov chain
- Obtain lim Pdn.
n→∞
- b) An infinite Markov chain on the set of non-negative integers has the transition matrix as follows:
pk0 = (k+1) / (k+2) and Pk, k+1 = 1 / (k+2)
- Find whether the chain is positive recurrent, null recurrent or transient.
- Find the stationary distribution, in case it exists.
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