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(MATH144)[2004](f)final~4660^_23675.pdf
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The Hong Kong University of Science & Technology

MATH244 C Applied Statistics

Final Examination C Spring 03/04

Answer ALL questions Date: 22 May 2004 (Sat)
Time allowed: 2 Hours
_______________________________________________________________________________________

PART I: Multiple Choice Questions (1 mark each)

1. Identical twins come from the same egg and hence are of the same sex. Fraternal twins have a 50-50 chance of being the same sex. Among twins the probability of a fraternal set is p and an identical set is q = 1 C p. If the next set of twins are of the same sex, what is the probability they are identical?
A.
B.
C. 1 D. q E.


2. Machine X has 3 independent components, two of which fail with probability p and one which fails with probability 0.5. The machine operates so long as at least two parts work. A second machine Y has only one component which fails with probability p. Let P[X] and P[Y] be the probabilities that machines X and Y fail, respectively. What is the relationship between P[X] and P[Y]?
A. P[X] =
P[Y] B. P[X] = P[Y] C. P[X] = 2P[Y]
D.
E. P[X] =
P[Y][1 C P [Y]]

3. Urn I contains 7 red and 3 black balls, and urn II contains 4 red and 5 black balls. After a randomly selected ball is transferred from urn I to urn II, 2 balls are randomly drawn from urn II without replacement. What is the probability that both balls drawn from urn II are red?
A.
B.
C.
D.
E.


4. are normal variables with variances and covariance between any pair of variables . What is the variance of



?
A.
B. 2n C.
D.
E.



/P.2
- 2 -

5. Let and be unbiased estimators of


. Suppose , , and


. What are the values of and for which is an unbiased estimator of



with minimum variance among unbiased estimators of this type?
A. = 1, = 0 B.


,
C.
,

D.
,
E.
,


6. Given two samples from a distribution with variance , one of size , the other of size , with (biased estimate) sample variances of and respectively, which of the following are unbiased estimates of ?






I.
II.
III.

A. All but I B. All but II C. All but III D. All E. None

7. Random samples of size 10 and 8 are drawn from normal populations P and Q, respectively. Unbiased estimates of the population variances are computed from these samples and denoted by and . If the variances of the population variances are equal, what is the probability



?
A. 0.025 B. 0.050 C. 0.950 D. 0.975 E. 0.990

8. If Var[X + Y] = 3, Var[X C Y] =1, E[X] = 1, and E[Y] = 2, then what is E[XY] ?
A.
B.
C.
D.
E. 3

9. Let X be the number of successes in a fixed number of repeated trials and let p be the probability of success in any one trial. If the mean and standard deviation of X are equal, which of the