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MATH 144 Applied Statistics
Final Examination (December 2008)

NAME: Signature: Student ID: Tutorial Session:

Instructions:
1.
Do NOT turn over the examination paper until you are told to do so.

2.
Turn o. all the communication devices during the examination.

3.
This is a closed book and close notes examination, but two 8.5 by 11 inch formula sheets are allowed.

4.
Check that there are 7 pages in addition to this cover page. Two draft papers, Table

A.3 and Table A.4 are provided at the end of this paper.

5.
Answer ALL questions in 3 hours.

6.
All answers should be either exact or correct to 4 decimal places.

7.
Cheating is a serious o.ense. Students who commit the o.ense may score no mark in the examination. Furthermore, more serious penalty may be imposed.


Question Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Total Marks
Marks

1. (a) (8 points) The joint probability density function of (X, Y ) is given by

y
-10 1 0
0.2 0.1 0.1 x 1
0.1 0.05 0.15 2
0.1 0.15 0.05
Find
(i)
the probability density function of Y ;

(ii)
E(Y );


(iii) P (X + Y< 1);
(iv) E(XY ).
(b)
(4 points) Let X B(100, 0.4) be a binomial random variable. Use normal approximation with continuity correction to .nd P (|X . E(X)| X ), where X is the standard deviation of X.

(c)
(3 points) Let X and Y be independent random variables. Assume that X N(0, 1) and Y N(1, 3). Find P (X>Y ).

(d)
(3 points) Consider one sample z-test of the hypothesis H0 : = 1 against H1 : < 1. Assume z =0.5, what is the P-value for the test?

(e)
(3 points) The one-sample t statistic from a random sample of n = 19 observations from a normal population for the two-sided test of


H0 : = 6 against .
H1 : =6
has the value t =2.2. What is the range of P-value? Can you reject the null hypothesis at the 0.1 level of signi.cance?
(f) (3 points) Let X1, ,Xn be a random sample from a normal population N(0,2)
and let W = n 1 in =1 Xi 2 . Prove that W is an unbiased estimator for 2 .
2. (a) (7 points) The joint density function of X and Y is given by

x + y for 0 <x< 1, 0 <y< 1

fX,Y (x, y)=
. .. ..
0 elsewhere

(i)
Find the probability density function of X;

(ii)
Find E(Y |X =0.5);


(iii) Find Cov(X, Y );
(iv) Are X and Y independent?
(b) (5 points) Let X1 and X2 be independent random variables, each with uniform distribution on (0, 1). Let Y = max(X1,X2).
(i)
Find the probability density function of Y ;

(ii)
Find E(Y ).


3. (10 points) The weight of a randomly selected can of a new soft drink is known to have a normal distribution with mean 12.1 ounces and a standard deviation of 0.1 ounce.
(i)
What is the probability that if a can is drawn at random, it weights between 11.9 and 12.3 ounces?

(ii)
What weight should be printed on the can so that the average weight of cans in a six pack is underweight for only 1% of