„¦BASIC MPH WRITTEN EXAMINATION IN BIOSTATISTICSÈ


¦PART IÈ


¦MAY 12, 1994:  9:00 AM - 12:00 PMÈ






‚INSTRUCTIONS:ƒ


a)™This is a ‚open bookƒ examination.


b)™Answer any ‚twoƒ questions during the three-hour time period.


c)™Put the answers to different questions on separate sets of paper.


d)™Put your code letter, ‚not ƒyour name, on each page.


e)™Return the examination with a signed statement of the honor pledge on a 

™page separate from your answers.


f)™You are required to answer only what is asked in the questions, not to tell all you ™

™know about the topics.…
þì1.™Let X¬1È, X¬2È, ..., X¬nÈ be a random sample from a N(Ûm;, 1) population.  Suppose that it ™
™is desired to test  H¬0È: Ûm; = 0  versus  H¬1È: Ûm; = Ûm;¬1È > 0,  where Ûm;¬1È is a known positive ™
™number.


4 pts.™™
a)™
™Show that the most powerful test of size Ûa; for testing  H¬0È  versus  H¬1È ™
þì™™
™involves a rejection region R of the form
™™
™™¦R = {Xµ_È : Xµ_È Û;2 k¬1Û,2Ûa;È} ,È
™™
™where Xµ_È = º1ÂnË   ÛS;µnȶi=1È X¬iÈ and where k¬1Û,2Ûa;È is chosen so that
™™
™ pr(Xµ_È Û;2 k¬1Û,2Ûa;È Û|8 H¬0È) = Ûa;.
þì
4 pts.™™
b)™
™For Ûa; = 0.025, what is the value of k¬.975È when n = 100 ?

2 pts.™™
™          c)™™
™Is the test developed in part (a) a uniformly most powerful (UMP) test 
™™
™of size Ûa; ?

7 pts.™™
d)™Suppose that we decide to reject H¬0È when Xµ_È Û;2 0.80.  
™™
™What is the „power… of this test when Ûm;¬1È = 1 and n = 25 ?

8 pts.™™
e)™As in part (d), again suppose that we decide to reject H¬0È when Xµ_È Û;2 
™™
™0.80.  What is the minimum sample size required to insure that we 
™™
™reject H¬0È: Ûm; = 0 with a probability of at least 0.975 when, 
™™
™in fact, Ûm;¬1È = 1 ?



2.™Suppose that the height X (in inches) for adult males in the United States is 
™assumed to be a N(70, 16) random variable, and that the height Y (in inches) for 
™United States adult females is assumed to have a N(65, 9) distribution.
™Assume that males and females are paired randomly without regard to height.

5 pts.™™
a)™In what proportion of pairs will there be a height difference of at least 
™™
™six inches ?

5 pts.™™
b)™What is the probability that the male in a pair will be taller than the 
™™
™female ?

5 pts.™™
c)™If the male and female heights for a pair differ by more than six inches, 
™™
™what is the probability that the male is taller?

10 pts.™™
d)™Now, suppose that X Û[2N(Ûm;¬mÈ, 16) and that Y Û[2N(Ûm;¬fÈ, 9),
™™
™ where Ûm;¬mÈ and Ûm;¬fÈ are „unknown… parameters.  It is desired to make a 
™™
™statistical inference about the unknown population mean height 
™™
™difference (Ûm;¬mÈ Û,2 Ûm;¬fÈ).  Let X¬1È, X¬2È, ..., X¬nÈ and Y¬1È, Y¬2È, ..., Y¬nÈ 
™™
™constitute random samples of size n from each of these two normal
™™
™populations.  Let Xµ_È = n«-1È ÛS;µnȶi=1ÈX¬iÈ and Yµ_È = ÛS;µnȶi=1ÈY¬iÈ.  
™™
™Find two random variables A and B, which are to be functions
™™
™ of (Xµ_È Û,2 Yµ_È) and V(Xµ_È Û,2 Yµ_È), such that
™™
™™™pr{A < (Ûm;¬mÈ Û,2 Ûm;¬fÈ) < B} = 0.90 .




3.™Suppose that we have obtained a sample of n pairs of observations (X¬1È, Y¬1È),
™(X¬2È,  Y¬2È), ..., (X¬nÈ,Y¬nÈ).  Suppose we are interested in the simple linear relationship:
™™
Y¬iÈ = Ûh;¬0È + Ûh;¬1ÈX¬iÈ + Ûe;¬iÈ ,   i = 1, 2, ..., n,
™where
™™
E(Ûe;¬iÈ) = 0,  i = 1, 2, ..., n
™™
Var(Ûe;¬iÈ) = Ûs;«2È, i = 1, 2, ..., n
™™
Cov (Ûe;¬iÈ, Ûe;¬jÈ) = 0, 1 Û:2 i < j Û:2 n
™™
X¬iÈ's are known, fixed constants

5 pts.™™
a)™The least squares estimators of Ûh;¬0È and Ûh;¬1È are defined to be the values of 
þì™™
™Ûh;¬0È and Ûh;¬1È which minimize the following:
™™
™™ÛS;µnȶi=1È (Y¬iÈ Û,2 Ûh;¬0È Û,2Ûh;¬1ÈX¬iÈ)«2È  .
þì™™
™Derive expressions for the least squares estimators of Ûh;¬0È and Ûh;¬1È.  Be
™™
™careful to show all work.

10 pts.™™
b)™Show whether or not the least squares estimators derived in (a) are 
™™
™unbiased estimators of Ûh;¬0È and Ûh;¬1È.  Be careful to show all work.

5 pts.™™
c)™Derive the variance of the least squares estimator of Ûh;¬1È.

5 pts.™™
d)™Suppose Ûe;¬iÈ Û[2 N(0, Ûs;«2È).  Using (c), explain how you would test:
™™
™™H¬0È: Ûh;¬1È = 0    vs    H¬AÈ: Ûh;¬1È ÛW20

™™
™Carefully state all assumptions.

™™