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The Final Exam of Probability & Statistics (I)
Department of Information Management
Date: 10 January 2005
09:30~11:30
NOTICE!
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You can use the calculator for some complicated computations. However, the Notebook
computers, PDAs, and Smart phones will not be allowed.
Answer the following questions in the closed-book context and honesty.
Any evidence of cheating will result in a failing grade for the course and further disciplinary
action through the appropriate Chi Nan procedures and committees.
I. Short answer
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7.
Brief the central limit theorem. (5%)
Compare the differences/similarities between the Poisson distribution and the exponential distribution.
(5%)
Compare the standard normal distribution and the t distribution. (5%)
What circumstances will drive you to compute the pooled variance estimator S p 2 in order to estimate
the difference in means of two normal populations? (5%)
A statistics student calculates a certain 95% confidence interval by a set of sample data and asserts that
the actual population mean will fall into this range with the probability of 0.95. Do you agree what this
student stated? (5%)
Brief the maximum likelihood approach for finding the parameter estimator. (5%)
(a) What is the so-called unbiased estimator? (5%) (b) Is the maximum likelihood estimator of the

n
^
normal population variance σ2 unbiased? (i.e., the MLE,  2 
 (x
i 1
i
 X )2
n
) (5%) (c) If it is biased,
please propose another unbiased estimator of σ2. (5%)
II. Computation problem
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If X is binomial with parameters n=150, p=0.6, compute the value of P{X≦80} with its normal
approximation making use of the continuity correction. (10%)
A random sample of 16 full professors at a large university yielded a sample mean annual salary of
$90,450 with a known standard deviation of $9,400. Determine a 95% upper interval of the average
salary of all full professors at that university (10%)
If the standard deviation in the second question is unknown and replaced by a sample standard
deviation of $9,400, please also determine a 95% upper interval of the average salary of all full
professors at that university. (5%) Compare this result with the answer of the second question. (5%)
A random sample of 100 items from a production line revealed 17 of them to be defective. Compute a
95 percent two-sided confidence interval for the probability that an item produced is defective. (10%)
What assumptions are you making? (5%)
A poll states with 95% confidence that the 50 percent of the votes concern the Internet terrorism. And
this report also reveals that it was surveyed from a successful sample with the size of 1068 so as to
reduce the margin of error within ±3%. Please prove that the sample size 1068 is necessary for such a
precise estimation. (10%)
If the survey result of the fifth question is 30 percent, how will the margin of error change? (5%)
Reference: Z0.95=1.645, Z0.975=1.96, Z0.995=2.58, t0.05,15=1.753, t.05,16=1.746, t.025,15=2.131