STAT 4101 Quiz, Spring 2020
First name: Last name:
Hawkid: @uiowa.edu
Please read the following instructions before you start the exam.
• There are three pages including the cover page.
• The quiz is from the noon on Feb 12th, 2020 to the noon on Feb 15th, 2020.
• The deadline
• There are 2 problems. The total is 10 ∗ 2 = 20 points.
• Show all work for each problem below. If you only present a correct answer without stating how you get it, no credit will be given.
• Calculator ready solutions are not sufficient to obtain full credits. A numer- ical answer is needed if a problem asks for it.
• You must stop writing immediately after the invigilator signals the end of the exam.
• Any distribution of this material is prohibited.
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1. (10 pts) Let
√1 θ with probability
2 X=√ 1
1 + θ with probability 2
for some θ > 0. Find an unbiased estimator of θ based on a single observation X.
Solution: We see that
E X2 = θ · 1 + (1 + θ) · 1 = θ + 1. 222
Therefore X2 − 1 is an unbiased estimator of θ. 2
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2. (10 pts) Suppose the visit duration of STAT 4101 office hours is exponentially distributed with the mean of β (minutes.) Suppose the average visit duration of 25 students is 14.5 minutes. Construct a 95% large-sample confidence interval for β.
Solution: Since X has mean β and variance β2, by central limit theorem, we see √25(X ̄ − β) ∼ N(0, 1),
β
approximately. Then a 95% large-sample confidence interval is √25(X ̄ − β)
which is equivalent to
P (−1.96 < β < 1.96) ≈ 0.95, P (1.689X ̄ < β < 5.208X ̄ ) ≈ 0.95,
By plugging in X ̄ = 14.5, we have (10.417,23.849), so we are 95% confidence that the true mean β is between 10.417 and 23.849 minutes.
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End of the exam!
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