Final Review
Q1. LetX1 , X2 , · · · , Xn be an independent random sample from the uniform distribution
U (0, θ).
(a) Find the probability density function of X(n) .
(b) Construct a pivotal variable based on the sample maximum X(n) .
(c) Use the result in part (a) and (b), to construct a 1 − α confidence interval for θ.
Q2. Let X1 , X2 , · · · , Xn be an independent random sample from Uniform[θ, 0] for θ < 0.
Let X(1) be the smallest value of the sample {Xi }ni=1 .
X(1)
(a) Show that T = is a pivotal quantity.
θ
(b) Give a 1 − α equal-tailed confidence interval for θ.
Q3. Suppose that 9 persons are randomly selected, and their height (in cm) is measured in the
morning and evening. Let X to denote the height measured in the morning, and Y to denote the
height measures in the evening. The observed values for this sample are given in the following
table:
Number (i) 1 2 3 4 5 6 7 8 9
Height in the morning (xi ) 170 175 160 161 165 172 170 180 174
Height in the evening (yi ) 170 174 158 160 163 172 168 177 172
Suppose that X ∼ N(µx , σx2 ), Y ∼ N(µy , σy2 ). Assume the variance for the height in the
morning and evening are equal.
(a) Find a 95% confidence interval of µx .
(b) Find a 95% confidence interval of µx − µy .
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(c) Is it reasonable to conclude that the height in the morning is higher than that in the evening
using 5% significance level?
Q4. Suppose 10 tires are randomly selected from Tire Factory A, and life times (unit: 104 km)
of this random sample are recorded as follows:
4.6, 5.0, 5.5, 4.2, 4.3, 4.7, 5.2, 4.1, 4.8, 5.3.
At the same time, 12 tires are randomly selected from Tire Factory B, and life times of this
random sample are recorded as follows:
4.5, 5.2, 5.3, 5.8, 5.3, 4.2, 4.2, 4.8, 5.5, 5.6, 5.6, 5.0.
Suppose that tire life times in the two factories follow some normal distributions with equal
variance. Answer the following questions at the significance level α = 0.1.
(a) Is it suitable for Factory A to claim that the average life time of tires it produces is not less
than 5 × 104 km? State the reason.
(b) Find a 90% confidence interval estimate for the variance of tire life times in Factory A,
2.
i.e., σA
(c) Find a 90% confidence interval estimate for the ratio of variance of tire life times in Factory
2 /σ 2 .
A and B, i.e., σA B
(d) Does Factory B produce higher quality tires than Factory A? State the reason.
Q5. Let X1 , X2 , · · · , Xn be an independent random sample from a Laplace(θ) distribution,
which has the density given by
1 |x|
f (x; θ) = exp − for − ∞ < x < ∞.
2θ θ
Here, θ > 0 is a parameter. Note that if X ∼ Laplace(θ), EX = 0, E|X| = θ and E(X 2 ) = 2θ2 .
(a) Find the MLE of θ on Ω1 = {θ : θ > 0}.
(b) Calculate the CRLB with respect to θ.
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(c) Is the MLE in (ii) the UMVUE of θ?
(d) Find the rejection region of the most powerful test for hypotheses:
H0 : θ = 1 versus H1 : θ = 2.
Q6. Suppose that X1 , X2 , · · · , Xn is an independent random sample from a distribution with
the following density:
e−(x−θ) , x ≥ θ,
f (x; θ) =
0, otherwise.
Here, −∞ < θ < ∞ is a parameter.
(a) Find a (non-trivial) sufficient statistic for θ.
(b) Based on the sufficient statistic in (i), find an unbiased estimator of θ (denoted by T1 ).
(c) Is T1 a consistent estimator of θ?
(d) Based on the sample mean X, find another unbiased estimator of θ (denoted by T2 ).
(e) Is T2 a consistent estimator of θ?
(f) Compare the efficiency of T1 and T2 .
Q7. Let X1 , X2 , · · · , Xn be an independent random sample from N (µ, 1). Suppose that a test
for the hypotheses:
H0 : µ = 0 versus H1 : 1 ≤ µ ≤ 2
( n )
X
has the rejection region Xi ≥ c for some constant c.
i=1
(a) Determine the value of c such that this test has size 0.05.
(b) Based on the value of c in (i), find the maximum and minimum power of this test under
H1 .
