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9. Confidence Interval Estimation 475

interval found in the Exercise 9.3.7 be used to test the scientist’s belief at 5%
level? If not, then find an appropriate 95% confidence interval for µ – µ in
2
1
the Exercise 9.3.7 and use that to test the hypothesis at 5% level.
9.3.9 Two neighboring towns wanted to compare the variations in the time
(minutes) to finish a 5k–run among the first place winners during each town’s
festivities such as the heritage day, peach festival, memorial day, and other
town–wide events. The following data was collected recently by the town
officials:
Town A (x ): 18 20 17 22 19 18 20 18 17
A
Town B (x ): 20 17 25 24 18 23
B
Assume that the performances of these first place winners are independent
and that the first place winning times are normally distributed within each
town’s festivities. Obtain a 90% two–sided confidence interval for σ /σ B
A
based on sufficient statistics for (µ , µ , σ , σ ). Can we conclude at 10%
B
A
A
B
level that σ = σ ?
B
A
9.3.10 Suppose that the random variables X , ..., X are iid having the
i1
ini
common pdf f(x; θ ), i = 1, 2, where we denote the exponential pdf f(x; a) =
i
a exp{–x/a}I(x > 0), a ∈ ℜ . Also, let the X ’s be independent of the X ’s.
+
–1
1j
2j
+
+
We assume that both the parameters θ , θ are unknown, (θ , θ ) ∈ ℜ × ℜ .
1
2
2
1
With fixed α ∈ (0, 1), derive a (1 – α) two-sided confidence interval for θ /θ 1
2
based on sufficient statistics for (θ , θ ). {Hint: Consider creating a pivot out
1 2
of
9.3.11 (Example 9.3.6 Continued) Suppose that the random variables X ,
i1
..., X are iid Uniform(0, θ ), i = 1, 2. Also, let the X ’s be independent of the
in
i
1j
X ’s. We assume that both the parameters are unknown, (θ , θ ) ∈ ℜ × ℜ .
+
+
2
1
2j
With fixed α ∈ (0, 1), derive a (1 – α) two-sided confidence interval for θ /
1
(θ + θ ) based on sufficient statistics for (θ , θ ).
1 2 1 2
9.4.1 (Example 9.4.5 Continued) Suppose that X , ..., X are iid random
i1
in
2
samples from the N(µ , σ ) population, i = 1, ..., 4. The observations X , ...,
i
i1
th
X refer to the i treatment, i = 1, ..., 4. Let us assume that the treatments are
in
independent too and that all the parameters are unknown. With fixed α ∈ (0,
1), derive the (1 – α) joint confidence intervals for estimating the parameters
θ = µ – µ , θ = µ + µ – 2µ .
1 1 2 2 2 3 4
9.4.2 (Example 9.4.4 Continued) Suppose that X , ..., X are iid random
i1
in
2
samples from the N(µ , σ ) population, i = 1, ..., 5. The observations X , ...,
i1
i
th
X refer to the i treatment, i = 1, ..., 5. Let us assume that the treatments
in
are independent too and that all the parameters are unknown. With fixed α ∈
(0, 1), derive the (1 – α) joint ellipsoidal confidence region for estimating the
parameters θ = µ – µ , θ = µ – µ and θ = µ + µ – 2µ .
1 1 2 2 2 3 3 3 4 5