Page 556 - Probability and Statistical Inference

11. Likehood Ratio and Other Tests 533

in the implementable form. {Hint: The LR test rejects H if and only if
0

11.3.7 Two types of cars were compared for the braking distances. Test
runs were made for each car in a driving range. Once a car reached the stable
speed of 60 miles per hour, the brakes were applied. The distance (feet) each
car travelled from the moment the brakes were applied to the moment the car
came to a complete stop was recorded. The summary statistics are shown
below:

Car Sample Size s
Make A n = 12 37.1 3.1
A
Make B n = 10 39.6 4.3
B
Assume that the elapsed times are distributed as N(µ , σ ) and N(µ , σ )
2
2
A
B
respectively for the make A and B cars with all parameters unknown. Test at
5% level whether the average braking distances of the two makes are signifi-
cantly different.
11.3.8 Verify the expressions of the MLE’s in (11.3.15).
11.3.9 Let the random variables X , ..., X be iid N(0, ), i = 1, 2, and that
i1
ini
the X ’s be independent of the X ’s. Here we assume that (µ , µ ) ∈ ℜ × ℜ are
2
1j
1
2j
known but (σ , σ ) ∈ ℜ × ℜ are unknown. With preassigned α ∈ (0, 1),
+
+
1
2
derive a level α LR test for H : σ = σ versus H : σ ≠ σ in the implementable
0
1
1
1
2
2
form.
11.3.10 Let the random variables X , ..., X be iid N(µ , ), i = 1, 2, and
i1 ini i
that the X ’s be independent of the X ’s. Here we assume that µ , µ ) ∈ ℜ ×
2
1
2j
1j
ℜ are known but (σ , σ ) ∈ ℜ × ℜ are unknown. With preassigned α ∈ (0,
+
+
2
1
1), derive a level α LR test for H : σ = σ versus H : σ ≠ σ in the
1
1
2
0
2
1
implementable form.
11.3.11 Let the random variables X , ..., X be iid N(µ , ), i = 1, 2, and
i1
i
ini
that the X ’s be independent of the X ’s. Here we assume that µ , µ , s , s are
2
1
1
2j
1j
2
2
all unknown and (µ , µ ) ∈ ℜ , (σ , σ ) ∈ ℜ . With preassigned α ∈ (0, 1)
+2
1
1
2
2
and a positive number D, show that the level α LR test for H : σ /σ = D
1
0
2
versus H : σ /σ ≠ D, would reject H if and only if > F n1 - 1,n2 - 1,α/
1
2
1
0
or < F . {Hint: Repeat the techniques from Section
2 n1 - 1,n2 -1,1 - α/2
11.3.2.}
11.3.12 Let the random variables X , ..., X be iid Exponential(θ ),
n1
i
ini
i = 1, 2, and that the X ’s be independent of the X ’s. Here we assume
1j
2j
that θ , θ are unknown and (θ , θ ) ∈ ℜ × ℜ . With preassigned α ∈
+
+
2
1
2
1
(0, 1), derive a level α LR test for H : θ = θ versus H : θ ≠ θ in the
0 1 2 1 1 2