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11. Likehood Ratio and Other Tests 535

Fieller-Creasy problem. Refer to Creasy (1954) and Fieller (1954). Both these
problems were elegantly reviewed by Kendall and Stuart (1979) and Wallace
(1980). In the Exercise 13.2.10, we have given the two-stage sampling tech-
nique of Chapman (1950) for constructing a fixed-width confidence interval
for µ – µ with the exact confidence coefficient at least 1 – α.}
1 2
11.4.1 A nutritional science project had involved 8 overweight men of
comparable background which included eating habits, family traits, health
condition and job related stress. An experiment was conducted to study the
average reduction in weight for overweight men following a particular regi-
men of nutritional diet and exercise. The technician weighed in each individual
before they were to enter this program. At the conclusion of the study which
took two months, each individual was weighed in again. It was believed that
the assumption of a bivariate normal distribution would be reasonable to use
for (X , X ).
2
1
Test at 5% level whether the true average weights taken before and after
going through the regimen are significantly different. At 10% level, is it pos-
sible to test whether the true average weight taken after going through the
regimen is significantly lower than the true average weight taken before going
through the regimen? The observed data is given in the adjoining table.

ID# of Weight (x , pounds) Weight (x , pounds)
1
2
Individual Before Study After Study
1 235 220
2 189 175
3 156 150
4 172 160
5 165 169
6 180 170
7 170 173
8 195 180

{Hint: Use the methodology from Section 11.4.1.}
11.4.2 Suppose that the pairs of random variables (X , X ) are iid bivariate
2i
1i
normal, N (µ , µ , ρ), i = 1, ..., n(≥ 2). Here we assume that µ , µ are
1
2
2
2
1
unknown but ρ are known where (µ, σ ) ∈ ℜ × ℜ , l = 1, 2 and –1 < ρ
+
l
l
< 1. With fixed α ∈ (0, 1), construct a level α test for a null hypothesis H : µ 1
0
= µ against a two-sided alternative hypothesis H : µ ≠ µ in the implementable
2
1
2
1
form. {Hint: Improvise with the methodology from Section 11.4.1.}
11.4.3 Suppose that the pairs of random variables (X , X ) are iid
2i
1i
bivariate normal, N (µ , µ , σ , σ , ρ), i = 1, ..., n(≥ 2). Here we assume
2
2
2 1 2