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11. Likehood Ratio and Other Tests 515

from the center is measured in inches. A player made 7 attempts to hit the
bull’s eye and the observed x values were recorded as follows:

2.5, 1.2, 3.0, 2.3, 4.4, 0.8, 1.6
Assume a normal distribution for X. We wish to test H : σ = 1 against H : σ
1
0
≠ 1 at 5% level. One obtains inches and s = 1.2164 inches. From
(11.2.21), we have the observed value of the test statistic:

With α = .05 and 6 degrees of freedom, one has and
Since lies between the two numbers 1.2373 and
14.449, we accept H or conclude that there is not enough evidence to reject
0
H at 5% level. !
0
Example 11.2.3 A preliminary mathematics screening test was given to a
group of twenty applicants for the position of actuary. This group’s test scores
(X) gave and s = 15.39. Assume a normal distribution for X. The
administrator wished to test H : σ = 12 against H : σ ≠ 12 at 10% level. From
0
1
(11.2.21), we have the observed value of the test statistic:

With α = .10 and 19 degrees of freedom, one has χ 2 19,.95 = 10.117 and χ 2 19,.05 =
30.144. Since lies outside of the interval (10.117, 30.144), one should
reject at 10% level. !

11.3 Two-Sample Problems

We focus on two independent normal populations and some associated likeli-
hood ratio tests. With fixed α ∈ (0, 1), first a level α LR test is derived for the
equality of means against a two-sided alternative hypothesis when the com-
mon population variance is unknown and come up with the customary two-
sided t-test which uses the pooled sample variance. Next, we derive a level a
LR test for the equality of variances against a two-sided alternative hypothesis
when the population means are unknown and come up with the customary
two-sided F-test.

11.3.1 Comparing the Means
Suppose that the random variables X , ..., X are iid N(µ , σ ), i = 1, 2,
2
i
i1
ini
and that the X ’s are independent of the X ’s. We assume that all three
1j 2j