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518 11. Likehood Ratio and Other Tests

From Example 4.5.2 recall that has the
Student’s t distribution with n + n – 2 degrees of freedom. Thus, in view of
2
1
(11.3.10), the implementable form of a level α LR test would be:

See the Figure 11.3.1. Note that the LR test rejects H when is
0
sizably different from zero with proper scaling.

Look at the Exercises 11.3.2-11.3.3 for a LR test of the equality
of means in the case of known variances.

Look at the Exercises 11.3.4-11.3.6 for a LR test to choose between
H : µ – µ = D versus H : µ – µ ≠ D.
0 1 2 1 1 2
Example 11.3.1 Weekly salaries (in dollars) of two typical high-school
seniors, Lisa and Mike, earned during last summer are given below:

Lisa: 234.26, 237.18, 238.16, 259.53, 242.76, 237.81, 250.95, 277.83
Mike: 187.73, 206.08, 176.71, 213.69, 224.34, 235.24

Assume independent normal distributions with unknown average weekly sala-
2
ries, µ for Lisa and µ for Mike, but with common unknown variance σ . At
L
M
5% level we wish to test whether the average weekly salaries are same for
these two students, that is we have to test H : µ = µ versus H : µ ≠ µ .
M
L
0
M
L
1
One has
so that the pooled sample variance
From (11.3.11), we find the observed value of the test statistic:

With α = .05 and 12 degrees of freedom, we have t 12,.025 = 2.1788. Since
|t | exceeds t 12,.025 , we reject the null hypothesis at 5% level. At 5% level,
calc
we conclude that the average weekly salaries of Lisa and Mike were signifi-
cantly different. !