HOMEWORK 5 due April 13


I. In the context of problem II of homework 4, 

For the parameter value (\lambda=2, \alpha=1.5).
Is it inside the 90% confidence region or outside?

What about 80% confidence region?

(which percentage confidence region has the 
parameter (\lambda=2, \alpha=1.5) lie exactly
on the border line?)

Find the P-value.

II. Suppose we have three groups of independent normal
observations:

group 1  with distribution   Normal $(\mu_1, \sigma_1 )$

group 2  with distribution   Normal $(\mu_2  \sigma_2 )$

group 3  with distribution   Normal $(\mu_3  \sigma_3 )$


One theory (the smaller model)
postulates that the means of the three groups
are equally spaced. That is 
\[
\mu_1 = \beta , \mu_2 = 2 \beta , \mu_3 = 3 \beta
\]
for some unknown $\beta$. So for this model, $\beta$ and the three
$\sigma$'s are the four parameters.

The bigger model is that all three $\mu$'s and three $\sigma$'s
are 6 unknown parameters.

We may put this into a hypothesis.

H_0 : the smaller model.

H_A : the bigger model minus the smaller model.


Find the P-value for this testing hypothesis problem given 
the following data, using the Wilks likelihood ratio method:

group 1:

 [1]  0.7929343  2.2774292  3.0844412 -0.3456977  2.4291247  2.5060559
 [7]  1.4252600  1.4533681  1.4355480  1.1099622  1.5228073  1.0016136
[13]  1.2237461  2.0644588  2.9594941  1.8897145  1.4889905  1.0888046
[19]  1.1628283  4.4158352  2.1340882  1.5093141  1.5594521  2.4595894
[25]  1.3062798  0.5517951  2.5747557  0.9763443  1.9848617  1.0640514
[31]  3.1022975  1.5244069

group 2:

 [1] 2.9358399 3.2481129 1.5563598 2.2485711 0.7299405 1.9885102 3.5585592
 [8] 3.3011537 6.1742444 2.3970359 2.7169530 3.5790655 2.5084899 2.5472285
[15] 2.3390227 2.1220212 3.2142578 3.2547251 1.2909531 3.1268861

group 3:.

 [1] 4.669332 4.782046 5.805229 6.675667 7.977381 5.071976 7.927092 4.610630
 [9] 6.787906 9.058789 5.958288 5.196440 5.990874 8.132501 4.633671 7.641393
[17] 7.595478 6.403767 6.008271 5.453438 5.560171 6.777944 8.484325 5.815922
[25] 4.331159