1. Describe how you would test one-sided and two sided hypotheses, using both finite sample and asymp totically valid testing procedures, in each of the follow ing cases. Use whatever procedures you feel are appro priate.
a. Testing hypotheses about the mean of a binomial distribution based on a random sample of size 1 from the binomial distribution.
b. Testing hypotheses about the mean of a Poisson dis tribution based on a random sample of size nfrom a Poisson population distribution.
2. The weekly proportion of storage tank capacity that is utilized at a regional hazardous liquid waste receiv ing site is an outcome of a Beta(8, 1) random variable. At the end of each week the storage tank is emptied. A random sample of a year's worth of observations of capacity utilization at the site produced the following in sequence rowwise:
|
0.148
|
0.501
|
0.394
|
0.257
|
|
0.759
|
0.763
|
0.092
|
0.155
|
|
0.409
|
0.257
|
0.586
|
0.919
|
|
0.278
|
0.076
|
0.019
|
0.513
|
|
0.123
|
0.390
|
0.030
|
0.082
|
|
0.935
|
0.607
|
0.075
|
0.729
|
|
0.664
|
0.802
|
0.338
|
0.539
|
|
0.055
|
0.984
|
0.269
|
0.069
|
|
0.679
|
0.382
|
0.549
|
0.028
|
|
0.770
|
0.296
|
0.105
|
0.465
|
|
0.194
|
0.675
|
0.696
|
0.068
|
|
0.091
|
0.132
|
0.156
|
0.050
|
|
0.477
|
0.754
|
0.164
|
0.527
|
a. Define a GLR size a test of Ho : e = 80 versus Ha : 8 =/: 80.
b. Test the hypothesis that the expected weekly stor age tank capacity utilized is equal to .5 at a sig nificance level of .05. If the exact distribution of the GLR test appears to be intractable, then use an asymptotic test.
c. Test that the observations might be interpreted as a random sample from some population distribution. Use a size .OS runs test.
d. Test that the observations can be interpreted as a random sample from a Beta(8, 1) population distri bution. Use a size .OS test.
e. Would the use of an LM test be tractable here? If so, perform an LM test of the hypothesis.