STATS 120C University of California Irvine Probability Statistics Questions

Please make the answers either handwritten or typed in .doc/pdf

1. Suppose we have the following unbalanced two-way ANOVA modeliid 2

yijk =?+?i+?j+?ij+?ijk ?ijk ?N(0,? )

where i = 1, 2, 3 and j = 1, 2 denote, respectively, the number of levels for factor A and the number of levels for factor B. Suppose that the number of observations (Ki j ) per group is:

Factor B FactorA j=1 j=2

i=145i=234i=343

e.g. we have recorded 4 observations at level 1 of factor A and level 1 for factor B, 5 observations at level 1 of factor A and level 2 for factor B, etc.

1. (a) State explicitly what are the constraints imposed on the ?i’s, the ?j’s and the ?i j ’s by the two-way ANOVA model in this situation.
2. (b) Provide an interpretation of the two-way ANOVA parameters ?i, ?jand ?i j in a way that is understandable by a non-statistician.
3. (c) DerivetheMLE’softhetwo-wayANOVAparameters?,?1,?2,?3,?1,?2,?11,?21,…,?32

2. Suppose that the following data represent the units of production turned out each day by 3 different machinists, each working on the same machine for three different days:

Machine

1. B1 15,15,17
2. B2 17,17,17
3. B3 15,17,16
4. B4 18,20,22

Machinist B 19,19,16 15,15,15 18,17,16 15,16,17

A

C 16,18,21 19,22,22 18,18,18 17,17,17

Using a 0.05 level of significance, test whether

1. (a) the four machines resulted in the same average production amount
2. (b) the machinist yield the same average production amount
3. (c) the effect of the machine on the average production amount does not depend on the machinist

For each of the tests above: state the null and alternative hypothesis, report the value of the test statistic (show how you obtained it), report the p-value, state your decision and state your conclusion.