1. (50 points) Cholesterol was measured in the serum samples of five randomly selected patients from a pool of patients. Two independently prepared repli- cate tubes were prepared for each patient for each of spectrophotometer from four brands. The objective of the study was to determine whether the relative cholesterol measurements for patients were consistent for four brands. The data mg/dl of cholesterol in the replicate samples from each patient on each brand.
Patient?Brand XXXXXXXXXX
XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX148.5?
XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX154.7?
XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX145.9?
XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX151.0?
(1) From the statement above identify the following experimental design com- ponents:
? Experimental unit?
? Treatments?
? Fixed or random levels of the treatments
. (2) Write the model to study cholesterol measurements on patients and brands. You need to fully specify model components and the distributions of ran- dom terms.?
. (3) Find out the estimates of the parameters in the model specified in part (2).?
. (4) Test if the main effects and interaction effect in the model are significantly different from zero at level a = 0.05.?? Specify hypotheses H0and H1for each effect.?? Test the hypotheses (reject or accept H0) and present the p-values for?each effect.?
. (5) Examine the following assumptions of the error term in the model.?? Independence: Show the test statistic and the decision?? Constant variance: Check with residual plots?? Normality: Test statistics and test under 5% significance level?
2. (50 points) A research specialist for a large seafood company investigated bac- terial growth on oysters and mussels subjected to three different storage tem- peratures. Nine cold storage units were available. Oysters and mussels were stored for two weeks in each of the cold storage units. A bacterial count was made from a sample of oysters and mussels at the end of two weeks. Hence, with 3 replicates, the first fixed treatment applied to a batch of 2 storage units and then the batch has been split into 2 storage units to apply the second fixed
2
treatment. The logarithm of bacterial count for each sample is shown in the following table
Temperature (?C) 0
5
10
Seafood Oyster Mussel XXXXXXXXXX XXXXXXXXXX.5780
XXXXXXXXXX XXXXXXXXXX3861
XXXXXXXXXX XXXXXXXXXX11.0329
. (1) From the statement above identify the following experimental design com- ponents:?? Experimental units? Treatments?
. (2) Write the model to study the bacterial counts on temperature and seafood with full specification of model components.?
. (3) Construct the analysis of variance (ANOVA) table with a column of the expected mean square (EMS). Use the auxiliary table to find the EMS for full credit.?
3
. (4) Test the significances of treatment effects in the model at the level a = 0.01.?? Specify hypotheses H0and H1for each effect.?? Test the hypotheses (reject or accept H0) and present the p-values for?each effect.?
. (5) Theresearchistoinvestigatebacterialgrowthonoystersandmusselsunder different storage temperatures. Briefly write a conclusion with respect to the research object.?