Analysis of Variance and Contrast Test for Conductivity Data and Adhesion Properties

1. A manufacturer of television sets is interested in the effect on tube conductivity of four different types of coatings for color picture tubes. A completely randomized experiment was conducted and the following conductivity data is obtained:

(a)Carry out an analysis of variance by hand (without using any computer software and compute SSTotal, SSTreatment, and SSError and complete the ANOVA table on the next page. You need to show all important steps in your computations.

(b)Based on the ANOVA results could you conclude that the mean conductivity of at least coating type is different from the others? Use 0.05 significance level. Give statistical reasons for your conclusion.

(d) Carry out a contrast test to determine if the average of the mean conductivities of coating types 1 and 2 is different from the average of the mean conductivities of types 3 and 4. Please do these computations by hand and not resort to the use of JMP or SAS. Use significance level 0.05. 

2. Aircraft primers are applied to aluminum surfaces for the purpose of improving paint adhesion. An engineer is interested in learning whether three types of primers differ in their adhesion properties. To answer this question, a one-way experiment was conducted with primer type as the single factor. The application of primer to a specimen (aluminum surface) usually takes 10 minutes. Since relative humidity and temperature can have an effect on adhesion, the experiment was conducted using an RCB design with 30 minute segments of time used as blocks. Within each thirty minute segment, three specimens were painted with the three primers in random order and the primers were assigned randomly to these three specimens. A new randomization was done in each block. The force required to peel the paint off each specimen was then measured after one week. The data from the experiment is given below:

(a)Write down the classical model that is appropriate for this experiment. Please label all symbols you use and indicate any distributional assumptions. You may assume that the block effect is random.

(b)see if there is strong evidence to suggest that the mean adhesive strength associated with the three primer types are not all equal. Please use the parameters of the classical model in specifying the hypotheses.

(c)Use JMP to analyze the data. Copy and paste the ANOVA results from JMP below.

(d)Using results in the above ANOVA table please test the hypotheses you specified in part (b) above. What is your statistical conclusion? Please re-state your results in lay terms.

(e)Carry out a mean comparison using the Tukey’s method. Please state your conclusions based on this mean comparison. Specifically, which primer type would you recommend and why?

3. An article in Biotechnology Progress (2001, Vol. 17, pp. 366-368) described an experiment to investigate Nisin extraction in aqueous two-phase solutions. A two-factor experiment in a CR design was conducted using factors A = concentration of PEG and B = concentration of Na2SO4. Data similar to that reported in the paper are shown below.

(a)Compute the ANOVA table for this experiment using JMP. Copy and paste it below (make sure you include SS due to main effects and interaction.)

(b)Test all relevant hypotheses in logical order and state your conclusions. Please copy and paste or write down the appropriate statistics you used from JMP output but please do not paste unrelated results.

(c) Please explain what post-ANOVA analysis you would carry out to determine the combination of factor levels that would maximize extraction percentage. You need not carry out this procedure, but just explain it.