Question

4. An experiment is designed study pigment dispersion in paint. Four different mixes of a particular to pigment studied. The procedure consists of preparing a particular mix and then applying that mix are panel by three application methods (brushing, spraying, and rolling). The response measured to a is the percentage reflectance of the pigment. Three days required to run the experiment, and the are data obtained follow. Analyze the data and draw conclusions, assuming that mixes and application methods are fixed. 2 Stat 341: Homework 5 SPRING, 2019 Mix Day App Method 2 3 4 64.5 66.3 74.1 66.5 2 68.3 69.5 73.8 70.0 3 70.3 73.1 78.0 72.3 2 1 65.2 65.0 73.8 64.8 2 69.2 70.3 74.5 68.3 3 71.2 72.8 79.1 71.5 3 66.2 66.5 72.3 67.7 2 69.0 69.0 75.4 68.6 3 70.8 74.2 80.1 72.4

Answer 1

(a) (3 pts) What design is this? Split-plot design (b)_(6 pts) State the statistical model and the corresponding assumptions yijk aj + (ya)ij + Bk + (aß) jk + Eijk, i = 1, ..,3,j = 1,... 4, k 1,..3 Ye-N(0.),a ΣΑ-Σ0ΣΟ aj = 0, (ya)i~N(0, 07a) (aß)jk (aß) jk k 0, and Eijk~N(0,02) (c) Make conclusions and check model adequacy. ANOVA table is shown below. Both Mix and Method factors are significant at alpha 0.05, and their interaction is slightly significant (pv-value 0.06). There is no unusual pattern noticed in QQ normality plot or residual plot Type 3 Tests of Fixed Effects Pr F Effect Num DF Den DF F Value Mix 6 135.77 <.0001 2 165.30 <.0001 Method 16 Mix*Method 2.49 0.0678 16

0Plet far Resid Nerl Quas Mun Sga a4 Naal Lire Pited (d) Now assume the application methods are random while the other terms are kept same as before. State the statistical model and the corresponding assumptions using the unrestricted method; reanalyze the data aj + (ya)ij + Bk + (aß)jk + Eijk, i = 1, ... ,3,j = 1, ... ,4, yijk k 1,3 -N(0,a)a Bк~N (0, о?), (ар)jк~ N (0, одр), аnd eijk~N(0,а?) aj = 0, (ya)~N(0, aa) The ANOVA analysis shows that only the fixed effect Mix is significant Type 3 Tests of Fixed Effects Effect Num DF Den DF F Value Pr > F 58.37 000 Mix 3 6 The covariance component estimates for all random effects are shown below, no effect is significant at alpha-0.05. From the QQ plot and residual plot all assumptions are met Covariance Parameter Estimates Estimate Standard Error Z Value Pr> Z Alpha Coy Parm Lower Upper 0.05 0.001987 Day 0.02216 0.09250 0.24 0.4053 1.767E25 Mix Day 0.02770 0.1655 0.17 0.4335 0.05 0.002525 1.934E54 0.05 Method 9.1146 9.2543 0.98 0.1623 2.4387 396.94 Mix*Method 0.3336 0.3315 1.01 0.1571 0.05 0.09094 12.6598

Covariance Parameter Estimates Cov Parm Estimate Standard Error Z Value Pr Z Alpha Lower Upper Residual 0.6718 0.2375 2.83 0.0023 0.05 0.3726 1.556 And the residual plots 0 Plot fer Resid 14 11 15 01 Neral Qasies 650 65 125 150 Nomal Le- Pdted