Question

2. An experiment was run in a semiconductor fabrication plant in an effort to increase yield. Five factors, each at two levels, were studied. The factors (and levels) were A aperture setting (small, large), B-exposure time (20% below nominal, 20% above nominal), C-development time (30s, 45s), D mask dimension (small, large), and E: etch time (145min, 15.5min). The unreplicated 25 design shown below was run. b 34 bd 32 be 35 bde 30 ab 55 abd 50 abe 52 abde 53 c 16 cd 18 ce 15 cde 15 ac 20 acd 2 ace 22 acde 20 bc = 40 bcd = 44 bce = 45 bcde = 41 abc = 60 abcd = 61 abce = 65 abcde-63 Stat 341: Homework 4 SPRING, 2019 (a) Construct a normal probability plot of the effect estimates. Which effects appear to be large? (b) Conduct an analysis of variance to confirm your findings for part (a (e) Write down the regression model relating yield to the significant process variables. (d) Analyze the residuals. Are there any obvious problems? (e) Interpret any significant interactions. Wht are your recommendationsregarding processoperating conditions?

Answer 1

Answer: Date:-12/05/2019 5 (a) First estimate the effects. For the 25 factorial experiment the table of signs is the following | B | C | D | E 4 | B | C | D | E abe ace bce ade + -bde cde acde bcde abcd+- abcde

The sign for each interaction can be obtained by multiplication of the signs of Factors that are in this interaction.

Use the table of signs to calculate the main effect of A. Add the effects with the corresponding A--1+a-bab-c+ac-bc+abc -d+ad-bd abd -cd+acd -bed +abcd -e +ae -be +abe - ce+ace-bceabce -de +ade -bde+abde -cde+acde-bcdeabcde) 16 (-7 + 9-34 + 55-16+20-40 + 60-8+10-32 + 50-18+21-44 + 61 16 -8+12-35 + 52-15+22-45 + 65-6+10-30 + 53-15+20-41+63) -i(189) 11.8125 16

All other effects are found similarly. The table of the effects for all factors and interactions is given below Effect | 11.8125| ABC |-0.4375 B 33.9375 ABD0.3125 C | 9.6875 | ΆΒΕ 1-0-1875 D-0.8125ACD -0.4375 E 0.4375ACE0.3125 ABI 7.9375 | 4DE | 0.8125 AC 0.4375 BCD 0.4375 AD |-0.0625 | ВСЕ | 0.9375 AE | 0.9375 | BDE | 0.1875 BC 0.0625 CDE-0.8125 BDI-0.6875 ABCD |-0.0625 BEI 0.5625 | ABCE. | 0.1875 CD 0.8125 ABDE 0.9375 CE 0.3125 ACDE-0.3125 DE -1.1875 BCDE-0.9375 ABCDE-0.1875 Effect 2

The normal probability plot of the effect estimates is shown below Normal Probability Plot of the Effects response is y, Alpha .05) Effect Type ◆ Not Signiert 95 Factor Nane ■AB 80 70 トート-+-+-+-+-+-+-+-+-+ : トー --+--+--+--+ | | | 0 5 10 15 20 Effect 25 30 35 Lenth's PSE 0.65625

Examining the magnitude of the effects and the normal probability plot of effects shows that and interaction AB are significan

(b) Carry out the analysis of variances. ss, (Contrast, (189)2 32 1 1 16.281

Compute all other sums of squares similarly. The result is summarized in the table below. Since the 3-way, 4-way and 5-way interactions are not significant, their sums of squares are pooled to be included into the error mean square. Also add all SS to find the total SS Sum of Source of variation Squares 1116.281 9214.031 750.781 5.281 1.531 504.031 1.531 0.031 7.031 0.031 3.781 2.531 5.281 0.781 11.281 AB AC AD AE BC BD BE CD CE DE

Find the total sum of squares. i,j 7+.+63+63) -11663.969 32

Find the sum of squares for error = 1 1 663.969-1 1 16.281-...-1 1 .281 = 39.754

Each Ss for a factor or interaction has 1 degree of freedom, SS has 32n-131 degrees of freedom, SS has 16 degrees of freedom (the sum of degrees of freedom for pooled interactions). Find each MS as the ratio of corresponding SS and DF. The result is summarized in the table below Sum ofDegrees Mearn Source of variation squares of freedomsquare 1116.281 9214.031 750.781 1116.281 9214.031 750.781 5.281 1.531 504.031 1.531 0.031 7.031 0.031 3.781 2.531 5.281 0.781 11.281 2.4846 5.281 1.531 AB AC AD AE BC BD BE CD CE DE Eror Total 504.031 1.531 0.031 7.031 0.031 3.781 2.531 5.281 0.781 11.281 39.754 16 3 1 11663.969

Find the value of F-statistic for each factor and interaction as a ratio of corresponding MS to MSE. The result is summarized in the table below. We use software to find the P-values SourceSum ofDegreesMean foP-value of variation squares of freedom square 1116.28:1 9214.031 750.781 5.281 1.531 504.031 1.531 0.031 7.031 0.031 3.781 2.531 5.281 0.781 11.281 39.754 1116.281 449.28 0.000 9214.031 3708.46 0.000 750.781 302.17 0.000 2.13 0.62 504.031 202.86 0.000 5.281 1.531 0.164 0.443 AB AC AD AE BC BD BE CD CE DE Error 0.62 0.443 0.922 0.112 0.922 0.235 0.328 0.164 0.585 0.049 1.531 0.031 7.031 0.031 3.781 2.531 5.281 0.781 11.281 2.4846 0.01 2.83 0.01 1.52 1.02 2.13 0.31 4.54 16 3 1 Total11663.969

We look for factors with small P-values. Such factors are A, B, C, and AB. This result and interaction AB should be considered important.

c) Write down the regression model relating yield to the significant process variables: Yajk = 30.53 + 5.9 IX, + 1 697 X, t 4.84 Xc + 7.94x,B

Residual Plots for Yield Normal Probability Plot of the Residuals Residuals Versus the Fitted Values -H 90 50 0.0 1.5 15 45 Fitted Valuc Residuals Versus the Order of the Data His togram of the Residuals 0.0 12 Observation Order Se ve que los residuales se distribuyen normalmente y que no hay patrones ni inconsistencia con la variabiliad.

(e) The only significant interaction is AB, the interaction of aperture setting and exposure time. Since the effects of A, B, and AB are positive, one should first of all use factors A and B in their high values to increase the yield. The interaction effect will only work on increasing the yield