3.36 An article in the Journal of the Electrochemical Society (Vol. 139, No. 2, 1992, pp. 524–532) describes an experiment to investigate the low-pressure vapor deposition of polysilicon. The experiment was carried out in a large-capacity reactor at Sematech in Austin, Texas. The reactor has several wafer positions, and four of these positions are selected at random. The response variable is film thickness uniformity. Three replicates of the experiment were run, and the data are as follows: Wafer Position Uniformity 1 2.76 5.67 4.49 2 1.43 1.70 2.19 3 2.34 1.97 1.47 4 0.94 1.36 1.65 (a) Is there a difference in the wafer positions? Use ???? = 0.05. (b) Estimate the variability due to wafer positions. (c) Estimate the random error component. (d) Analyze the residuals from this experiment and comment on model adequacy.

The obtained p-value is 0.0077 which is less than the level of significance, 0.05 which results in the rejection of the null hypothesis. Therefore, it can be concluded that there is a difference in the wafer positions.

Step 3

(b) The variability due to wafer positions can be estimated as,

Step 1

According to the provided data, three replicates were run of an experiment. Minitab is used to do the analysis and obtaining the residual plots.

Steps are as;

1. Enter the data into worksheet.

2. Go to Stat>ANOVA>One-way.

3. Select the response as uniformity and factor as wafer position.

4. Click graphs and tick normal probability plot of residuals and residual versus fits.

5. Click ok twice.

(c) The random error component can be estimated as;

Step 2

(d) The obtained normal probability plot and residual plots are;

The obtained normal probability plot of residuals shows that there is not much deviation from normality as majority of the observations are near the straight line. Although, there are few outliers too.

Step 3

The variability in film thickness seems to depend on wafer position. The wafer position number 1 appear to have greater variation in uniformity than the other positions. It provides the evidence against the assumption of equal variances at all factor levels.

It can also be observed that the variability in the residuals is high for higher fitted values which shows that the residuals depends on the fitted values. Thus, the residual analysis indicates model inadequacy.