Question: Tablet Hardness. Tablet hardness (for medicine) is measured in standard Cobb units (SCU) and specifications for hardness are 17 ± 5 SCU. After analyzing the tablet production process, engineers concluded that tablet press compression level and powder moisture level had large effects on final tablet hardness. Management was using a trial-and- error method to adjust compression and moisture levels. The engineering team decided to adopt a more systematic approach to finding a good combination of compression and moisture. Low and high settings were identified for both factors for purposes of experimentation. The following summaries were obtained from an experiment where the four different combinations of compression and moisture content were used to produce tablets. Given in the table are means and standard deviations of hardness for a number of batches made under each treatment combination. (Batch hardness, y, was defined by testing and averaging test results for ten tablets from the batch. The ns in the following table are numbers of batches. The were obtained by averaging the y's determined from each batch. The s's are the standard deviations of the ys for a given compression/moisture combination.)
(a) How many experimental factors were there in this study? Name them.
(b) Give a model equation for the "normal distributions with a common variance" description of batch hardness that uses "factorial effects" notation.
(c) Find the pooled estimate of the (supposedly common) variance of batch hardness under a fixed set of processing conditions. What are the associated degrees of freedom?
(d) Plot the sample means in interaction plot format. Put compression on the horizontal axis. Does it appear that there are strong interactions between compression and moisture? Why or why not?
(e) Find individual 95 % two-sided confidence intervals for the four treatment combination means.
(f) Enhance your plot in (d) by drawing error bars around the means based on your confidence limits from (e). What is suggested by these about the statistical detectability of interaction (lack of parallelism) in this study?
(g) Find the fitted effects for each effect in the model given in (b) using Yates algorithm.
(h) Find a 99 % two-sided confidence interval for the two-factor interaction at the high levels of both factors.
(i) Find a 99 % two-sided confidence interval for the high moisture main effect. Find a 99 % two-sided confidence interval for the high compression main effect.
(j) Using the Bonferroni inequality, find a set of 94 % simultaneous con- fidence intervals for the three effects estimated in (h) and (i).
(k) Based on your intervals from (j) and the enhanced interaction plot from (f) discuss what has been learned about how compression and moisture impact tablet hardness.
(l) Which combination of compression and moisture seems best? Defend your answer.
(m) Using the information given in this problem estimate the fraction of high compression/low moisture batches that have measured "batch hardness" within 1 SCU of the target value of 17 SCU. Adopt a normal model and assume your estimated average for high compression/low moisture is reasonable for the true average and the pooled estimate of variance is reasonable as well. Do you have appropriate information here to estimate the fraction of individual tablets made under these same conditions with tested hardness inside the 17 ± 5 SCU specifications (for individuals)? Why or why not?