Data set: Cups
Have you ever wondered how paper cups are made? During the process, different temperatures, adhesives, pressure settings, paper stocks, types of machine, machine speeds, and many other variables can impact the quality of the cup that is made. Some cup-making machines can produce over 300 cups per minute. However, it can take a few hours to find the optimal running conditions. This can lead to a significant amount of wasted time and materials, as operators adjust the machine set-tings until good-quality cups are produced.
A manager of a manufacturing company, shift managers, cup machine operators, and a lone statistician decided to identify which factors were most influential in keeping their cups from leaking. Over 30 possible factors were identified, but after some thoughtful discussions the group settled on six variables that should be tested for their effects on leaking cups. One of the six factors of interest was which paper supplier to use. Since the company was considering changing suppliers, funds were available to do some product testing before a purchase was made. However, each trial (each run of production under specified factor conditions) would cost the company thousands of dollars in lost production time, material costs, and employee costs. The following is a list of questions representing the main factors in this study.
• Should the side-seam temperature be set at 70% or 90% when the cup is folded and the sides are sealed together?
• Should the side seam be sealed with an additional adhesive? This adds to production cost, but could be worthwhile if it improves the quality of the cup.
• Should the bottom temperature be set at 80% or 94% when the bottom is attached to each cup?
• Should the bottom pressure be set at 1000 or 1120 when the bottom is attached to each cup?
• Which supplier, Royal or Imperial, should be used? The suppliers have very similar prices.
• Should the paper stock come with an additional coating? This coating adds to production cost, but could be worthwhile if it improves the quality of the cup. The company agreed to conduct 32 tests, but wanted to test all six factors and all corresponding two-way interactions. Fractional factorial designs are very useful for this type of exploratory data analysis. The details of fractional factorial designs are beyond the scope of this text. However, balanced data are a key concept behind these designs. For example, in the Cups data, every factor has two levels and each level has 16 observations. In addition, within the first factor level (the 16 observations where side-seam temperature is set to 70%), every other factor is still balanced (every other factor has 8 observations at each level).
a. Use software to conduct an ANOVA to analyze the Cups data set. This ANOVA should include six terms corresponding to the six main factors and 15 interaction terms.
b. Create a main effects plot comparing the effects of each main factor. Explain how the p-values correspond to what you see in the main effects plot.
c. Create a graph of all interaction plots. Explain how the p-values correspond to what you see in the interaction plots.
d. Create a probability plot or histogram of the residuals to determine if the residuals are consistent with data from a normal distribution. Do residuals appear to follow the normal distribution?
e. State your conclusions for this study. Provide an answer to each of the six key questions listed above, using p-values and plots. In addition, provide an interpretation of any significant interaction effects. Your explanations should be understandable to both managers and machine operators. Assume you are the lone statistician involved in this study. Be sure to clearly identify the population for which these conclusions hold and also state whether causation has been shown.