In the statistical model in Equation (4.5), the following assumptions need to be validated about the random error terms, ɛijk, before any formal hypothesis test can be developed:
• The error terms are independent and identically distributed.
• The error terms follow a normal probability distribution, denoted as ɛ ̴ N(0, σ2).
the second assumption includes an equal variance assumption about the random errors from the different factor- level groups: σ112 = σ122 = σ132 = σ212 = σ222 = σ232.
a. The independence assumption implies that there is no relationship between one observation and the next. The identically distributed assumption means that each observation sampled within each brand/ water combination is from a population with the same mean and variance. If all 26 paper towels were sampled from one roll to assess the Brand C/5 drops factor combination, would you be concerned about violating the independence and/or the identically distributed assumption? Why or why not?
b. Calculate the sample means and standard deviations of all six factor- level combinations. Clearly, some groups have much larger variation than others. In addition, the variation within each group increases as the average breaking strength increases. To address this issue, a transformation of the data can often be used that will “ stabilize” the variances so that the equal variance assumption is reasonable on this new scale. (Chapter 2 describes transformations in more detail.)
c. Transform the response variable using natural log (Strength) and √Strength. Did both trans-formations improve the equal variance assumption?