Question: It looks rather like the rate of increase in number of good pieces with number of hours training is decreasing, rather than staying constant (linear).

a) Fit a quadratic regression equation relating number of acceptable pieces to number of hours training.

b) State the equation.

c) Plot the equation along with the data.

d) Is the quadratic term significant?

e) Do you think that the quadratic equation is sufficient to describe the relationship between the number of acceptable pieces and the number of hours training?

f) What are the Regression SS and df for the quadratic model?

g) What are the Residual SS and df for the quadratic model?

h) Comparing the SS and df for the linear and quadratic models, is the increase in Regression SS and df equal to the decrease in Residual SS and df?

i) Now use the One-Way Anova procedure to analyze how the mean number of acceptable pieces varies according to number of hours training

j) Under Contrasts, Select Polynomial, Degree = Linear.

k) Compare the SS and df labeled Linear Term: Contrast to the Regression SS and df for the linear regression in part 2.

l) Compare the Within Groups SS and df to the Residual SS and df for the linear regression.

m) Since there are replicated measurements at each level of hours of training, it is possible to assess the lack of fit of a linear model to the data. This is done by testing the Deviation of the data from the linear model. SS Deviation divided by df for deviation gives MS for Deviation. The test of

H₀: no significant deviation from linear model

is given by an F statistic with MS(Deviation) in the numerator and MS(Within Groups) in the denominator. What do you conclude about the fit of a linear model to the data?

n) Note that a straight line (first-degree polynomial) is determined by two points. A quadratic equation can be found which will pass exactly through 3 points. In our example, we have four means (corresponding to the four levels of training). A cubic equation can fit these four points exactly. Note also that the Between Groups SS has 3 degrees of freedom. Why does SS(Deviation) have 2 degrees of freedom?