Refer to the data on the number of acceptable pieces of a glass product as a function of the number of hours of training by the glassworker. 28 workers were randomly divided into four groups of seven workers each; one group of workers was then trained for 6 hours, another for 8, another for 10, and the last group for 12 hours.

This data set was previously used to illustrate the relationship between regression and analysis of variance models. In that, we saw that the four levels of training could be indicated by three dummy variables, and that analyzing the number of acceptable pieces as a regression on three dummy variables produced the same results as modeling the number of good pieces as an analysis of variance on a single factor with four levels.

The previous analyses the treated the factor (number of hours training) as categorical. However, the factor levels are actually numerical . That being the case, we could just as well analyze the data as a regression problem.

1. Plot the number of acceptable pieces against the number of hours training. Use a scatter plot.
2. Treating the number of hours training as a numerical variable,
   1. fit a simple linear regression equation relating number of acceptable pieces to number of hours of training.
   2. State the equation.
   3. Plot the equation along with the data.
   4. Is there a significant linear relationship?
   5. Is there a strong linear relationship?
   6. Do you think that the simple linear equation is adequate to describe the relationship between number of acceptable pieces and hours training?
   7. What are the Sum of Squares and Degrees of Freedom for Regression?
   8. What are the Sum of Squares and Degrees of Freedom for Residual?
      
      
3. 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).
   1. Fit a quadratic regression equation relating number of acceptable pieces to number of hours training.
   2. State the equation.
   3. Plot the equation along with the data.
   4. Is the quadratic term significant?
   5. 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?
   6. What are the Regression SS and df for the quadratic model?
   7. What are the Residual SS and df for the quadratic model?
   8. 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?
      
   9. Now use the One-Way Anova procedure to analyze how the mean number of acceptable pieces varies according to number of hours training
   10. Under Contrasts, Select Polynomial , Degree = Linear.
   11. Compare the SS and df labeled Linear Term: Contrast to the Regression SS and df for the linear regression in part 2.
   12. Compare the Within Groups SS and df to the Residual SS and df for the linear regression.
   13. 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
       
       H ₀ : 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?
   14. 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?
       
4. Run the One-Way Anova analysis again, but this time under Contrasts, select Polynomial, Degree = Quadratic .
   1. Is there a significant quadratic effect?
   2. Is there a significant cubic effect?
      
      
5. Now use the General Linear Model procedure to perform the same analysis as in part 5. Use the drop-down menu to see the various kinds of contrasts GLM provides and choose the one which corresponds to the analysis in part 5.
6. In the One-Way Anova, write contrasts to compare mean number of acceptable pieces
   1. At 8 and 6 hours
   2. At 10 and 8 hours
   3. At 12 and 10 hours.
7. Perform the same analysis as in part 7, but use GLM. (Don’t worry about the directions of the comparisons.)

Price: $24.17

Solution: The downloadable solution consists of 12 pages, 1217 words and 8 charts.
Deliverable: Word Document