1 Visual and musculoskeletal problems associated with the use of visual display ter-

Example 12.1 Visual and musculoskeletal problems associated with the use of visual display ter-

  minals (VDTs) have become rather common in recent years. Some researchers have focused on vertical gaze direction as a source of eye strain and irritation. This direc- tion is known to be closely related to ocular surface area (OSA), so a method of measuring OSA is needed. The accompanying representative data on

  y 5 OSA (cm 2 ) and x5 width of the palprebal fissure (i.e., the horizontal width of

  the eye opening, in cm) is from the article “Analysis of Ocular Surface Area for Comfortable VDT Workstation Layout” (Ergonomics, 1996: 877–884). The order in which observations were obtained was not given, so for convenience they are listed in increasing order of x values.

  Thus (x 1 ,y 1 ) 5 (.40, 1.02), (x 5 ,y 5 ) 5 (.57, 1.52) , and so on. A Minitab scatter plot is

  shown in Figure 12.1; we used an option that produced a dotplot of both the x values and y values individually along the right and top margins of the plot, which makes it easier to visualize the distributions of the individual variables (histograms or boxplots are alternative options). Here are some things to notice about the data and plot:

  • Several observations have identical x values yet different y values (e.g.,

  x 8 5x 9 5 .75 , but y 8 5 1.80 and y 9 5 1.74 ). Thus the value of y is not

  determined solely by x but also by various other factors. • There is a strong tendency for y to increase as x increases. That is, larger values

  of OSA tend to be associated with larger values of fissure width—a positive relationship between the variables.

  Figure 12.1 Scatter plot from Minitab for the data from Example 12.1, along with dotplots of x and y values

  12.1 The Simple Linear Regression Model

  • It appears that the value of y could be predicted from x by finding a line that is rea-

  sonably close to the points in the plot (the authors of the cited article superimposed such a line on their plot). In other words, there is evidence of a substantial (though not perfect) linear relationship between the two variables.

  ■ The horizontal and vertical axes in the scatter plot of Figure 12.1 intersect at

  the point (0, 0). In many data sets, the values of x or y or the values of both variables differ considerably from zero relative to the range(s) of the values. For example, a study of how air conditioner efficiency is related to maximum daily outdoor tem- perature might involve observations for temperatures ranging from 80°F to 100°F. When this is the case, a more informative plot would show the appropriately labeled axes intersecting at some point other than (0, 0).

  Example 12.2 Arsenic is found in many ground-waters and some surface waters. Recent health

  effects research has prompted the Environmental Protection Agency to reduce allow- able arsenic levels in drinking water so that many water systems are no longer com- pliant with standards. This has spurred interest in the development of methods to remove arsenic. The accompanying data on x 5 pH and y5 arsenic removed () by a particular process was read from a scatter plot in the article “Optimizing Arsenic Removal During Iron Removal: Theoretical and Practical Considerations” (J. of Water Supply Res. and Tech., 2005: 545–560).

  Figure 12.2 shows two Minitab scatter plots of this data. In Figure 12.2(a), the soft- ware selected the scale for both axes. We obtained Figure 12.2(b) by specifying scal- ing for the axes so that they would intersect at roughly the point (0, 0). The second plot is much more crowded than the first one; such crowding can make it difficult to ascertain the general nature of any relationship. For example, curvature can be over- looked in a crowded plot.

  Figure 12.2 Minitab scatter plots of data in Example 12.2

  CHAPTER 12 Simple Linear Regression and Correlation

  Large values of arsenic removal tend to be associated with low pH, a negative or inverse relationship. Furthermore, the two variables appear to be at least approxi- mately linearly related, although the points in the plot would spread out somewhat about any superimposed straight line (such a line appeared in the plot in the cited article).

  ■