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Regression Help (page 2)

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By — McGraw-Hill Professional
Updated on Aug 26, 2011

Scatter Plots

When we examine Fig. 6-8A, it appears there is a relationship between temperature and rainfall for the town of Happyton. In general, as the temperature increases, so does the amount of rain. There is also evidently a relationship between temperature and rainfall in Blissville, but it goes in the opposite sense: as the temperature increases, the rainfall decreases. How strong are these relationships? We can draw scatter plots to find out.

In Fig. 6-9A, the average monthly rainfall is plotted as a function of the average monthly temperature for Happyton. One point is plotted for each month, based on the data from Table 6-1A. In this graph, the independent variable is the temperature, not the time of the year. There is a pattern to the arrangement of points. The correlation between temperature and rainfall is positive for Happyton. It is fairly strong, but not extremely so. If there were no correlation (that is, if the correlation were equal to 0), the points would be randomly scattered all over the graph. But if the correlation were perfect (either +1 or –1), all the points would lie along a straight line.

Figure 6-9B shows a plot of the average monthly rainfall as a function of the average monthly temperature for Blissville. One point is plotted for each month, based on the data from Table 6-1B. As in Fig. 6-9A, temperature is the independent variable. There is a pattern to the arrangement of points here, too. In this case the correlation is negative instead of positive. It is a fairly strong correlation, perhaps a little stronger than the positive correlation for Happyton, because the points seem more nearly lined up. But the correlation is far from perfect.

Regression Curves

The technique of curve fitting, which we learned about in Chapter 1, can be used to illustrate the relationships among points in scatter plots such as those in Figs. 6-9A and B.

Examples, based on ''intuitive guessing,'' are shown in Figs. 6-10A and B. Fig. 6.10A shows the same 12 points as those in Fig. 6-9A, representing the average monthly temperature and rainfall amounts for Happyton (without the labels for the months, to avoid cluttering things up). The dashed curve represents an approximation of a smooth function relating the two variables. In Fig. 6-10B, a similar curve-fitting exercise is done to approximate a function relating the average monthly temperature and rainfall for Blissville.

In our hypothetical scenarios, the data shown in Tables 6-1A and B, Figs. 6-8A and B, Figs. 6-9A and B, and Figs. 6-10A and B are all based on records gathered over 100 years. Suppose that we had access to records gathered over the past 1000 years instead! Further imagine that, instead of having data averaged by the month, we had data averaged by the week. In these cases we would get gigantic tables, and the bar graphs would be utterly impossible to read. But the scatter plots would tell a much more interesting story. Instead of 12 points, each graph would have 52 points, one for each week of the year. It is reasonable to suppose that the points would be much more closely aligned along smooth curves than they are in Figs. 6-9A and B or Figs. 6-10A and B.

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