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Previously, we considered situations where the response (or dependent) variable was determined completely by just one other variable. Now we will consider cases where the one response variable depends on several explanatory (or independent) variables. In addition, we will examine the nature of this dependence.
For each of the following response variables, identify at least three explanatory variables that influence the given variable, and for each explanatory variable determine whether it is catagorical or quantitative.
(a) The cost of a 1000 mile car trip.
(b) The length of time a traffic light is set to remain yellow.
(c) The time it takes to travel from Boston to Miami.
(d) A college student's GPA.
(a) the price of gasoline - quantitative, make/model of car - catagorical, amount of road tolls - quantitative
(b) the speed limit of the roads it's on - quantitative, number of nearby traffic lights - quantitative, amount of traffic on the roads - this could be either quantitative or catagorical depending on what values are assigned the variables (numbers vs. "high," "medium," "low," etc.)
(c) the mode (air, car, bus, train) - catagorical, when traveling - catagorical, number of stops- quantitative
(d) amount of time studying- quantitative, types of courses- catagorical, how much time watching TV or playing video games- quantitative
It is interesting to note that many quantitative variables could be converted to catagorical, as explained in (b), by assigning them "high," "medium," or "low."
Functions with multiple explanatory variables can be described in the same four ways as functions with one explanatory variable: with words, tables, formulas, and graphs.
For functions with multiple explanatory variables, we sometimes investigate the behavior of one variable by holding the other variables constant.
For example, let's consider wind-chill factor, which depends on two explantory variables: air temperature and wind speed. The following table is from the Mount Washington Observatory
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Suppose we fix wind speed at 10 miles per hour. The graph of wind chill as a function of temperature is:
Now, suppose we fix the temperature at 10 degrees Fahrenheit. The graph of wind chill as a function of wind speed is:
