The table gives the price of an umbrella, the amount of rainfall,and the number

Question

The table gives the price of an umbrella, the amount of rainfall,and the number of umbrellas purchased.

Umbrellas (number per day)
Price (dollars per umbrella)
0 2
(mm of rainfall)
10 10 7
8
12
20
4
7
8
30
2
4
7
40
1
2
4
Draw graphs to show the relationship between:
a) The price and the number of umbrellas purchased, holding theamount of rainfall constant.
b) The number of umbrellas purchased and the amount of rainfall,holding the price constant.
c) The amount of rainfall and the price, holding the number ofumbrellas purchased constant.
Umbrellas (number per day)
Price (dollars per umbrella)
0 2
(mm of rainfall)
10 10 7
8
12
20
4
7
8
30
2
4
7
40
1
2
4

Explanation / Answer

I assume you're supposed to answer this problem using aprogram like Excel or STATA. . . All three sub-parts are basically the same question, just withdifferent variables. This question is designed to push yourunderstanding of multivariate regression a bit. . . Let's start with the basics. Do you understand how tosolve this problem, if it took out the words "holding SOMETHINGconstant"? That is, can you graph the relationship betweenprice and number of umbrellas purchased? . . There are different ways to interpret the words "therelationship." My best guess is that they're looking for asimple linear relationship, but that could be wrong. If yourclass has been covering linear regression lately, then that's whatthey're looking for. To find a simple linear relationshipbetween prices and umbrellas, you would do a 2-stepprocedure:    1) Regress P = a + b*U + e, where P is the price,U is the number of umbrellas sold, a and b are parameters, and e isthe error term (which you can't observe).    2) Graph your predicted relationship. Youcould do this by hand, using your estimate of a as the Y-interceptand your estimate of b as the slope. Alternatively, you couldgraph it by computer. Some programs have a built-in way ofdoing this for you, but that varies by program. To graphit by computer without using a built-in shortcut:     i) For each observation, create a FITTEDVALUE (aka a PREDICTED VALUE) by taking that observation's value ofU and plugging into the equation P^ = a^ + b^*U, where a^ and b^are your estimates of a and b from step 1. We put a ^ on P toindicate that it's a predicted value, not the true value.    ii) Make a Line Graph with U on the X-axis and P^on the Y axis. How to do this varies by program. WARNING: you might need to sort the data by your X-variable first(though it won't matter for a completely straight line like thisone, it's a good habit to get into).     Aside that might help later: this works okhere, because we're graphing a straight line, so we need only twopoints. However, if you had a non-linear function (say,because you were fitting a quadratic), you could end up with a"curve" that was rather blocky-looking. You can make thecurve a lot smoother by creating a ton of values of your X-variable(0, 0.1, 0.2, 0.3, 0.4, etc), then finding predicted values ofthose, and then plotting your made-up X's and your predictedY^'s. That "fills in" the gaps between the X-values that youactually observe. . . . OK, so now we've talked about how to graph the relationshipbetween P and U, without worrying about rainfall. But how dowe hold rainfall constant? . . Terms like "holding the amount of rainfall constant" or"controlling for rainfall" just mean that we're sticking thevariable "rainfall" into our regression. So our step 1changes to become 1) P = a + b*U + c*R + e . . Now the coefficient b tells us the effect of a 1-unit increasein Umbrellas Sold on the price, HOLDING RAINFALL CONSTANT. . . Now we can go ahead and graph this relationship--except thatthere's a catch. Our equation relating P and U now has R init, and we cannot simply ignore that. To get our fittedvalues P^, we have to specify a value of R that we're holding Rconstant at, and then plug that value into our equation. Usually we do this at the mean value of R. So find mean-R,and then calculate . . P^ = a^ + b^*U + mean-R*c^ .     = (a^ + mean-R*c^) + b^*U . where I grouped a^ and mean-R*c^ together because they're bothconstants for this calculation. . If you omit the mean-R*c^ part, your line will still be thecorrect shape (i.e. the correct slope), but it will have the wrongY-intercept, so you'll be predicting the wrong number of Y's. This is a subtlety that most students will probably miss, unlessyour teacher has been quite emphatic.

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