AP Statistics / Mr. Hansen
10/15/2005

Name: _________________________

Programmed Learning:
Transformations To Achieve Linearity

 

0.

Before you do the programmed learning exercises #1-12, read the following summary of the procedure for performing a general curve fitting using a “transformation to achieve linearity.” That way, you will know approximately where you are heading.

 

Steps:

  • Store X values in one list (say, L1) and Y values in another (say, L2).
  • Make an educated guess concerning the function that relates L1 to L2.
  • Apply the inverse of that function to L2, so as to form L3.
  • Do a LSRL fit of L1 to L3 (STAT CALC 8).
  • Write an equation that expresses the inverse result as being approximately equal to the LSRL model.
  • Solve for y.
  • Write answer as =whatever.
  • Store a column of these  values somewhere (say, in L4).
  • Compute residuals manually and store in another column (say, L5). The formula is y – . In calculator notation, this becomes L2–L4®L5. Remember, however, that you cannot use calculator notation for the AP exam or for any written work in this class. (Question #12 below is a rare exception to that rule.)
  • Make a residual plot and analyze it.
  • Make a scatterplot of X and Y with the  curve overlaid.

 

 

 

Questions #1-12 below will walk you through this process, step by step. Each step is small but crucial. Read carefully.

Consider the following set of ordered (x, y) pairs:

{(2, 1.4), (3, 1.7), (5, 2.2), (8, 2.75), (9, 2.88), (12, 3.51), (16, 4.02), (20, 4.55)}

This is a better data set than the small set we were using in last Thursday’s class. As we did on Thursday, place the explanatory column in L1, the response column in L2, and the square of the L2 values in L3.

 

 

 

Have you done this? _____________

 

 

1.

We speculate that the relationship between L1 and L2 is approximately a square-root relationship. Why are we squaring L2 to get L3? (This is a 6- or 7-word answer. If you can’t remember the reason, then skip it and move on.) __________________________________________

 

 

2.

Perform a LSRL fit between L1 and L2. Write the equation of the mathematical model.

____________________________________________________________________

 

 

3.

Sketch the residual plot and write two sentences regarding your conclusions.

 

 

 

4.

Because we think L1 and L2 have approximately a square-root relationship, perhaps the LSRL model in #2 is inappropriate. Accordingly, let us perform a LSRL fit between L1 and L3 instead. Remembering that L3 contains y2 values, we therefore have . . .

y2 » _______________________ + _______________________x

 

 

5.

Solve the approximate equation in #4 for y. Remember to use the “»” sign instead of an “=” sign.

 

 

 

 

 

 

 

 

6.

Replace the “y »” that you used in #5 with the notation “” (notice the double hat to distinguish this model from the single-hat model you found in #2). Rewrite the equation of your mathematical model in this way.

____________________________________________________________________

 

 

7.

List several of the built-in regression capabilities of your calculator other than linear or exponential.

_____________________ , _____________________ , _____________________ , _____________________

 

 

8.

In #6, you created a “custom” mathematical model that does not exist as one of the built-in features of your calculator. Accordingly, your calculator cannot compute the RESID list automatically. We will need to construct the residuals manually. However, first we must construct a list of  values, and we may use list L4 for that purpose. Note: If you were wise enough to perform question #4 as STAT CALC 8 L1,L3,Y1 then Y1 already contains the LSRL fit, and you can simply define L4 as Ö(Y1(L1)). If you were not so wise, then you will have to punch in some horrible gobbledygook like Ö(–.4333623492631+1.0426239839214L1) for L4.

Have you created your L4 column? _____________

 

 

9.

Create residuals in L5 by using the formula for residual. Hint: Your  values are in L4 from #8, and your y values are in L2.

Have you created your L5 column? _____________

 

 

10.

Sketch the new residual plot. (This is simply a STAT PLOT with L1 on the horizontal axis and L5 on the vertical axis.)

 

 

 

 

 

 

 

 

11.

Is your residual plot in #10 better or worse than the one in #3? Explain briefly.

 

 

 

 

 

 

12.

Write calculator keystroke instructions that would allow a fellow student at this point to create a scatterplot for the original X and Y data, with the curve from #6 overlaid.