SPH 7460 Final Exam  December 18, 2012                                    page 1 of 4

Three Problems - 4 pages               Name: _________________________________________
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1.  A linear transformation T: R^2 --> R^2 is such that:

      | 1 |   |  1  |      | 0 |   | 1/2 |
    T |   | = |     | ,  T |   | = |     |.
      | 0 |   | 1/2 |      | 1 |   |  1  |

    a)  Sketch the image of the unit square under the transformation T.



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    b)  Find the matrix corresponding to T.












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    c)  What is the determinant of T ?


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SPH 7460 Final Exam  December 18, 2012                                    page 2 of 4

                                      Name: _________________________________________
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1., Continued

   d)  Find the eigenvalues of T.













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   e)  Why is the product of the eigenvalues the same as the determinant of T ?




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   f)  Write a SAS program to compute the eigenvalues and eigenvectors of T.




 

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SPH 7460 Final Exam  December 18, 2012                                    page 3 of 4

                                      Name: _________________________________________
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2.  Given a data file with 1000 observations of the random variable X, write a SAS macro
    which will compute the bootstrap estimate of the 95 % confidence interval for the
    standard deviation of X.  The call to the macro should look like:

           %boot95(datafile, xvar, xstddev, nboots, cl95, cu95),

    where datafile is the data file, xvar is the variable X of interest, xstddev is the
    sample xstddev of xvar, nboots is the number of bootstrap samples, and cl95 and cu95
    are the lower and upper 95% confidence limits for the true standard deviation.









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SPH 7460 Final Exam  December 18, 2012                                    page 4 of 6

                                      Name: _________________________________________
=====================================================================================

3.  Concentrations C in the blood of a toxic substance are a function of a person's
    body surface area A and the time t after ingesting the toxic substance;
    specifically,

         C = (f * A + g * exp(-h * t))^2 + e

    where f, g, and h are unknown parameters and e is measurement error; e ~ N(0, sigma^2).

    a) Given a dataset D that has C, A, and t for each of N people, write a SAS PROC NLIN
       program which will produce the least-squares estimates of f, g, and h.



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      b) Describe how you would test the hypothesis that f = 0.




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