39 Assignment 2

Due 11:59pm Wednesday June 12

You may choose to complete the assignment working closely with another student in this class. If you choose to do this, you only need to submit one joint assignment (i.e., one file for both of you). Please make sure to put both of your names on the assignment.

Although you may complete this assignment jointly, you may participate in a learning group of any size to complete the assignment, but please only submit your own work (i.e., do not copy another group or persons work).

Please save the file as Yourlastnames_Assignment2 and upload it to Canvas. Your two-person group only needs to submit a single file, but make both of your names are listed. Make sure to show all of your work and computer code, so that your mathematical and numerical results are easily reproducible. Also, please note that about 15% of this assignment grade will involve making the document you submit look professional (e.g., no typos, plots and figures that are easy to read and have labels, no awkward or sloppy formatting, etc).

  1. Using your own words, write 3-5 sentences explaining what a mathematical model is.

  2. Using your own words, write 3-5 sentences explaining what a statistical model is.

  3. Explain why quantities that we estimate (e.g., a slope parameter), predict, or forecast (e.g., your retirement income) should not be presented as a single value (e.g., a point estimate or prediction).

  4. For the matrices \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\) below, use a computer to obtain

    1. \(\mathbf{A}+\mathbf{C}\) ,
    2. \(\mathbf{A}-\mathbf{C}\) ,
    3. \(\mathbf{AB}\) ,
    4. \((\mathbf{A}\text{+}\mathbf{\mathbf{\mathbf{B}}})^{-1}\mathbf{(C}+\mathbf{B})\).

\[\mathbf{A}=\left[\begin{array}{ccc} 1 & 2 & 3\\ 2 & 5 & 6\\ 3 & 6 & 9 \end{array}\right]\] \[\mathbf{B}=\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right]\]

\[\mathbf{C}=\mathbf{A}^{'}\]

  1. Which of the models below are a linear statistical models? Please justify your answer using words.
    1. \(\mathbf{y}=\beta_{0}+\beta_{1}\mathbf{x}^{2}\)
    2. \(\mathbf{y}= \mathbf{\beta}_{1}e^{\mathbf{x}}+\boldsymbol{\varepsilon}\)
    3. \(\mathbf{y}=\beta_{0}+e^{\beta_{1}\mathbf{x}}+\boldsymbol{\varepsilon}\)
    4. There was a typo here. The question with a typo was \(\mathbf{y}=\beta_{0}+e^{\log(\beta_{1})\mathbf{x}}+\boldsymbol{\varepsilon}\). What I intended to put was \(\mathbf{y}=\beta_{0}+e^{\log(\beta_{1})}\mathbf{x}+\boldsymbol{\varepsilon}\).
  2. For all four models in part a-d of problem 5, assume that \(\hat{\beta_{0}} = -0.34\) and \(\hat{\beta_{1}} = 1.04\). For values of \(x\) between 0 and 3 (i.e., \(0<x<3\)) plot the expected (predicted) value of \(y\) (i.e., plot the line or curve for each model).