Homework 1   January 10, 2017

Please staple your homework pages in the upper left corner.   [50-70 pts]

Due:  (a)  in class Tuesday January 17
      (b)  in mailbox of TA Daniel Cross by close of business 1/17


1.  6.3                    [ 5 pts]
2.  6.6                    [ 5 pts]
3.  6.9 (a-e)              [25 pts]
4.  For the linear model   [15 pts]

          Y_i = beta1 + beta2 * x_i + eps_i   i=1,...,n
                                              eps_i ~ N(0,sigma^2) i.i.d.

    Show  ( sum y_i,  sum y_i^2,  sum   x_i * y_i  ) sufficient
    for  theta = ( beta1, beta2, sigma^2 ).  Note:  The values x_i
    are not random, but chosen by the experimenter.

5.  [20 pts]  [Required for 519 students;  extra credit 419 students]

    In Tukey's Exploratory Data Analysis book, Tukey demonstrates the
    box-and-whiskers plot on Lord Rayleigh's data measuring the
    weight of nitrogen gas obtained by various means.  Discrepancies in
    the results led to his discovery of element argon.  Rayleigh made 24
    measurements from 1892-1894, with mean of 2.30584 and standard
    deviation of 0.00537.  It is common to assume such measurements
    of a fundamental quantity are normally distributed.

    Plot a contour plot of the log-likelihood and the L2E criteria as
    a function of the mean and the log10( standard-deviation ) over
    intervals ( 2.296, 2.314 ) and log10( 0.0001, 0.01 ), respectively.
    Note the location of the sample mean and standard deviation on each.
    Compare and comment on your findings.

    Hints:
       (1)  In R, use    x = read.csv( "rayleigh.csv", header=T )
       (2)  Picking contour levels to display a bit tricky, given
            the very large range of function values.  Try several.