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.