Statistical Methods

Homework 4:

Note 1: Homeworks are due at the beginning of the "due date" class. Late hw will not be accepted, except in extraordinary circumstances.

Note 2: If you can't solve some of the problems, please come to the office hours, or email us a very specific and short question. Email will not work well or at all for involved or unclear questions.

For this homework, please complete the following problems:

• Problem 1

  For independent random variables X and Y, we know that f(x,y) = f1(x)f2(y) where f1 is the marginal density of X and f2 is the marginal density of Y. Using this, show that:
  

  • a) E(XY) = E(X)E(Y)
    
  • b) Var(X+Y) = Var(X)+Var(Y)
    
  
  
• Problem 2

  You have a random sample of size n, X1,X2,...Xn. Using the result in part (b) above, find the variance of the sample mean of that sample.
  
  

• Problem 3

  At time t=0, a lab technician starts an experiment where 20 identical components are tested. The lifetime distribution of each component is exponential with mean lifetime of w hours. The technician leaves the test facility right away, and comes back in 24 hours to find 15 components still running (and 5 of them failed). Based on this sample, find the MLE for w.
  
  

• Problem 4

  Please use a software package to solve this problem. Write down all the commands used.

  Aphid infestation of fruit trees is usually controlled either via pesticides or via ladybug innundation. In a particular area, 2 different (and well isolated) groves, with 15 fruit trees each, are selected for an experiment. The trees in both groves are of the same age, roughly the same size, and can be assumed to be independent. One grove is sprayed with pesticides, and one is flooded with ladybugs. The fruit yield (in pounds) for each tree is given below:

  
  Pesticide trees:
  55.57109 36.50319 47.80090 33.34822 36.16251
  35.28337 41.50154 44.18931 40.81439 33.88648
  44.90427 49.97089 22.85414 27.84301 38.49843
  
  

  Ladybug trees:
  45.44505 35.52320 46.97865 45.76921 41.66216
  54.69599 58.77678 49.08538 48.53812 70.17137
  51.86253 39.59365 42.10194 47.39945 39.04648
  
  

  • a) Plot the histograms of the yields in the two groves
    
  • b) Comment on the histogram shapes. Which densities do they resemble? In particular, do they appear normal?
    
  • c) Find the sample means of yields for the two groves
    
  • d) Using your conclusion in (b) and answer in (c), provide the two 95% confidence intervals for the true mean yields for trees under the two treatments.
    
  • e) Interpret the confidence intervals you constructed.
    
  • f) Find the sample variances for the yields in the two groves
    
  • g) We have learned how to construct confidence intervals for the variance of a normal distribution -- we do that using a chi-square distribution. For the grove(s) with approximately normal yield, construct the 95% confidence interval for the true variance of yield.