Assigned

Due

• Download the R statistical language. 
  http://www.r-project.org/ . 
  Click on the link under "Download" on the left of the screen, choose a mirror site, then follow directions to download and install R.

  Type the following in R, to make sure it's working.  You should observe three random numbers between 0 and 1, and then the same numbers put in ascending order.

              x <- runif(3, min=0, max=1)
           x
           y <- sort(x)
           y
   

• p. 236 #5.1.1, 5.1.2ac, 5.1.3bcd, 5.1.4 bcd(confidence interval part)
• p. 247 #5.2.5, 5.2.6, 5.2.8, 5.2.14 (note typo: midrange is (Y1+Y3)/2), 5.2.23b, 5.2.25,  5.2.28a

• p. 201 #4.1.7, 4.1.10, 4.1.12, 4.1.25, 4.1.26
• p. 207 #4.2.2, 4.2.4

• p 218 # 4.3.1, 4.3.2, 4.3.5, 4.3.7, 4.3.11, 4.3.16
• p 225 # 4.4.1, 4.4.8, 4.4.9, 4.4.10
• The following set of commands generates 2000 random uniform numbers, rearranges them into 2 rows and 1000 columns, then plots the histogram of the column means (2 uniform numbers per mean).

  > x <- runif(2000, min=0, max=1)
  > dim(x) <- c(2,1000)
  > hist(colMeans(x))
   

  Use similar commands to  display histograms for n=5, 10, 100.  Then generate exponential random variables with rate 2, and display histograms of sample means for n=2, 5, 10, 100.  Comment.  Then do the same, but using Cauchy random variables instead of exponential.  Comment again.  Why don't the Cauchy random variables behave the same as the others?  (You will need these commands:   x <- rexp(2000, rate=2)               x <- rcauchy(2000)   )

• p 270 #5.5.3, 5.5.4, 5.5.8, 5.5.9, 5.5.11
• p277 #5.6.4, 5.6.5, 5.6.6, 5.6.8
• In class, S was a binomial random variable with n=100, and unknown p. We found the exact values of the power function for a test that H0:p=.8 vs H1:p<.8.  We decided to reject if S<=74, which gave us a significance level of 0.0875.  We found power function values of 0.4465 for p=.75, 0.8369 for p=.7, and 0.9988 for p=.6. 

Suppose we want to run this test many times to see for ourselves the rejection rates.  In R, go to File, then New Script.  Type the following into the new window that opens:

x <- rbinom(1000, 100, .8)
y <- (x<=74)
sum(y)
 

The first command creates 1000 observations of a binomial random variable with n=100, p=.8.  (After running the commands, try typing x by itself at the carat prompt, and you will see the list.) The second tests each against the cutoff, and makes a list of TRUE and FALSE values. (See this by typing y at the prompt). The third counts the TRUE values (a TRUE is recorded as a 1, a FALSE as a 0, so adding them counts the TRUEs). 

Go to Edit, then Run All. Repeat a few times.  Then change the value of p to 0.75, 0.7, 0.6, and whatever other values you want.  How does the behavior of the test change?

p 294 # 5.8.1, 5.8.2, 5.8.5, 5.8.7

p330 # 6.2.1

p 350 #6.4.1, 6.4.2

p379 # 7.2.1, 7.2.2, 7.2.4, 7.2.8