Homework Project 1

Homework Project 1

Note: For all homework, use R for calculations. Do not give raw computer output as your main answer to any question; do include R code in an appendix. Do not report numbers to more significant digits than is warranted. It is recommended to use the options:

options(digits=4, show.signif.stars=FALSE)

Remember that providing a clear and reasonable justification of your answers is at least as important as getting the answer right.

Problem 1.1

1. Enter the following R commands:

set.seed(201801)

U <- matrix(runif(1000 * 200), nrow=1000, ncol=200) z <- apply (U, 2, mean)

1. These commands create a 1000x200 (a thousand rows and two hundred columns) matrix of random draws, called U and a vector z of length 200 which contains the means of each column of U. Now enter the command hist(U[,1]). This command takes the first column of U (a column vector of length 1000) and makes a Print out this histogram and describe what it looks like. What distribution is the runif command drawing from?
2. Now enter the command hist(z). This command makes a histogram from the vector z. Print out this Describe what it looks like and how it differs from the one above. Based on the histogram, what distribution do you think z follows?
3. You generated U and z with the same random draws, so how can they have different distributions? What’s going on here?

Problem 1.2

An investigator does not know the formula for the area of a rectangle.  Taking an empirical approach, she creates 20 typical rectangles using the following R code set.seed(201801)

length <- rnorm(20, mean=4, sd=0.5) width <- rnorm(20, mean=3, sd=0.5)

1. For each rectangle, calculate area and perimeter, and construct a scatterplot for the area (y-axis) versus perimeter (x-axis)
2. Fit a simple linear regression using the area as the response, and add the fitted line to the
3. What is the correlation coefficient? Is this a good fit?
4. For a new rectangle with length = 1 and width = 1, predict the value of area using the fitted regression.
5. Does the regression make sense to build the relationship between area and perimeter?

Problem 1.3

The ratdrink data from the faraway package consist of 5 weekly measurements of body weight for 27 rats. The first 10 rats are on a control treatment while 7 rats have thyroxine added to their drinking 10 Rats have thiouracil added to their water.

1. Construct three scatterplots (treat = control, thiouracil, or thyroxine) using lattice to show the relationship between wt and weeks for each subject.
2. Fit a random intercept model to show the effect of the treatment
3. Fit a random intercept, random slope model to show the effect of the treatment
4. Compare the two models in b) and c) and comment on which model fit the data.