Homework Assignment for Lesson 1

1. Suppose that you are planning to run an experiment with one treatment factor having four levels: ''none'', ''low'', ''medium'', and ''high'', and you have the resources to conduct the experiment on 20 experimental units. Assign at random 20 experimental units to the 4 levels of the treatment, so that each treatment is assigned 5 units. Your answer should include your SAS code used.

2. Repeat question 1 to obtain a second experimental design assigning the 20 units to the 4 levels of the treatment.

3. Suppose that you are planning to run an experiment with one treatment factor having three levels. It has been determined that $r_1=3$, $r_2=r_3=5$. Assign at random 13 experimental units to the 3 treatments so that the first treatment is assigned 3 units and the other two treatments are each assigned 5 units.

4. Visit http://www.tylervigen.com/spurious-correlations (or some other website of your choosing) and find an example of two observed quantities that are correlated, but you think are not causally related. Clearly show the data (you could download an image), and describe why you think the two quantities are not causally related. Give an example of another factor (not measured) which you think could have a causative link with one or both of the quantities shown. Give some explanation for why this not measured factor could be causally linked to one or both of the quantities.

5. Let $X \sim N(2,6)$ and $Y \sim N(-3,2)$ and $Z \sim N(0,1)$. All three random variables are independent of each other. Do the following. Show all work.

  1. What is the distribution of $W=X+Y+Z$ ? What are $E(W)$ and $Var(W)$?
  2. What is the distribution of $Q=2Y$?
  3. What is the distribution of $P=-2X+4$?
  4. Find $a$ and $b$ so that $M=a+bX$ is distributed as a standard Normal distribution.

6. Use SAS to simulate 1000 iid random variables $\{X_i\}$ with $X_i \sim N(-2,3)$. Plot a histogram of your simulated values.

  1. Also simulate 1000 iid random variables $\{Y_i\}$ with $Y_i \sim (3,1)$. Plot a histogram of your simulated values.
  2. Finally, plot a histogram of $\{Z_i\}$, where $Z_i=X_i+Y_i$.
  3. Is $Z_i$ independent of $X_i$? Explain your answer.
  4. Compare the theoretical mean and variance of $Z_i$ with the sample mean and variance of the $Z_i$s.

7.

  1. Let $X_i \sim N(0,1),\ i=1,2,3$ be 3 iid standard normal random variables.
  2. What is the distribution of $Q_1=X_1^2$?
  3. What is the distribution of $Q_2=X_2^2+X_3^2$?
  4. What is the distribution of $Q_3=X_1^2+X_2^2+X_3^2$?
  5. What is the distribution of $G=\frac{2Q_1}{Q_2}$?
  6. Explain why $\frac{Q_3/3}{Q_2/2}$ is NOT an $F$-distributed random variable.