Problem 1
Re-express the following expressions. In one case the expression may become significantly more complex.
-
-
-
-
Problem 2
Suppose that
, and
are variables and
are constants. Using the equation
compute the effect of changing
from
to
, holding
constant. Write in a logical form that makes it most
apparent what differences are being computed.
Problem 3
Name two substantially different statistical tests that would be useful for the each of the following hypotheses, assuming that needed assumptions hold.
- The population mean systolic blood pressure for treated and untreated patients is the same.
- The population mean systolic blood pressure for patients on placebo, drug A, and drug B are all equivalent.
- There is no association between systolic blood pressure and total serum cholesterol.
- The chance of a patient getting a stroke is the same for both sexes. (extra credit)
Problem 4
A randomized clinical trial is done to compare two treatments. What role if any does prediction play in this study?
Problem 5
Do all calculations by hand or by using low-level software functions (i.e., do not use any regression
functions or menus). Considering the following data:
x |
1 |
2 |
3 |
4 |
5 |
y |
98 |
198 |
315 |
380 |
530 |
- Compute least squares estimates a and b for simple linear regression
- Compute the predicted values and residuals from each observation
- For the fitted a and b and for the 4 other combinations of them obtained by multiplying and dividing b by 0.9 and by adding and subtracting 10 from a, compute the fitting criterion. Describe the patterns you see in the various values of the criterion. You may want to define an R function in order to save work, e.g.:
sse <- function(x, y, a, b) {
yhat <- a + b * x
sum((y - yhat)^2)
}
x <- 1:5
y <- c(98,198,315,380,530)
a <- ...
b <- ...
sse(x, y, a, b)
sse(x, y, a-10, b)
...