Assignment 4, due Oct 24, 2014
1. Suppose v = (x, y)T is a random vector with mean vector = (2, 1)T , and variance-covariance matrix
=
If a = (2, 3)T , and A =
1
2
2 1
1 4
2
4
(a) Calculate the mean of aT v.
(b) Calculate the variance of aT v.
(c) Calcula
Assignment 5, due Oct 31, 2014
1. For the stress data set, test whether there is any linear association between the response BP and the
predictors AGE and ST RESS, given that W EIGHT is already in the model. Use, AND INCLUDE output
from one or more lm ts.
Assignment 6, 2014, Due Nov. 28
1. For the stackloss dataset, consider the model
stack.loss = 0 + 1 Air.F low + 2 W ater.T emp + 3 W ater.T emp2 +
(a) Obtain the leverage values for this model and plot them against the case number (order). Discuss the va
Assignment 1, due Sept. 19 2014
1. Find the equation of the line which passes through the points (1,1) and
(3,5).
2. Suppose you are given three data points (1,4), (2,6) and (3,7) and the
line y = 2 + 3x. Give the three residuals and their sum of squares.
Assignment 1, due Sept. 19 2014
(4)
1. Find the equation of the line which passes through the points (1,1) and
(3,5).
slope = rise/run, so slope = (5-1)/(3-1) = 2
y-intercept is -1, because a decrease in 1 in the horizontal direction
leads to a change o
Simple Linear Regression
1. Model and Parameter Estimation
(a) Suppose our data consist of a collection of n pairs (xi , yi ), where xi is an observed value
of variable X and yi is the corresponding observation of random variable Y . The simple
linear reg
Hypothesis Testing
The basic ingredients of a hypothesis test are
1
2
3
4
5
the
the
the
the
the
null hypothesis, denoted as Ho
alternative hypothesis, denoted as Ha
test statistic
data
conclusion.
The hypotheses are usually statements about the values of
Lecture 1. Introduction
An example multiple regression data set
Following Hald cement data, example 10.1, p338, and Table B21, p573
of Mongomery, Peck and Vining.
1
2
3
4
5
6
7
8
9
10
11
12
13
i
1
2
3
4
5
6
7
8
9
10
11
12
13
y x1 x 2 x 3 x 4
78.50
7 26
6
Assignment 2, due Sept. 29 2014
1. Suppose X and Y are random variables with x = 10, x = 2, y = 4,
y = 1, and Cov[X, Y ] = 1.5. Calculate
(a) E[2X Y ]
(b) V ar[2X Y ]
(c) Cor[2X, Y ], where Cor stands for correlation.
(d) Find coecients a and b so that Co
Why least squares?
When there is only one predictor variable, we refer to it as x
(rather than x1 ), and the data are (xi , yi ), i = 1, . . . , n
To fit a line y = 0 + 1 x to some data we commonly use
least squares
This minimizes the sum of squares of ve
Derivation of formulae for intercept and slope
We wish to minimize
S(0 , 1 ) =
X
(yi 0 1 xi )2 .
with respect to 0 and 1 .
There are two variables 0 and 1 , and so both partial
derivatives can be set to zero and solved.
(A partial derivative with respect
STAT 3340 Assignment 1, Fall 2016 - due Sept 23
1. Find the equation of the line which passes through the points (1,1) and (4,5).
2. Suppose you are given three data points (1,4), (2,6) and (3,7) and the line y = 1 + 4x. Give
the three residuals and their
STAT 3340 Assignment 1, Fall 2016 - due Oct 21
The length of a species of fish is to be represented as a function of the age and water temperature.
The fish are kept in tanks at 25, 27, 29 and 31 degrees Celsius. After birth, a test specimen is chosen
at
Stat/Math 3340, Assignment 3, due Friday, October 28
1. Data was collected on the degree to which oceanic phytoplankton is inhibited by exposure to
ultraviolet radiation. The following shows a plot of inhibition (y) vs uv concentration (x1), with
separate
Stat/Math 3340, Assignment4, due Friday, Nov 25
1. The data set fish on the class website (available either as a .csv file) has data on fish lengths,
age and water temperature. Using the lm command, fit the linear model
y = 0 + 1 x1 + 2 x2 +
where x1 is
Confidence intervals (CIs) I
There are two equivalent ways of defining confidence intervals
for a parameter .
First Method I
The first defines a confidence interval to be the realization of
a random interval which contains the true value of the
parameter
Review of random variables, means and variances
We model observations using random variables and probability
functions.
Discrete random variables take on discrete values (in
one-to-one correspondence with the integers).
The probability mass function assig
Assignment 3, due Oct. 10, 2014
1. In a study of 12 subjects, a correlation of r = .62 was found between
their income and the number of miles driven in a month. Assess whether
the correlation is signicantly dierent from 0.
(a) State the two hypotheses.
(b
Assignment 2, due Sept. 29 2014
1. Suppose X and Y are random variables with x = 10, x = 2, y = 4,
y = 1, and Cov[X, Y ] = 1.5. Calculate
(3)
(4)
(4)
(a) E[2X Y ].
E[2X Y ] = E[2X] E[Y ] = 2E[X] 4 = 2(10) 4 = 16
(b) V ar[2X Y ] .
V ar[2X Y ] = 4V ar[X]
Assignment 3, due Oct. 10, 2014
1. In a study of 12 subjects, a correlation of r = .62 was found between
their income and the number of miles driven in a month. Assess whether
the correlation is signicantly dierent from 0.
(2)
(4)
(a) State the two hypoth
Assignment 4, due Oct 24, 2014
1. Suppose v = (x, y)T is a random vector with mean vector = (2, 1)T , and variance-covariance matrix
2 1
1 4
=
If a = (2, 3)T , and A =
(2)
1
2
2
4
(a) Calculate the mean of aT v.
Using the formula E(a v) = a E(v)
E(a v) =
Assignment 5, due Oct 31, 2014
1. For the stress data set, test whether there is any linear association between the response BP and the
predictors AGE and ST RESS, given that W EIGHT is already in the model. Use, AND INCLUDE output
from one or more lm ts.
Assignment 6, 2014 Solutions
1. For the stackloss dataset, we have been using the model
stack.loss = 0 + 1 Air.F low + 2 W ater.T emp + 3 W ater.T emp2 +
(a) Obtain the leverage values for this model and plot them against the case number (order). Discuss
1
Name: 6W1; _ _ Student Number: _ _
STATISTICS 3340/MATH 3340 Midterm Test, Friday Nov. 1, 2013
Please answer the questions in the space provided. Justify your answers.
One sheet of formulae/notes, letter size. is permitted.
1. The model ,I/ = i0 + Jim +
(2)
(4)
(3)
STATISTICS 3340/MATH 3340 Final Exam. Mon. Dec. 16. 2013
Please answer the questions in the space provided. Justify your answers.
Three sheets of formulae/notes, letter size, are permitted.
1. A random vector v = (in. my has mean p. = c(1.2) a
Student Number: _
STATISTICS 3340/ MATH 3340 Midterm Test 1. Wed. . Oct 2, 2013
Please answer the questions in the space provided. Justify your answers.
One sheet of formulae/notes, letter size, are permitted.
1. To assess the benefit of the use of CT flu
STAT 3340 Assignment 3 solutions
(out of 45 points)
1. Data was collected on the degree to which oceanic phytoplankton is inhibited by exposure to
ultraviolet radiation. The following shows a plot of inhibition (y) vs uv concentration (x1),
with separate
STAT 3340 Assignment 2 solutions
(out of 50 points)
The length of a species of fish is to be represented as a function of the age and water temperature.
The fish are kept in tanks at 25, 27, 29 and 31 degrees Celsius. After birth, a test specimen is chose