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ORBS7290 Regression Analysis
Assignment 2
Submission deadline: 11:59pm, 18/05/2024
1. Consider to it the data set f(xi, yi), i = 1, · · · , ng by the regression model
yi = β0 + β1xi + β2xi(3) + εi , (1)
where β0 , β1 and β2 are unknown constants, fεig are i.i.d. with E(εi) = 0 and Var(εi)=σ2 . The model (1) can be expressed as the following standard form
y = XB + ε (2)
(a) Write down y , X , B and ε .
(b) Find E(ε) and Cov(ε).
(c) Find E(y) and Cov(y).
2. Consider to it the data set f(xi, yi), i = 1, · · · , ng by the regression model
yi = βxi(2) + εi ,
Find the least squares estimator of β .
3. Show that the residuals from a linear regression model can be expressed as
e = (I — H)ε,
where H = X(X' X)-1 X' . Furthermore, calculate the expectation and covariance matrix of the residual vector e, i.e., E(e) and Cov(e).
4. The data shown below present the average number of surviving bacteria in a canned food product and the minutes of exposure to 300o F heat.
|
Number of Bacteria |
Minutes of Exposure |
|
175 108 95 82 71 50 49 31 28 17 16 11 |
1 2 3 4 5 6 7 8 9 10 11 12 |
(a) Plot a scatter diagram. Does it seem likely that a straight-line model will be adequate?
(b) Fit the straight-line model. Compute the summary statistics and the residual plots. What are your conclusions regarding model adequacy?
(c) Identify an appropriate transformed model for these data. Fit this model to the data and conduct the usual tests of model adequacy.
5. Fitness data give various measures of heart and pulse rate taken on men in a physical itness course. The goal is to predict the rate of oxygen consumption (which is difficult and expensive to measure) from the other variables. The following factors were considered:
X1: Age in year
X2: Weight in kilograms
X3: Time to run 1 2/1miles
X4: Resting pulse rate
X5: Pulse rate at begin of run
X6: Pulse rate at end of run
Y : Oxygen consumption in milliliters (ml) per kilogram (kg) body weight per minute. The data is given in the table below.
(a) Consider the following multiple linear model:
y = β0 + β1x1 + β2x2 + β3x3 + β4x4 + β5x5 + β6x6 + ε, (3)
where the random error ε N (0, σ2 ). Estimate the coefficients of the model and σ2 by the least squares method.
(b) Applying the forward selection, backward elimination, stepwise regression to model (3) and give your recommended models by using other criteria.
(c) Consider the following quadratic regression model:
y = β0 + βixi + βijxixj + ε, (4)
where the random error ε N (0; σ2 ). Estimate the coe伍cients of the model and σ2 by the least squares method.
(d) Applying the forward selection, backward elimination, stepwise regression to model (4) and give your recommended models by using other criteria.