1. For the lack of fit test discussed in class show that X = PZX.
2. Consider the data set in question 5. Use the original settings of x1, x2, x3, x4 and x5. Answer the following questions.
(a) Examine the data for any indication of the multicollinearity.
(b) Use R to fit a multiple regression model with all five x as the independent variables. Find a 95% prediction interval when x1 = 37.5, x2 = 2.2, x3 = 10, x4 = 5 and x5 = 100.
(c) Fit a multiple regression models with x1 and x2 as the independent variables. Then perform a lack of fit test for the fitted model.
3. A researcher studied the effects of the charge rate on the life of a new type of power cell in a preliminary small-scale experiment. The charge rate (X) was controlled at three levels (.6, 1.0, and 1.4). Factors pertaining to the discharge of the power cell were held at fixed levels. The life of the power cell (Y ) was measured in terms of the number of discharge-charge cycles that a power cell underwent before it failed. The data obtained in the study are contained in the following table.
cell y x
1 150.00 0.60
2 86.00 1.00
3 49.00 1.40
4 288.00 0.60
5 157.00 1.00
6 131.00 1.00
7 184.00 1.00
8 109.00 1.40
9 279.00 0.60
10 235.00 1.00
11 224.00 1.40
(a) Fit the following polynomial to the data.
y = βo + β1x + β2x2 + ε
(b) Now consider x as a factor and fit a one-way ANOVA to the data.
(c) Compare the SSE for the models in (a) and (b). What does this suggest for the lack of fit test?
4. A solid-fuel rocket propellant loses weight after it is produced. The following data are available:
months since weight loss,
production, x y (kg)
0.25 1.42
0.50 1.39
0.75 1.55
1.00 1.89
1.25 2.43
1.50 3.15
1.75 4.05
2.00 5.15
2.25 6.43
2.50 7.89
(a) Fit the following polynomial to the data.
yi = β0 + β1x + β2x2 + ε
(b) Apply the Gram-Schmidt orthogonalization procedure to the columns of X in (a) and then fit a regression model to the data.
(c) Fit the following polynomial to the data. Note that you may use package “far” in R for the Gram-Schmidt orthogonalization.
yi = β0 + β1(x − x‾) + β2(x − x‾)2 + ε.
(d) Evaluate the variance inflation factors for the models in (a), (b), and (c). What can you conclude about the impact of orthogonalization and centering the x’s in a polynomial model on multicollinearity?
(e) Use the models in (a) and (c) to construct a 95% prediction interval for weight
loss when x = 2.75.
5. An experiment was conducted to investigate the effect of five control variables on the selective H2SO4 hydrolysis of waxy maize starch granules. These variables were x1 = temperature, x2 = acid concentration, x3 = starch concentration, x4 = hydrolysisduration (time), and x5 = stirring speed. The measured response, Y = hydrolysis yield (wt%), was calculated as the ratio between the weight of freeze-dried hydrolyzed particles and the initial weight of native granules for an aliquot of 50mL taken in the 250 mL of hydrolyzed suspensions. The original and coded settings of x1, x2, x3, x4, x5 are given in the following table:
Note that the coded settings of the three equally-spaced levels of each variable are −1, 0, and 1. A complex design was used to measure the response Y at the specified combinations of the levels of the five factors. The resulting data set (using the coded settings of the control variables) is presented in Table below.
(a) Use the data in the following table to fit the model,
Y = β0 + i=1Σ5 βixi + β6x3x5 + β7x42 + ε
where ∼ N(0, σ2).
(b) Give values of the least-squares estimates of the model’s parameters and their standard errors.
(c) Obtain individual confidence intervals on the models parameters using a 95% confidence coefficient for each interval.
(d) Obtain Scheff ´es simultaneous 95% confidence intervals on the model’s parameters.
(e) Test the hypothesis
H0 : β1 + β2 + β4 = 0
β2 + 3β7 = 2
against the alternative hypothesis, Ha : H0 is not true, and state your conclusion at the α = 0.05 level.
(f) For the hypothesis in part (f), compute the power of the test under the alternative hypothesis,
H0 : β1 + β2 + β4 = 1
β2 + 3β7 = 4
given that σ2 = 1.