Exercise 8
1.The following data, which are given in coded units, were obtained from a tool-life testing investigation in which X1 = measure of the feed, X2 = measure of the speed,
Y = measure of the tool life:
X1 |
X2 |
Y |
-1 |
-1 |
61 |
1 |
-1 |
42 |
-1 |
1 |
58 |
1 |
1 |
38 |
0 |
0 |
50 |
0 |
0 |
51 |
(a)Obtain the least squares estimates of the parameters Beta0, Beta1, and Beta2 in the model:
Mu = Beta0 + Beta1 * X1 + Beta2 * X2
where Mu = the mean value of Y.
(b)Which variable influences tool life more over the region of experimentation, feed or speed?
(c)Obtain a 95% confidence interval for Beta2, assuming that the standard deviation of the observations Sigma = 2.
(d)Carry out any further analysis that you think is appropriate.
(e)Is there a simpler model that will adequately fit the data?
2.Using the data below, determine the least square estimates of the coefficients of this model:
Mu = Beta0 + Beta1 * X1 + Beta2 * X2 + Beta3 * X3
Where Mu is the mean value of the drying rate.
Temperature (deg C) |
Feed rate (kg/min) |
Humidity (%) |
Drying rate (coded units) |
300 |
15 |
40 |
3.2 |
400 |
15 |
40 |
5.7 |
300 |
20 |
40 |
3.8 |
400 |
20 |
40 |
5.9 |
300 |
15 |
50 |
3.0 |
400 |
15 |
50 |
5.4 |
300 |
20 |
50 |
3.3 |
400 |
20 |
50 |
5.5 |
Coding for the variables is as follows:
Xi |
i = 1 temperature (deg C) |
i = 2 feed rate (kg/min) |
i = 3 humidity (%) |
-1 |
300 |
15 |
40 |
+1 |
400 |
20 |
50 |
The coding equation for temperature is
X1 = 1/50 ( temperature in units of deg.C - 350 deg.C)
3.Mr. A and Mr. B, both newly employed, pack crates of material. Let X1 and X2 be indicator variables showing which parker is on duty, and Y be the number of crates packed. On 9 successive working days these data were collected:
Day t |
A X1t |
B X2t |
Yt |
1 |
1 |
0 |
48 |
2 |
1 |
1 |
51 |
3 |
1 |
0 |
39 |
4 |
0 |
1 |
24 |
5 |
0 |
1 |
24 |
6 |
1 |
1 |
27 |
7 |
0 |
1 |
12 |
8 |
1 |
0 |
27 |
9 |
1 |
1 |
24 |
Denote by Beta1 and Beta2 the number of crates packed per day by A and B, respectively.
First entertain the model
Mu = Beta1 * X1 + Beta2 * X2
Carefully stating what assumptions you make, do the following:
{3(Yhatu - Ybaru)2}u , and an “error” term, sum of {{(Yui -Ybaru)2}i}u , where u and i run from 1 to 3.(Note: The above design consists of three distinct sets of conditions, (X1, X2) = (0,1), (1,0), (1,1), which are designated by the subscripts u = 1, 2, 3, respectively. The subscripts i = 1, 2, 3 are used to designate the three individual observations included in each set.)
X1, X2, Yui, Yhatu, Ybaru, Yhatu – Ybaru, and Yui -Ybaru, and make relevant plots.
Look for a time trend in the residuals.
(g) At this point it was suggested that the presence of B might stimulate A to do
more (or less) work by an amount Beta12, while the presence of A might stimulate B to do more (or less) work by an amount Beta21. Furthemore, there might be a time trend. A new model was therefore tentatively entertained:
Mu = ( Beta1 + Beta12 * X2 ) * X1 + ( Beta2 + Beta21 * X1 ) * X2 + Beta4 * t
that is,
Mu = Beta1 * X1+ Beta2 * X2 + ( Beta12 + Beta21 )* X1 * X2 + Beta4 * t
The Beta12 and Beta21 terms cannot be separately estimates, but the combined effect Beta12+ Beta21 = Beta3 can be estimated from this model:
Mu = Beta1 * X1+ Beta2 * X2 + Beta3 * X3 + Beta4 * t
where X1 * X2 = X3. Fit this model, examine the residuals, and, if the model appears adequate, report least squares estimates of the parameters together with their standard errors.
(h)What do you think is going on? Consider other ways in which the original
model could have been wrong.