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)

 

 

  1. What are the coding equations for feed rate and humidity?
  2. The design is a 23 factorial. What are the relations between the estimates of Beta0, Beta1, Beta2, Beta3 and the factorial effects calculated in the usual way?

 

 

 

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:

  1. Obtain least squares estimates of Beta1 and Beta2 .
  2. Compute the residuals {Yt-Yhatt}.
  3. Calculate the sums of {X1t(Yt -Yhatt)}t and {X2t (Yt -Yhatt)}t , where t runs from 1 to 9.
  4. Show an analysis of variance appropriate for contemplation of the hypothesis that A and B pack at a rate of 30 crates per day, the rate laid down by the packers’ union.
  5. Split the residual sum of squares into a lack-of-fit term, SL = sum of
  6. {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.)

  7. Make a table of the following quantities, each having 9 values:

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.