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 β₀, β₁, and β₂ in the model:

η = β₀ + β₁X₁ + β₂X₂

where η = 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 β₂, assuming that the standard deviation of the observations σ = 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:

η = β₀ + β₁X₁ + β₂X₂ + β₃X₃

where η 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:

X_i

i = 1

temperature

(deg C)

i = 2

feed rate

(kg/min)

i = 3

humidity

(%)

-1

300

400

The coding equation for temperature is

X₁ = 1/50 ( temperature in units of deg.C - 350 deg.C)

What are the coding equations for feed rate and humidity?

The design is a 2³ factorial. What are the relations between the estimates of β₀, β₁, β₂, β₃and the factorial effects calculated in the usual way?

3.Mr. A and Mr. B, both newly employed, pack crates of material. Let X₁ and X₂be 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 X_1t	B X_2t	Y_t
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 β₁ and β₂ the number of crates packed per day by A and B, respectively.

First entertain the model

η = β₁X₁ + β₂X₂

Carefully stating what assumptions you make, do the following:

Obtain least squares estimates of β₁ and β₂.

Compute the residuals {Y_t-Ŷ_t}.

Calculate Σ_t X_1t (Y_t-Ŷ_t) and Σ_t X_2t (Y_t-Ŷ_t) , where t runs from 1 to 9.

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.

Split the residual sum of squares into a lack-of-fit term, S_L = Σ_u 3(Ŷ_u - Ϋ_u)² and an “error” term Σ_u Σ_i(Y_ui-Ϋ_u)² ,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.)

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

X₁, X₂, Y_ui, Ŷ_u, Ϋ_u, Ŷ_u – Ϋ_u, and Y_ui-Ϋ_u, and make relevant plots.

Look for a time trend in the residuals.

at this point it was suggested that the presence of B might stimulate A to do

more (or less) work by an amount β₁₂, while the presence of A might stimulate B to do more (or less) work by an amount β₂₁. Furthemore, there might be a time trend. A new model was therefore tentatively entertained:

η = ( β₁+ β₁₂X₂ ) * X₁+ ( β₂ + β₂₁X₁ ) * X₂+ β₄* t

that is,

η = β₁X₁+ β₂X₂+ ( β₁₂+ β₂₁ )* X₁* X₂ + β₄* t

The β₁₂ and β₂₁ terms cannot be separately estimates, but the combined effect

β₁₂+ β₂₁ = β₃ can be estimated from this model:

η = β₁* X₁+ β₂* X₂ + β₃* X₃ + β₄* t

where X_1*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.

What do you think is going on? Consider other ways in which the original model could have been wrong.