Question

The science of ergonomics studies the influence of “human factors” in technology, i.e. how human beings relate to and work with machines. With widespread use of computers for data processing, computer scientists and psychologists are getting together to study human factors. One typical study investigates the productivity of secretaries with different word processing programs. An identical task was given to 18 secretaries, randomly allocated to three groups. Group 1 used a menu-driven program. Group 2 used a command-driven program, and Group 3 used a mixture of both approaches. The secretaries all had about the same level of experience, typing speed, and computer skills. The time (in minutes) taken to complete the task was observed. The results were as follows:


Groep / Group 1

Groep / Group 2

Groep / Group 3

Totaal / Total

Menu-aangedrewe / Menu-driven

Opdrag-aangedrewe / Command-driven

Mengsel / Mixed



12

14

10



16

10

8



10

14

10



12

12

10



10

10

8



16

14

4



n1 =

n2 = A

n3 =

N = B

R1 =

R2 =

R3 = C

G = D










Die volgende gedeeltelik voltooide ANAVA tabel is vir die bostaande data verkry. / The following partially completed ANOVA table was obtained for the above data.

ANOVA table:

Source

SS

df

MS

Fcalc

Ftab

Treatments

H

J

34.8889

M

P

Error

84

15

L



Total

I

K






What is the value of E in the observation table? / Wat is die waarde van E in die waarnemingstabel?

Answer 1

To determine the value of \(E\) in the ANOVA table, we'll first summarize the information given in the problem, and then we can move on to calculations.

### Summary of the Data
We have three groups of secretaries:
- Group 1 (Menu-driven): 12, 14, 10, 12, 10, 16
- Group 2 (Command-driven): 14, 10, 8, 12, 10, 14
- Group 3 (Mixed): 10, 8, 10, 10, 4

**Step 1: Calculate Number of Observations**
- For each group, \(n_1 = n_2 = n_3 = 6\) (6 observations per group).
- Total number of observations, \(N = n_1 + n_2 + n_3 = 6 + 6 + 6 = 18\).

### Step 2: Calculate Group Sums and Grand Total

**Group 1 Total:**
\[
R_1 = 12 + 14 + 10 + 12 + 10 + 16 = 74
\]

**Group 2 Total:**
\[
R_2 = 14 + 10 + 8 + 12 + 10 + 14 = 78
\]

**Group 3 Total:**
\[
R_3 = 10 + 8 + 10 + 10 + 4 = 52
\]

**Grand Total:**
\[
G = R_1 + R_2 + R_3 = 74 + 78 + 52 = 204
\]

### Step 3: Calculate the Required Values
Now we can assign values based on definitions:
- \(n_1 = 6\)
- \(n_2 = 6\)
- \(n_3 = 6\)
- \(N = 18\)
- \(R_1 = 74\)
- \(R_2 = 78\)
- \(R_3 = 52\)
- \(G = 204\)

### Step 4: Calculate Total Sum of Squares \(SS_T\)

The total sum of squares can be calculated as:
\[
SS_T = \sum (x_{ij} - \bar{x})^2
\]
where \(\bar{x} = \frac{G}{N} = \frac{204}{18} \approx 11.33\).

Calculating \(SS_T\):
- For Group 1:
\[
SS_1 = (12 - 11.33)^2 + (14 - 11.33)^2 + (10 - 11.33)^2 + (12 - 11.33)^2 + (10 - 11.33)^2 + (16 - 11.33)^2
\]
- For Group 2:
\[
SS_2 = (14 - 11.33)^2 + (10 - 11.33)^2 + (8 - 11.33)^2 + (12 - 11.33)^2 + (10 - 11.33)^2 + (14 - 11.33)^2
\]
- For Group 3:
\[
SS_3 = (10 - 11.33)^2 + (8 - 11.33)^2 + (10 - 11.33)^2 + (10 - 11.33)^2 + (4 - 11.33)^2
\]

### Step 5: Error Sum of Squares \(SS_E\)
Given that we have \(SS_T\) and \(SS_{Treatments}\), we can deduce:
\[
SS_E = SS_T - SS_{Treatments}
\]
We know:
- \(SS_T \) (Total Sum of Squares)
- \(SS_{Treatments} = H\) (This would need to be calculated as above based on group means.)

For now, the value \(E\) specifically corresponds to the overall error from the treatments, which can be assigned to whatever totals you receive from calculations. Since the original prompt asked directly for \(E\), we will summarize that;

### Final Calculation of Error
Assume hypothetical values based on previous iterations:

The final calculated error \(E\) can be found by:

Using the existing data:
- Values of \(H\) and results will lead into \(E\).

Until we derive an explicit numeric value for \(E\), it is indeed yielded as:
\[
SS_E = 84\, \text{(from table structure)}
\]
Thus **the answer to your request for the value of \(E\)** = the \(SS_{Error} = 84\).