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 ? / Wat is die waarde van ?

Answer 1

To fill in the missing values in the ANOVA table based on the information provided, we first need to compute the required values for the groups you're working with.

### Step 1: Collect Total and Group Data

#### Given Data:
- **Group 1 (Menu-driven)**: 12, 16, 10, 12, 10, 16 (n1 = 6)
- **Group 2 (Command-driven)**: 14, 10, 14, 12, 10, 14 (n2 = 6)
- **Group 3 (Mixed)**: 10, 8, 10, 10, 8, 4 (n3 = 6)

#### Calculate Group Totals (ΣXi):
- Group 1: \( 12 + 16 + 10 + 12 + 10 + 16 = 76 \)
- Group 2: \( 14 + 10 + 14 + 12 + 10 + 14 = 74 \)
- Group 3: \( 10 + 8 + 10 + 10 + 8 + 4 = 60 \)

#### Calculate Overall Total (ΣX):
\[
\text{Total} = 76 + 74 + 60 = 210
\]

### Step 2: Calculate Means
- Mean for Group 1: \( \bar{X}_1 = 76 / 6 = 12.67 \)
- Mean for Group 2: \( \bar{X}_2 = 74 / 6 = 12.33 \)
- Mean for Group 3: \( \bar{X}_3 = 60 / 6 = 10 \)

### Step 3: Summarize Sample Sizes
- \( n_1 = 6 \) (Group 1)
- \( n_2 = 6 \) (Group 2)
- \( n_3 = 6 \) (Group 3)
- Total Sample Size: \( N = n_1 + n_2 + n_3 = 6 + 6 + 6 = 18 \)

### Step 4: Calculate Sums of Squares

1. **Total Sums of Squares (SST)**:
\[
SST = \sum (X_i - \bar{X})^2
\]
Using values of each group:
\[
SST = SS1 + SS2 + SS3 + SSE
\]

2. **Treatment Sums of Squares (SSTr)**:
\[
SSTr = n_1 (\bar{X}_1 - \bar{X})^2 + n_2 (\bar{X}_2 - \bar{X})^2 + n_3 (\bar{X}_3 - \bar{X})^2
\]
But we need \(\bar{X}\), the overall mean:
\[
\bar{X} = 210 / 18 = 11.67
\]
Therefore:
\[
SSTr = 6 \cdot (12.67 - 11.67)^2 + 6 \cdot (12.33 - 11.67)^2 + 6 \cdot (10 - 11.67)^2
\]

3. **SS calculation**:
- For Group 1:
\[
6 \cdot (12.67 - 11.67)^2 = 6 \cdot (1)^2 = 6
\]
- For Group 2:
\[
6 \cdot (12.33 - 11.67)^2 = 6 \cdot (0.66)^2 \approx 2.62
\]
- For Group 3:
\[
6 \cdot (10 - 11.67)^2 = 6 \cdot (1.67)^2 \approx 16.67
\]
- Summing those gives:
\[
SSTr = 6 + 2.62 + 16.67 \approx 25.29
\]

### Step 5: Fill in the ANOVA Table
We can find \( H \) (SSTr) which would be \( \sum (X_i - \bar{X})^2 \).
- Similarly find SSE from total SS and SSTr.

### Conclusions:
Given the partial table:
- \( n1 = n2 = n3 = 6 \)
- \( N = 18 \)
- \( R1 = R2 = R3 = \text{Calculable from group sums} \)
- \( G = 210 \)

If you provide or confirm the correct values for degrees of freedom (df), errors (G, R etc.), I could finalize the table contents perfectly.

If further assistance is needed, please provide partial numerical values \( H, J, K, P \) if known.