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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:



Group 1

Group 2

Group 3

Total

Menu-driven

Command-driven

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

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 H in the ANOVA table?

1 answer

To find the value of \( H \) in the ANOVA table, we first need to calculate the sum of squares for treatments (SS_Treatment). The treatments in this case are the three groups using different types of word processing programs.

Steps to Calculate SS_Treatment (H):

  1. Calculate the total mean (Grand Mean):

    • First, we calculate the total time taken to complete the task for all secretaries (combined from all groups).
    • Then, we divide that total by the total number of observations (N) to get the Grand Mean.

  2. Calculating the sum of squares for each treatment group:

    • For each group, we calculate the sum of squares for that group, which is given by the formula:

    \[ SS_{Group} = n_{Group} \times (\text{Mean}_{Group} - \text{Grand Mean})^2 \]

    where \( n_{Group} \) is the number of observations in that group, \( \text{Mean}_{Group} \) is the mean time for that group, and the Grand Mean is the overall mean time across all groups.

  3. Calculate total sum of squares for treatments (H):

    • Finally, we sum the SS values for each individual group.

Data for Each Group:

  • Group 1 (Menu-driven): Times = [12, 16, 10, 12, 10, 16]
  • Group 2 (Command-driven): Times = [14, 10, 14, 12, 10, 14]
  • Group 3 (Mixed): Times = [10, 8, 10, 10, 8, 4]

Step 1: Calculate Group Means:

  • Group 1 Mean: \[ \text{Mean}_{G1} = \frac{12 + 16 + 10 + 12 + 10 + 16}{6} = \frac{76}{6} \approx 12.67 \]

  • Group 2 Mean: \[ \text{Mean}_{G2} = \frac{14 + 10 + 14 + 12 + 10 + 14}{6} = \frac{74}{6} \approx 12.33 \]

  • Group 3 Mean: \[ \text{Mean}_{G3} = \frac{10 + 8 + 10 + 10 + 8 + 4}{6} = \frac{60}{6} = 10 \]

Step 2: Calculate the Grand Mean:

Total time for all groups: \[ \text{Total} = 12+16+10+12+10+16 +14+10+14+12+10+14 +10+8+10+10+8+4 = 178 \]

Total observations (N = 18): \[ \text{Grand Mean} = \frac{Total}{N} = \frac{178}{18} \approx 9.89 \]

Step 3: Calculate SS for Each Group:

\[ SS_{G1} = 6 \times (12.67 - 10)^2 = 6 \times (2.67)^2 = 6 \times 7.1289 \approx 42.7734 \ SS_{G2} = 6 \times (12.33 - 10)^2 = 6 \times (2.33)^2 = 6 \times 5.4289 \approx 32.5734 \ SS_{G3} = 6 \times (10 - 10)^2 = 6 \times (0)^2 = 0 \]

Step 4: Calculate Total SS (H):

Now sum these SS for treatments:

\[ H = SS_{G1} + SS_{G2} + SS_{G3} \ H = 42.7734 + 32.5734 + 0 \ H \approx 75.3468 \approx 75.347 ( rounding is acceptable) \]

Thus, the value of \( H \) in the ANOVA table is approximately 75.347.