Q 1.87. - Elementary Statistics

Question

In a designed experiment, there are two factors, say, Factor A and Factor B. Factor A has three levels, say, $a_1, a_2 \mathrm{and} a_3$; Factor B has four levels, say, $b_1, b_2, b_3 \mathrm{and} b_4$.

(a)  Construct a schematic for the treatments similar to Table 1.8 on page 27. 

(b)  Use part ( a ) to determine the number of treatments. 

(c)  Could you have determined the answer for part ( b ) without constructing the schematic in part ( a )? Explain your answer.

Step-by-Step Solution

Answer

Part (a)

Part (b) 12

Part (c) 12

Part (a) Step 1. Given information.

The given statement is:

In a designed experiment, there are two factors, say, Factor A and Factor B. Factor A has three levels, say, $a_1, a_2 \mathrm{and} a_3$; Factor B has four levels, say, $b_1, b_2, b_3 \mathrm{and} b_4$.

Part (a) Step 2. Construct a schematic for the treatments similar to Table 1.8 on page 27.

Construct a schematic for the treatments as shown below:

Part (b) Step 1. Determine the number of treatments.

There are twelve combinations based on the schematic for the treatments obtained in part (a).

As a result, there are a total of twelve treatments.

Part (c) Step 1. Determine the number of treatments without constructing the schematic.

The number of levels for factor A is three, while factor B has four levels.

The number of treatments will equal the product of factor A and factor B :

$$\begin{gathered} =\mathrm{Number} \mathrm{of} \mathrm{levels} \mathrm{of} \mathrm{factor} A \times \mathrm{Number} \mathrm{of} \mathrm{levels} \mathrm{of} \mathrm{factor} B \\ =3\times 4 \\ =12 \end{gathered}$$

As a result, The total number of levels is 12.