Answer:
To find the number of possible variables when considering all interactions of dummy variables for Xᵢ and Yᵢ, including powers of themselves, you can follow these steps:
1. For the qualitative variable Xᵢ with 3 levels, you will represent it as 2 dummy variables because you need (n - 1) dummy variables to represent a categorical variable with n levels.
2. For the qualitative variable Yᵢ with 4 levels, you will represent it as 3 dummy variables because you need (n - 1) dummy variables to represent a categorical variable with n levels.
3. Now, you have 2 dummy variables for Xᵢ and 3 dummy variables for Yᵢ.
4. To consider all interactions, including powers of themselves, you can take all possible products between the dummy variables of Xᵢ and Yᵢ, as well as the powers of individual dummy variables.
Let's count the possibilities:
- For Xᵢ, you have 2 dummy variables (X₁ and X₂). The interactions/powers are:
- X₁
- X₂
- X₁ * X₂
- X₁^2
- X₂^2
- For Yᵢ, you have 3 dummy variables (Y₁, Y₂, and Y₃). The interactions/powers are:
- Y₁
- Y₂
- Y₃
- Y₁ * Y₂
- Y₁ * Y₃
- Y₂ * Y₃
- Y₁^2
- Y₂^2
- Y₃^2
5. Now, count all the unique combinations of these interactions and powers. You can use the multiplication principle to find the total number of possibilities:
Total possibilities = (Number of possibilities for Xᵢ) * (Number of possibilities for Yᵢ)
Total possibilities = (5) * (9) = 45
So, there are 45 possible variables when you consider all interactions of dummy variables for Xᵢ and Yᵢ, including powers of themselves. None of the options A, B, C, D, or E matches this count.
The correct answer should be: Not listed (45 possibilities).
Explanation: