Which one of the following can be considered as appropriate pair of values of regression coefficient of y on x and regression coefficient of x on y?

Options

A. `(1,1)`
B. `(-1,1)`
C. `(-(1)/(2),2)`
D. `((1)/(3),(10)/(3))`

Explanation

To solve the problem of identifying the appropriate pair of values for the regression coefficients of y on x and x on y, we need to follow these steps:

### Step 1: Understand the properties of regression coefficients
The regression coefficients \( b_{y|x} \) (regression of y on x) and \( b_{x|y} \) (regression of x on y) must satisfy two conditions:
1. The product of the two coefficients should be less than or equal to 1:
\[ b_{y|x} \times b_{x|y} \leq 1 \]
2. The signs of the coefficients should be the same (both positive or both negative).

### Step 2: Analyze the given options
We will analyze each option provided to see if they meet the conditions stated above.

1. **Option 1: (1, 1)**
- Product: \( 1 \times 1 = 1 \) (satisfies the first condition)
- Signs: Both are positive (satisfies the second condition)
- **Conclusion**: This option is valid.

2. **Option 2: (-1, 1)**
- Product: \( -1 \times 1 = -1 \) (satisfies the first condition)
- Signs: Different signs (does not satisfy the second condition)
- **Conclusion**: This option is invalid.

3. **Option 3: (0.5, 0.5)**
- Product: \( 0.5 \times 0.5 = 0.25 \) (satisfies the first condition)
- Signs: Both are positive (satisfies the second condition)
- **Conclusion**: This option is valid.

4. **Option 4: (10/3, 1/3)**
- Product: \( \frac{10}{3} \times \frac{1}{3} = \frac{10}{9} \) (this is greater than 1, does not satisfy the first condition)
- Signs: Both are positive (satisfies the second condition)
- **Conclusion**: This option is invalid.

### Step 3: Finalize the answer
From our analysis, the valid pairs of regression coefficients are:
- Option 1: (1, 1)
- Option 3: (0.5, 0.5)

However, since the question asks for an appropriate pair, the most straightforward and commonly accepted pair is (1, 1).

### Final Answer:
The appropriate pair of values of regression coefficients of y on x and regression coefficients of x on y is **(1, 1)**.

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