Question 1182549
Here's the analysis of the Agri-Data:

**Dependent and Independent Variables:**

*   **Dependent Variable (Y):** Rapeseed & Mustard Production (Metric Tonnes) - This is what we are trying to predict or explain.
*   **Independent Variable (X):** Rapeseed & Mustard Area (Hectare) - This is the variable we are using to predict production.

**Part A: Line of Regression Y on X**

The line of regression Y on X has the form: Y = a + bX

Where:

*   b (slope) = \[Σ(Xi - X̄)(Yi - Ȳ)] / \[Σ(Xi - X̄)²]
*   a (intercept) = Ȳ - bX̄

Here's the calculation:

1.  **Calculate the means:**
    *   X̄ (Mean Area) = (873 + 407 + 325 + 848 + 226) / 5 = 535.8
    *   Ȳ (Mean Production) = (437 + 193 + 154 + 466 + 139) / 5 = 277.8

2.  **Calculate Σ(Xi - X̄)(Yi - Ȳ):**

| District      | Area (X) | Production (Y) | (X-X̄) | (Y-Ȳ) | (X-X̄)(Y-Ȳ) |
|---------------|----------|----------------|-------|-------|-------------|
| Uttar Kashi   | 873      | 437            | 337.2 | 159.2 | 53669.44    |
| Chamoli       | 407      | 193            | -128.8| -84.8 | 10922.24    |
| Rudra Prayag  | 325      | 154            | -210.8| -123.8| 26095.04    |
| Tehri Garhwal | 848      | 466            | 312.2 | 188.2 | 58712.04    |
| Dehradun      | 226      | 139            | -309.8| -138.8| 43004.24    |
| **Total**       |          |                |       |       | **192303.00**|

3.  **Calculate Σ(Xi - X̄)²:**

| District      | Area (X) | (X-X̄) | (X-X̄)² |
|---------------|----------|-------|--------|
| Uttar Kashi   | 873      | 337.2 | 113703.84|
| Chamoli       | 407      | -128.8| 16589.44|
| Rudra Prayag  | 325      | -210.8| 44436.64|
| Tehri Garhwal | 848      | 312.2 | 97468.84|
| Dehradun      | 226      | -309.8| 95976.04|
| **Total**       |          |       | **368174.80**|

4.  **Calculate b:**
    b = 192303 / 368174.80 ≈ 0.522

5.  **Calculate a:**
    a = Ȳ - bX̄ = 277.8 - (0.522 * 535.8) ≈ -2.25

Therefore, the line of regression Y on X is: **Y = -2.25 + 0.522X**

**Part B: Coefficient of Determination (r²)**

r² = \[Σ(Ŷi - Ȳ)²] / \[Σ(Yi - Ȳ)²] where Ŷi are the predicted Y values from the regression line.

1.  **Calculate the predicted Y values (Ŷi) using the regression equation:**

| District      | Area (X) | Production (Y) | Ŷi = -2.25 + 0.522X | (Ŷi - Ȳ)² | (Yi - Ȳ)² |
|---------------|----------|----------------|----------------------|----------|----------|
| Uttar Kashi   | 873      | 437            | 453.75              | 2744.76 | 25344.64|
| Chamoli       | 407      | 193            | 209.69              | 278.57 | 7196.84|
| Rudra Prayag  | 325      | 154            | 167.35              | 205.81 | 15376.84|
| Tehri Garhwal | 848      | 466            | 440.79              | 1742.40 | 35436.64|
| Dehradun      | 226      | 139            | 115.61              | 397.02 | 19264.84|
| **Total**       |          |                |                      | **5369.36** | **102519.80**|

2.  **Calculate r²:**
    r² = 5369.36 / 102519.80 ≈ 0.052

**Interpretation of b (byx):**

The slope, b = 0.522, means that for every one-hectare increase in the area under rapeseed and mustard cultivation, the production is predicted to increase by 0.522 metric tonnes, on average.

**Interpretation of r²:**

The coefficient of determination, r² = 0.052, means that only 5.2% of the variation in rapeseed and mustard production can be explained by the variation in the area under cultivation. This indicates a very weak linear relationship between area and production. Other factors are likely playing a much larger role in determining production.