SOLUTION: Here is the Agri-Data, mention the name of dependent and independent variables. Part A: Prepare the line of regression Y on X. Part B: Calculate r2 (coefficient of determination

Question 1182550: Here is the Agri-Data, mention the name of dependent and independent variables.
Part A: Prepare the line of regression Y on X.
Part B: Calculate r2 (coefficient of determination). Interpreted the byx.
District_2, Rapeseed & Mastard Production in Metric Tonnes, Rapeseed & Mustard Area (Hectare)
Uttar Kashi: 437, 873
Chamoli: 193, 407
Rudra Prayag: 154, 325
Tehri Garhwal: 466, 848
Dehradun: 139, 226

Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
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.

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