document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182549>Question 1182549</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Here is the Agri-Data, mention the name of dependent and independent variables. <BR>\n" );
document.write( "Part A: Prepare the line of regression Y on X. (5 Marks)<BR>\n" );
document.write( "Part B: Calculate r2 (coefficient of determination). Interpreted the byx.\r<BR>\n" );
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document.write( "District_2 / 	Rapeseed & Mustard Production in Metric tonnes / Rapeseed & Mustard Area ( Hectare)  	<BR>\n" );
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document.write( "Uttar Kashi/    437/ 873<BR>\n" );
document.write( "Chamoli/        193/407<BR>\n" );
document.write( "Rudra Prayag/   154/325<BR>\n" );
document.write( "Tehri Garhwal/  466/848<BR>\n" );
document.write( "Dehradun/       139/226 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849923 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849923\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's the analysis of the Agri-Data:\r<BR>\n" );
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document.write( "**Dependent and Independent Variables:**\r<BR>\n" );
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document.write( "*   **Dependent Variable (Y):** Rapeseed & Mustard Production (Metric Tonnes) - This is what we are trying to predict or explain.<BR>\n" );
document.write( "*   **Independent Variable (X):** Rapeseed & Mustard Area (Hectare) - This is the variable we are using to predict production.\r<BR>\n" );
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document.write( "**Part A: Line of Regression Y on X**\r<BR>\n" );
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document.write( "The line of regression Y on X has the form: Y = a + bX\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "*   b (slope) = \[Σ(Xi - X̄)(Yi - Ȳ)] / \[Σ(Xi - X̄)²]<BR>\n" );
document.write( "*   a (intercept) = Ȳ - bX̄\r<BR>\n" );
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document.write( "Here's the calculation:\r<BR>\n" );
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document.write( "1.  **Calculate the means:**<BR>\n" );
document.write( "    *   X̄ (Mean Area) = (873 + 407 + 325 + 848 + 226) / 5 = 535.8<BR>\n" );
document.write( "    *   Ȳ (Mean Production) = (437 + 193 + 154 + 466 + 139) / 5 = 277.8\r<BR>\n" );
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document.write( "2.  **Calculate Σ(Xi - X̄)(Yi - Ȳ):**\r<BR>\n" );
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document.write( "| District      | Area (X) | Production (Y) | (X-X̄) | (Y-Ȳ) | (X-X̄)(Y-Ȳ) |<BR>\n" );
document.write( "|---------------|----------|----------------|-------|-------|-------------|<BR>\n" );
document.write( "| Uttar Kashi   | 873      | 437            | 337.2 | 159.2 | 53669.44    |<BR>\n" );
document.write( "| Chamoli       | 407      | 193            | -128.8| -84.8 | 10922.24    |<BR>\n" );
document.write( "| Rudra Prayag  | 325      | 154            | -210.8| -123.8| 26095.04    |<BR>\n" );
document.write( "| Tehri Garhwal | 848      | 466            | 312.2 | 188.2 | 58712.04    |<BR>\n" );
document.write( "| Dehradun      | 226      | 139            | -309.8| -138.8| 43004.24    |<BR>\n" );
document.write( "| **Total**       |          |                |       |       | **192303.00**|\r<BR>\n" );
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document.write( "3.  **Calculate Σ(Xi - X̄)²:**\r<BR>\n" );
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document.write( "| District      | Area (X) | (X-X̄) | (X-X̄)² |<BR>\n" );
document.write( "|---------------|----------|-------|--------|<BR>\n" );
document.write( "| Uttar Kashi   | 873      | 337.2 | 113703.84|<BR>\n" );
document.write( "| Chamoli       | 407      | -128.8| 16589.44|<BR>\n" );
document.write( "| Rudra Prayag  | 325      | -210.8| 44436.64|<BR>\n" );
document.write( "| Tehri Garhwal | 848      | 312.2 | 97468.84|<BR>\n" );
document.write( "| Dehradun      | 226      | -309.8| 95976.04|<BR>\n" );
document.write( "| **Total**       |          |       | **368174.80**|\r<BR>\n" );
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document.write( "4.  **Calculate b:**<BR>\n" );
document.write( "    b = 192303 / 368174.80 ≈ 0.522\r<BR>\n" );
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document.write( "5.  **Calculate a:**<BR>\n" );
document.write( "    a = Ȳ - bX̄ = 277.8 - (0.522 * 535.8) ≈ -2.25\r<BR>\n" );
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document.write( "Therefore, the line of regression Y on X is: **Y = -2.25 + 0.522X**\r<BR>\n" );
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document.write( "**Part B: Coefficient of Determination (r²)**\r<BR>\n" );
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document.write( "r² = \[Σ(Ŷi - Ȳ)²] / \[Σ(Yi - Ȳ)²] where Ŷi are the predicted Y values from the regression line.\r<BR>\n" );
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document.write( "1.  **Calculate the predicted Y values (Ŷi) using the regression equation:**\r<BR>\n" );
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document.write( "| District      | Area (X) | Production (Y) | Ŷi = -2.25 + 0.522X | (Ŷi - Ȳ)² | (Yi - Ȳ)² |<BR>\n" );
document.write( "|---------------|----------|----------------|----------------------|----------|----------|<BR>\n" );
document.write( "| Uttar Kashi   | 873      | 437            | 453.75              | 2744.76 | 25344.64|<BR>\n" );
document.write( "| Chamoli       | 407      | 193            | 209.69              | 278.57 | 7196.84|<BR>\n" );
document.write( "| Rudra Prayag  | 325      | 154            | 167.35              | 205.81 | 15376.84|<BR>\n" );
document.write( "| Tehri Garhwal | 848      | 466            | 440.79              | 1742.40 | 35436.64|<BR>\n" );
document.write( "| Dehradun      | 226      | 139            | 115.61              | 397.02 | 19264.84|<BR>\n" );
document.write( "| **Total**       |          |                |                      | **5369.36** | **102519.80**|\r<BR>\n" );
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document.write( "2.  **Calculate r²:**<BR>\n" );
document.write( "    r² = 5369.36 / 102519.80 ≈ 0.052\r<BR>\n" );
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document.write( "**Interpretation of b (byx):**\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "**Interpretation of r²:**\r<BR>\n" );
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document.write( "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.<BR>\n" );
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