document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195287>Question 1195287</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manager wants to predict the cost (Y) of travel for salespeople based on the number of days (X) spent on each sales trip. Using 22 observations the linear equation Y = 40 + 12X is fitted using Ordinary Least Squares (OLS) technique and the R2 value is 0.5.<BR>\n" );
document.write( "Required;	<BR>\n" );
document.write( "1.	What is the sum of all squared errors of regression?<BR>\n" );
document.write( "2.	What is the value of the Adjusted R2 statistic?<BR>\n" );
document.write( "3.	Given that the Total Sum of Squares (SST) is 300, what is the value of an unbiased estimator of σ2, the variance of the random term?<BR>\n" );
document.write( "4.	What is the value of the F-statistic used to test for the overall relevance of the model?<BR>\n" );
document.write( "5.	The manager wishes to test (at 5% level of significance) the hypothesis that X has a significant positive effect on Y. Which of the following best approximates the critical t-value for the test?<BR>\n" );
document.write( "please help me solve this question<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848434 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848434\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Sum of Squared Errors (SSE)**\r<BR>\n" );
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document.write( "* **R-squared (R²)** is the coefficient of determination, which represents the proportion of the total variation in the dependent variable (Y) that is explained by the independent variable (X). <BR>\n" );
document.write( "* **Formula:** R² = 1 - (SSE / SST) <BR>\n" );
document.write( "    * where:<BR>\n" );
document.write( "        * SSE: Sum of Squared Errors<BR>\n" );
document.write( "        * SST: Total Sum of Squares\r<BR>\n" );
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document.write( "* **Rearrange to solve for SSE:**<BR>\n" );
document.write( "    * SSE = SST * (1 - R²)<BR>\n" );
document.write( "    * SSE = 300 * (1 - 0.5) <BR>\n" );
document.write( "    * SSE = 300 * 0.5 <BR>\n" );
document.write( "    * SSE = 150\r<BR>\n" );
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document.write( "**2. Adjusted R-squared**\r<BR>\n" );
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document.write( "* **Formula:**<BR>\n" );
document.write( "    * Adjusted R² = 1 - [(1 - R²) * (n - 1) / (n - k - 1)]<BR>\n" );
document.write( "        * where:<BR>\n" );
document.write( "            * n: number of observations (22)<BR>\n" );
document.write( "            * k: number of independent variables (1 in this case)\r<BR>\n" );
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document.write( "* **Calculate:**<BR>\n" );
document.write( "    * Adjusted R² = 1 - [(1 - 0.5) * (22 - 1) / (22 - 1 - 1)]<BR>\n" );
document.write( "    * Adjusted R² = 1 - (0.5 * 21 / 20)<BR>\n" );
document.write( "    * Adjusted R² = 1 - 1.05 <BR>\n" );
document.write( "    * Adjusted R² = -0.05 \r<BR>\n" );
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document.write( "* **Note:** The adjusted R-squared can be negative in some cases, especially when the model has a poor fit.\r<BR>\n" );
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document.write( "**3. Unbiased Estimator of σ²**\r<BR>\n" );
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document.write( "* **Mean Squared Error (MSE):** <BR>\n" );
document.write( "    * MSE = SSE / (n - k - 1) <BR>\n" );
document.write( "    * MSE = 150 / (22 - 1 - 1) <BR>\n" );
document.write( "    * MSE = 150 / 20 <BR>\n" );
document.write( "    * MSE = 7.5\r<BR>\n" );
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document.write( "* **Unbiased Estimator of σ²:** MSE = 7.5\r<BR>\n" );
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document.write( "**4. F-statistic**\r<BR>\n" );
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document.write( "* **Formula:** F = (Regression Mean Square / Residual Mean Square) <BR>\n" );
document.write( "    * Regression Mean Square (MSR) = SSR / k <BR>\n" );
document.write( "        * SSR (Sum of Squares Regression) = SST * R² = 300 * 0.5 = 150<BR>\n" );
document.write( "        * MSR = 150 / 1 = 150<BR>\n" );
document.write( "    * Residual Mean Square (MSE) = 7.5 (calculated earlier)\r<BR>\n" );
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document.write( "* **F-statistic:** F = 150 / 7.5 = 20\r<BR>\n" );
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document.write( "**5. Critical t-value**\r<BR>\n" );
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document.write( "* **Hypotheses:**<BR>\n" );
document.write( "    * H₀: β₁ = 0 (No significant relationship between X and Y)<BR>\n" );
document.write( "    * H₁: β₁ > 0 (X has a significant positive effect on Y) <BR>\n" );
document.write( "* **Significance Level:** α = 0.05<BR>\n" );
document.write( "* **Degrees of Freedom:** n - 2 = 22 - 2 = 20<BR>\n" );
document.write( "* **One-tailed test:** Since we are testing for a positive effect.\r<BR>\n" );
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document.write( "* **Using a t-distribution table or statistical software:** <BR>\n" );
document.write( "    * The critical t-value for a one-tailed test with α = 0.05 and 20 degrees of freedom is approximately 1.725.\r<BR>\n" );
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document.write( "**In Summary:**\r<BR>\n" );
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document.write( "1. SSE = 150<BR>\n" );
document.write( "2. Adjusted R² = -0.05<BR>\n" );
document.write( "3. Unbiased Estimator of σ² = 7.5<BR>\n" );
document.write( "4. F-statistic = 20<BR>\n" );
document.write( "5. Critical t-value ≈ 1.725\r<BR>\n" );
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document.write( "I hope this helps! Let me know if you have any other questions.<BR>\n" );
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