document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198680>Question 1198680</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A corporation owns several companies. The strategic planner for the corporation believes dollars spent on advertising can to some extent be a predictor of total sales dollars. As an aid in long-term planning, she gathers the following sales and advertising information from several of the companies for 2005 (RS.in lakhs).\r<BR>\n" );
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document.write( "Advertising	Sales<BR>\n" );
document.write( "12.5	        148<BR>\n" );
document.write( "3.7	         55<BR>\n" );
document.write( "21.6	        338<BR>\n" );
document.write( "60.0	        994<BR>\n" );
document.write( "37.6	        541<BR>\n" );
document.write( "6.1	        89<BR>\n" );
document.write( "16.8	       126<BR>\n" );
document.write( "41.2	        379\r<BR>\n" );
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document.write( "Develop the equation of the simple regression line to predict sales from advertising expenditures using these data. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848242 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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=848242\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ## Linear Regression for Predicting Sales from Advertising Expenditures\r<BR>\n" );
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document.write( "Here's how to develop the equation of the simple regression line to predict sales from advertising expenditures using the given data:\r<BR>\n" );
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document.write( "**1. Calculate the Mean Values:**\r<BR>\n" );
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document.write( "* Mean Advertising (x̄) = Σ(Advertising) / n <BR>\n" );
document.write( "    * Σ (summation symbol) represents the sum of all advertising values.<BR>\n" );
document.write( "    * n = number of companies (data points) = 8\r<BR>\n" );
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document.write( "* Mean Sales (ȳ) = Σ(Sales) / n\r<BR>\n" );
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document.write( "**2. Calculate Deviations from the Mean:**\r<BR>\n" );
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document.write( "* Deviation from mean for Advertising (xᵢ - x̄) for each company (i = 1 to 8)<BR>\n" );
document.write( "* Deviation from mean for Sales (yᵢ - ȳ) for each company\r<BR>\n" );
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document.write( "**3. Calculate the Covariance (Σ(xᵢ - x̄)(yᵢ - ȳ))**\r<BR>\n" );
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document.write( "* Sum the product of deviations from the mean for advertising and sales across all companies.\r<BR>\n" );
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document.write( "**4. Calculate the Variance of Advertising (Σ(xᵢ - x̄)²)**\r<BR>\n" );
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document.write( "* Sum the squared deviations from the mean for advertising across all companies.\r<BR>\n" );
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document.write( "**5. Calculate the Slope (β₁):**\r<BR>\n" );
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document.write( "* β₁ = Σ(xᵢ - x̄)(yᵢ - ȳ) / Σ(xᵢ - x̄)²\r<BR>\n" );
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document.write( "**6. Calculate the Y-intercept (β₀):**\r<BR>\n" );
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document.write( "* β₀ = ȳ - β₁(x̄)\r<BR>\n" );
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document.write( "**7. Form the Regression Equation:**\r<BR>\n" );
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document.write( "* Sales (ŷ) = β₀ + β₁(Advertising)\r<BR>\n" );
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document.write( "**Steps to follow using the provided data:**\r<BR>\n" );
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document.write( "1. **Calculate Mean Values:**\r<BR>\n" );
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document.write( "   * You'll need to calculate the average advertising expenditure and average sales for the 8 companies.\r<BR>\n" );
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document.write( "2. **Calculate Deviations from the Mean:**\r<BR>\n" );
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document.write( "   * Subtract the mean advertising expenditure from each individual advertising expenditure.<BR>\n" );
document.write( "   * Do the same for sales.\r<BR>\n" );
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document.write( "3. **Calculate Covariance and Variance:**\r<BR>\n" );
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document.write( "   * Multiply the corresponding deviations from the mean (advertising and sales) for each company and sum them up.<BR>\n" );
document.write( "   * Square the deviations from the mean for advertising and sum them up.\r<BR>\n" );
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document.write( "4. **Calculate Slope (β₁) and Y-intercept (β₀):**\r<BR>\n" );
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document.write( "   * Use the formulas mentioned above to find β₁ and β₀.\r<BR>\n" );
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document.write( "5. **Form the Regression Equation:**\r<BR>\n" );
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document.write( "   * Plug in the calculated β₀ and β₁ values to get the final equation for predicting sales based on advertising expenditure.\r<BR>\n" );
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document.write( "**Note:** This process requires calculations. You can perform them manually using a spreadsheet or use statistical software to obtain the results.\r<BR>\n" );
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document.write( "Once you have the equation, you can estimate sales for a given advertising expenditure by plugging that value into the equation.<BR>\n" );
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