document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176480>Question 1176480</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. A manufacturer of basic commodities wants to know the relationship of advertising costs and sales. The data for a particular period on the two variables were recorded as follows:\r<BR>\n" );
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document.write( "Advertising costs (P1,000 )	<BR>\n" );
document.write( "20<BR>\n" );
document.write( "40<BR>\n" );
document.write( "30<BR>\n" );
document.write( "50<BR>\n" );
document.write( "25<BR>\n" );
document.write( "30<BR>\n" );
document.write( "40<BR>\n" );
document.write( "35<BR>\n" );
document.write( "45<BR>\n" );
document.write( "20<BR>\n" );
document.write( "25<BR>\n" );
document.write( "40	\r<BR>\n" );
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document.write( "Sales (P1,000)<BR>\n" );
document.write( "385<BR>\n" );
document.write( "400<BR>\n" );
document.write( "489<BR>\n" );
document.write( "580<BR>\n" );
document.write( "410<BR>\n" );
document.write( "475<BR>\n" );
document.write( "510<BR>\n" );
document.write( "500<BR>\n" );
document.write( "525<BR>\n" );
document.write( "360<BR>\n" );
document.write( "420<BR>\n" );
document.write( "480\r<BR>\n" );
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document.write( "	Required:<BR>\n" );
document.write( "		a. Plot the data on a scatter diagram.<BR>\n" );
document.write( "		b. Find the equation of the regression line to predict sales from advertising costs.<BR>\n" );
document.write( "		c. Estimate the sales, if the advertising cost is P55,000.00.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850576 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=850576\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**a. Plot the Data on a Scatter Diagram**\r<BR>\n" );
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document.write( "* **X-axis:** Advertising Costs (P1,000)<BR>\n" );
document.write( "* **Y-axis:** Sales (P1,000)\r<BR>\n" );
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document.write( "Plot the following points:\r<BR>\n" );
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document.write( "* (20, 385)<BR>\n" );
document.write( "* (40, 400)<BR>\n" );
document.write( "* (30, 489)<BR>\n" );
document.write( "* (50, 580)<BR>\n" );
document.write( "* (25, 410)<BR>\n" );
document.write( "* (30, 475)<BR>\n" );
document.write( "* (40, 510)<BR>\n" );
document.write( "* (35, 500)<BR>\n" );
document.write( "* (45, 525)<BR>\n" );
document.write( "* (20, 360)<BR>\n" );
document.write( "* (25, 420)<BR>\n" );
document.write( "* (40, 480)\r<BR>\n" );
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document.write( "**b. Find the Equation of the Regression Line**\r<BR>\n" );
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document.write( "We need to find the equation of the form ŷ = a + bx, where:\r<BR>\n" );
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document.write( "* x = Advertising costs<BR>\n" );
document.write( "* y = Sales<BR>\n" );
document.write( "* b = slope<BR>\n" );
document.write( "* a = y-intercept\r<BR>\n" );
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document.write( "First, calculate the necessary sums:\r<BR>\n" );
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document.write( "* Σx = 20 + 40 + 30 + 50 + 25 + 30 + 40 + 35 + 45 + 20 + 25 + 40 = 400<BR>\n" );
document.write( "* Σy = 385 + 400 + 489 + 580 + 410 + 475 + 510 + 500 + 525 + 360 + 420 + 480 = 5434<BR>\n" );
document.write( "* Σx² = 400 + 1600 + 900 + 2500 + 625 + 900 + 1600 + 1225 + 2025 + 400 + 625 + 1600 = 14300<BR>\n" );
document.write( "* Σy² = 148225 + 160000 + 239121 + 336400 + 168100 + 225625 + 260100 + 250000 + 275625 + 129600 + 176400 + 230400 = 2499696<BR>\n" );
document.write( "* Σxy = (20\*385) + (40\*400) + (30\*489) + (50\*580) + (25\*410) + (30\*475) + (40\*510) + (35\*500) + (45\*525) + (20\*360) + (25\*420) + (40\*480) = 7700 + 16000 + 14670 + 29000 + 10250 + 14250 + 20400 + 17500 + 23625 + 7200 + 10500 + 19200 = 190295<BR>\n" );
document.write( "* n = 12 (number of data points)\r<BR>\n" );
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document.write( "Now, calculate b (slope):\r<BR>\n" );
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document.write( "* b = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²)<BR>\n" );
document.write( "* b = (12\*190295 - 400\*5434) / (12\*14300 - 400²)<BR>\n" );
document.write( "* b = (2283540 - 2173600) / (171600 - 160000)<BR>\n" );
document.write( "* b = 109940 / 11600<BR>\n" );
document.write( "* b ≈ 9.4776\r<BR>\n" );
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document.write( "Next, calculate a (y-intercept):\r<BR>\n" );
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document.write( "* a = (Σy - bΣx) / n<BR>\n" );
document.write( "* a = (5434 - 9.4776\*400) / 12<BR>\n" );
document.write( "* a = (5434 - 3791.04) / 12<BR>\n" );
document.write( "* a = 1642.96 / 12<BR>\n" );
document.write( "* a ≈ 136.9133\r<BR>\n" );
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document.write( "Therefore, the regression equation is:\r<BR>\n" );
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document.write( "* ŷ = 136.9133 + 9.4776x\r<BR>\n" );
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document.write( "**c. Estimate the Sales if the Advertising Cost is P55,000.00**\r<BR>\n" );
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document.write( "* x = 55 (since the advertising cost is in P1,000)<BR>\n" );
document.write( "* ŷ = 136.9133 + 9.4776\*55<BR>\n" );
document.write( "* ŷ = 136.9133 + 521.268<BR>\n" );
document.write( "* ŷ ≈ 658.1813\r<BR>\n" );
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document.write( "Therefore, the estimated sales are approximately P658,181.30.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );