document.write( "Question 1197140: Linear Regression
\n" ); document.write( "The table below shows the value,
\n" ); document.write( "V
\n" ); document.write( " , of an investment (in dollars)
\n" ); document.write( "n
\n" ); document.write( " years after 1990.\r
\n" ); document.write( "\n" ); document.write( "n
\n" ); document.write( " 1 3 7 12 14 19
\n" ); document.write( "V
\n" ); document.write( " 21120 20148.8 19440 18581.2 17480 15276\r
\n" ); document.write( "\n" ); document.write( "Determine the linear regression equation that models the set of data above, and use this equation to answer the questions below. Round to the nearest hundredth as needed.
\n" ); document.write( "Based on this regression model, the value of this investment was $
\n" ); document.write( " in the year 1990.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Based on the regression model, the value of this investment is
\n" ); document.write( "Select an answer\r
\n" ); document.write( "\n" ); document.write( "at a rate of $
\n" ); document.write( " per year. \n" ); document.write( "

Algebra.Com's Answer #830270 by math_tutor2020(3817) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "I'll use x and y in place of n and V
\n" ); document.write( "x = n
\n" ); document.write( "y = V\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "x = number of years after 1990
\n" ); document.write( "y = value of an investment (in dollars)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Given Data:\n" ); document.write( "\n" ); document.write( "

x	y
1	21120
3	20148.8
7	19440
12	18581.2
14	17480
19	15276

There are n = 6 rows of data to represent the sample size.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "We'll form the following columns:
\n" ); document.write( "x^2
\n" ); document.write( "xy\r
\n" ); document.write( "\n" ); document.write( "The x^2 column is where we square each x value
\n" ); document.write( "eg: 19 squares to 19^2 = 19*19 = 361\r
\n" ); document.write( "\n" ); document.write( "The xy column has us multiply each x and y value together (separately per row).
\n" ); document.write( "Eg: 3*20148.8 = 60446.4 in the second row of this column.\r
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

x	y	x^2	xy
1	21120	1	21120
3	20148.8	9	60446.4
7	19440	49	136080
12	18581.2	144	222974.4
14	17480	196	244720
19	15276	361	290244

Use of a spreadsheet program is strongly recommended.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Then compute the following sums
\n" ); document.write( "P = sum of the x values = 56
\n" ); document.write( "Q = sum of the y values = 112046
\n" ); document.write( "R = sum of the x^2 values = 760
\n" ); document.write( "S = sum of the xy values = 975584.8\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The regression line is of the form y = mx+b
\n" ); document.write( "m = slope = $\"%28n%2AS+-+P%2AQ%29%2F%28n%2AR+-+P%5E2%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "b = y intercept = $\"%28Q%2AR+-+P%2AS%29%2F%28n%2AR+-+P%5E2%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Let's find the slope
\n" ); document.write( " $\"m+=+%28n%2AS+-+P%2AQ%29%2F%28n%2AR+-+P%5E2%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"m+=+%286%2A975584.8+-+56%2A112046%29%2F%286%2A760+-+56%5E2%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"m+=+-295.693258426966\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"m+=+-295.69\"$ approximately\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now compute the y intercept
\n" ); document.write( " $\"b+=+%28Q%2AR+-+P%2AS%29%2F%28n%2AR+-+P%5E2%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"b+=+%28112046%2A760+-+56%2A975584.8%29%2F%286%2A760+-+56%5E2%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"b+=+21434.1370786517\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"b+=+21434.14\"$ approximately\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The regression line is y = -295.69x + 21434.14 approximately
\n" ); document.write( "As a quick shortcut, you can use technology (eg: calculator or spreadsheet) to calculate the equation for the regression line.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "x = 0 corresponds to the year 1990
\n" ); document.write( "Plug it in to find that y = 21434.14 which is exactly the y intercept.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Based on the regression model, the estimated value of the investment was about $21434.14 in the year 1990.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The slope tells us how much the investment goes up or goes down.
\n" ); document.write( "In this case, a negative slope means the investment is decreasing in value. As x goes up y goes down.
\n" ); document.write( "The slope is roughly m = -295.69 to represent a decrease of $295.69 per year.\r
\n" ); document.write( "
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\n" ); document.write( "\n" ); document.write( "----------------------------------------------------------------------------\r
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\n" ); document.write( "\n" ); document.write( "Answers:
\n" ); document.write( "$21434.14 was the approximate value of the investment in the year 1990 (estimate according to the regression line)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The investment decreases at a rate of about $295.69 per year (estimate according to the regression line)
\n" ); document.write( " \n" ); document.write( "

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