SOLUTION: Need help!!! Company A is a microcomputer producer. The following data represents Company A’s yearly sales volume and its advertising expenditures over a period of 8 years. Sale

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Question 868166: Need help!!!
Company A is a microcomputer producer. The following data represents Company A’s yearly sales volume and its advertising expenditures over a period of 8 years. Sales in millions of dollars and advertising is in $10,000.
1993 15 32
1994 16 33
1995 18 35
1996 17 34
1997 16 36
1998 19 37
1999 19 39
2000 24 42
Using the method of least squares, what is the estimated regression line between sales and adverting, and the predicted sales in dollars, with an advertising expenditure of $400K actual dollars as scaled to 40.
a. Sales=-10.4211 =.7895 Advertising: Sales $315,790
b. Advertising= 16.7143 + 1.0714 Sales: Sales $59.57 Million
c. Sales=.7895 + 10.421 Advertising: Sales $416.63 Million
d. Sales= -10.4211 + .7895 Advertising: Sales= $21.16 Million
e. None of the Above
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi

Do Like the Excel Scatter Plot for Obtaining a Regression - no muss no fuss

 .

However, should know how the method of least squares works..same result

but might be a need to know.

x	y	xy	x^2

32	15	480	1024

33	16	528	1089

35	18	630	1225

34	17	578	1156

36	16	576	1296

36	16	576	1296

39	19	741	1521

42	24	1008	1764

287	141	5117	10371

141 = 8a + 287b		

5117= 287a + 10371b

a= 1322362/74073 and b =-469/4073   	

y = a + bx	




 
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables




  

  

  system%288%2Ax%2B287%2Ay=141%2C287%2Ax%2B1037%2Ay=5117%29

  

  

  

  First let A=%28matrix%282%2C2%2C8%2C287%2C287%2C1037%29%29. This is the matrix formed by the coefficients of the given system of equations.

  

  

  Take note that the right hand values of the system are 141 and 5117 which are highlighted here: 

  system%288%2Ax%2B287%2Ay=highlight%28141%29%2C287%2Ax%2B1037%2Ay=highlight%285117%29%29

  

  

  

  These values are important as they will be used to replace the columns of the matrix A.

  

  

  

  

  Now let's calculate the the determinant of the matrix A to get abs%28A%29=%288%29%281037%29-%28287%29%28287%29=-74073. Remember that the determinant of the 2x2 matrix A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 is abs%28A%29=ad-bc. If you need help with calculating the determinant of any two by two matrices, then check out this solver.

  

  

  

  Notation note: abs%28A%29 denotes the determinant of the matrix A.

  

  

  

  ---------------------------------------------------------

  

  

  

  Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5Bx%5D (since we're replacing the 'x' column so to speak).

  

  

  A%5Bx%5D=%28matrix%282%2C2%2Chighlight%28141%29%2C287%2Chighlight%285117%29%2C1037%29%29

  

  

  

  Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%28141%29%281037%29-%28287%29%285117%29=-1322362. Once again, remember that the determinant of the 2x2 matrix A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 is abs%28A%29=ad-bc

  

  

  

  To find the first solution, simply divide the determinant of A%5Bx%5D by the determinant of A to get: x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%28-1322362%29%2F%28-74073%29=1322362%2F74073

  

  

  

  So the first solution is x=1322362%2F74073

  

  

  

  

  ---------------------------------------------------------

  

  

  We'll follow the same basic idea to find the other solution. Let's reset by letting A=%28matrix%282%2C2%2C8%2C287%2C287%2C1037%29%29 again (this is the coefficient matrix).

  

  

  

  

  Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5By%5D (since we're replacing the 'y' column in a way).

  

  

  A%5Bx%5D=%28matrix%282%2C2%2C8%2Chighlight%28141%29%2C287%2Chighlight%285117%29%29%29

  

  

  

  Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%288%29%285117%29-%28141%29%28287%29=469.

  

  

  

  To find the second solution, divide the determinant of A%5By%5D by the determinant of A to get: y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28469%29%2F%28-74073%29=-469%2F74073

  

  

  

  So the second solution is y=-469%2F74073

  

  

  

  

  ====================================================================================

  

  Final Answer:

  

  

  

  

  So the solutions are x=1322362%2F74073 and y=-469%2F74073 giving the ordered pair (1322362/74073, -469/74073)

  

  

  

  

  Once again, Cramer's Rule is dependent on determinants. Take a look at this 2x2 Determinant Solver if you need more practice with determinants.