Mark Gershon, owner of a musical instrument distributorship, thinks that damand
ID: 410552 • Letter: M
Question
Mark Gershon, owner of a musical instrument distributorship, thinks that damand for guitars may be related to the number of television appearances by the popular group Maroon 5 during the previous month. Gershon has collected the data shown in the following table: Maroon 5 TV Appearances Demand for Guitars Tnis exercise condains only parls b, , and d b) Using the least-squares regression method, the equation for forecasting is fround your responaes to four decimal plsces. c) The estimate for guitar salas if Maroon 5 performed on TV S timessales fround yaur response to two decimal places) The correlation coeflicient ( forthismel (round your responae to four deaimel placesj The coeficient of deteminationotis moelround your respense to four deoimal pleces). The percentage o variation in sales that can be explained by TV appearances % round your response to two dec na'p 2) for cesExplanation / Answer
The least-squares regression method Calculation
Observation (n=6)
Maroon 5 TV Appearances (x)
Demand for Guitars (y)
x^2
X y
1
3
3
9
9
2
4
5
16
20
3
7
7
49
49
4
5
4
25
20
5
7
11
49
77
6
7
6
49
42
Mean
5.5
6
x bar
Y bar
Sum
33
36
197
217
x
y
x^2
x y
slop b = ( x y - n*x bar*y bar)/( x^2- n*xbar^2) =
1.23
Intercept a = y bar - b * x bar=
-0.74
Equation is y = a + b x
Where a is intercept and b is slop
Therefore y = -0.74 + 1.23 *x
b) Using the least-squares regression method, the equation for forecasting is
Y = -0.7419 + 1.2258 * x
c) The estimate for guitar sales if Maroon 5 performed on TV 9 times
Y = -0.7419 + 1.2258 * x
Here we have x = 9
Therefore Y = -0.7419 + 1.2258 * 9 = 10.29
d) The correlation coefficient (r) for this model
r = (n x y - x y)/ sqrt [(nx^2-(x) ^2) * (ny^2-(y) ^2)]
Observation (n=6)
Maroon 5 TV Appearances (x)
Demand for Guitars (y)
x^2
xy
y^2
1
3
3
9
9
9
2
4
5
16
20
25
3
7
7
49
49
49
4
5
4
25
20
16
5
7
11
49
77
121
6
7
6
49
42
36
Mean
5.5
6
x bar
ybar
Sum
33
36
197
217
256
x
y
x^2
xy
y ^2
r = (6* 217 – 33 * 36)/ sqrt [ (6*197-33^2)*(6*256-36^2)]
= 0.7631
The coefficient of determination (r squared) for this model
The coefficient of determination (r squared) = r^2 = 0.7631^2 = 0.5823
About 58.23% of variation in sale of guitar can be explained by TV appearances
Observation (n=6)
Maroon 5 TV Appearances (x)
Demand for Guitars (y)
x^2
X y
1
3
3
9
9
2
4
5
16
20
3
7
7
49
49
4
5
4
25
20
5
7
11
49
77
6
7
6
49
42
Mean
5.5
6
x bar
Y bar
Sum
33
36
197
217
x
y
x^2
x y
slop b = ( x y - n*x bar*y bar)/( x^2- n*xbar^2) =
1.23
Intercept a = y bar - b * x bar=
-0.74
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