Respuesta :
Answer:
a)
Price=150.414-16.626 Age
b)
$34.032
c)
We can't estimate the price of 18 year old car using the above regression equation because the price of 18 year old car falls outside the scope of the model
Step-by-step explanation:
Age(X) Price(Y)
8 18
3 94
6 50
9 21
2 145
5 42
6 36
3 99
a)
The regression equation can be written as
y=a+bx
[tex]b=\frac{sum(x-xbar)(y-ybar)}{sum(x-xbar)^{2} }[/tex]
a=ybar-bxbar
xbar=sumx/n
xbar=(8+3+6+9+2+5+6+3)/8=5.25
ybar=sumy/n
ybar=(18+94+50+21+145+42+36+99)/8=63.125
x y x-xbar y-ybar (x-xbar)^2 (x-xbar)(y-ybar)
8 18 2.75 -45.125 7.5625 -124.09375
3 94 -2.25 30.875 5.0625 -69.46875
6 50 0.75 -13.125 0.5625 -9.84375
9 21 3.75 -42.125 14.0625 -157.96875
2 145 -3.25 81.875 10.5625 -266.09375
5 42 -0.25 -21.125 0.0625 5.28125
6 36 0.75 -27.125 0.5625 -20.34375
3 99 -2.25 35.875 5.0625 -80.71875
Total 43.5 -723.25
[tex]b=\frac{sum(x-xbar)(y-ybar)}{sum(x-xbar)^{2} }[/tex]
b=-723.25/43.5
b=-16.626
a=ybar-bxbar
a=63.125-(16.626)*5.25
a=150.414
Thus, the required regression equation is
y=150.414-16.626 x
Price=150.414-16.626 Age
b)
For predicting price of 7 years old car we put x=7 in the estimated regression equation
y=150.414-16.626 x
y=150.414-16.626(7)
y=150.414-116.382
y=34.032
The predicted price of a 7 year old car is $34.032.
c)
We can't estimate the price of 18 year old car using the above regression equation because the price of 18 year old car falls outside the scope of the model. The estimate of price is only valid for 2 to 9 year old car.
The Least Square Regression line equation, gives the best line that describes the relationship between the price and the age of the car model.
The correct responses are;
- (a) The regression line equation is; [tex]\overline y = \mathbf{-16.6264 \cdot \overline x + 150.4163}[/tex]. Please find attached the graph of the regression line.
- (b) The price of a 7 year old car is $3,403.15.
- (c) The price of an 18 year old car is $-14,885.89.
Reasons:
(a) The given data is presented in the following table;
[tex]\begin{tabular}{c|c|c|c|c|c|c|c|c|}Age&8&3&6&9&2&5&6&3\\Price&18&94&50&21&145&42&36&99\end{array}\right][/tex]
The regression equation is given as follows;
[tex]\overline y = b \cdot \overline x + a[/tex]
Where;
[tex]\overline y[/tex] = The price of the car = 63.125
[tex]\overline x[/tex] = The age of the car = 5.25
[tex]b = \mathbf{\dfrac{\sum \left(x_i - \bar x\right) \times \left(y_i - \bar y\right) }{\sum \left(x_i - \bar x\right )^2 }}[/tex]
Therefore;
[tex]\displaystyle b = \frac{-723.25}{43.5} \apporx -16.6264[/tex]
[tex]a = \mathbf{\overline y - b \cdot \overline x}[/tex]
Therefore;
a = 63.125 - (-16.6264 × 5.25) = 150.4163
Therefore;
The equation for the regression line is; [tex]\overline y[/tex] = -16.6264·[tex]\overline x[/tex] + 150.4163
The graph of the data points showing the regression line created with MS Excel is attached.
(b) The price of a 7 year old car is therefore;
[tex]\overline y[/tex] = -16.6264 × 7 + 150.4163 = 34.0315
The price of a 7 year old car is $34.0315 × 100 = $3,403.15
(c) The price of an 18 year old car is therefore;
Price, [tex]\overline y[/tex] = -16.6264 × 18 + 150.4163 = -148.8589
The price of an 18 year old car is $-148.8589 × 100 = $-14,885.89
Therefore, the price of an 18 year old car can be taken as 0
Learn more about regression line equation here:
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