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The owner of Maumee Ford-Mercury-Volvo wants to study the relationship between the age of a car and its selling price. Listed below is a random sample of 12 used cars sold at the dealership during the last year.Car Age (years) Selling Price ($1000) Car Age (years) Selling Price ($1000)1 9 12.0 7 9 8.92 8 10.9 8 11 9.73 10 4.6 9 10 9.74 12 4.3 10 12 2.75 9 5.6 11 6 10.66 8 13.5 12 6 8.9(a) Determine the regression equation. (Round your answers to 3 decimal places. Negative values should be indicated by a minus sign.)(b) Estimate the selling price of a 12-year-old car (in $1000). (Round your answer to 3 decimal places.)(c) Interpret the regression equation (in dollars). (Round your answer to nearest dollar amount.)

Respuesta :

Answer:

a) [tex]m=-\frac{45.1}{43.666}=-1.0328[/tex]

Nowe we can find the means for x and y like this:

[tex]\bar x= \frac{\sum x_i}{n}=\frac{100}{12}=9.167[/tex]

[tex]\bar y= \frac{\sum y_i}{n}=\frac{101.4}{12}=8.45[/tex]

And we can find the intercept using this:

[tex]b=\bar y -m \bar x=8.45-(-1.0328*9.167)=17.918[/tex]

So the line would be given by:

[tex]y=-1.0328 x +17.918[/tex]

b) For this case we can replace in the lineal model the value of x =12 and we got:

[tex]y=-1.0328 *12 +17.918 =5.523[/tex]

c) For this case we can conclude that every increase in the age of the car decrease the selling price in about 1.0328*1000= 1032.8$ dollars from the original price.  

Explanation:

a) Regression equation.  

X represent the age an Y the Selling price

For this case we need to calculate the slope with the following formula:

[tex]m=\frac{S_{xy}}{S_{xx}}[/tex]

Where:

[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}[/tex]

[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}[/tex]

So we can find the sums like this:

[tex]\sum_{i=1}^n x_i =110[/tex]

[tex]\sum_{i=1}^n y_i =101.4[/tex]

[tex]\sum_{i=1}^n x^2_i =1052[/tex]

[tex]\sum_{i=1}^n y^2_i =982.32[/tex]

[tex]\sum_{i=1}^n x_i y_i =884.4[/tex]

With these we can find the sums:

[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=1052-\frac{110^2}{12}=43.667[/tex]

[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=884.4-\frac{110*101.4}{12}=-45.1[/tex]

And the slope would be:

[tex]m=-\frac{45.1}{43.666}=-1.0328[/tex]

Nowe we can find the means for x and y like this:

[tex]\bar x= \frac{\sum x_i}{n}=\frac{100}{12}=9.167[/tex]

[tex]\bar y= \frac{\sum y_i}{n}=\frac{101.4}{12}=8.45[/tex]

And we can find the intercept using this:

[tex]b=\bar y -m \bar x=8.45-(-1.0328*9.167)=17.918[/tex]

So the line would be given by:

[tex]y=-1.0328 x +17.918[/tex]

Part b

For this case we can replace in the lineal model the value of x =12 and we got:

[tex]y=-1.0328 *12 +17.918 =5.523[/tex]

Part c

For this case we can conclude that every increase in the age of the car decrease the selling price in about 1.0328*1000= 1032.8$ dollars from the original price.