Given the values shown in the table, let be "x" the Vehicle weight (in hundreds of lbs.) and "y" the City MPG (Miles per gallon).
1. Given the points:
[tex](27,25),(35,19),(39,16),(32,21),(40,15),(23,29),(18,31),(37,15),(17,46),(23,26),(37,17),(30,26),(23,29),(32,19),(20,33),(30,21)[/tex]
You can plot them on a Coordinate Plane:
2. Notice the following line:
Notice that the points are closed to the red line that has a negative slope. Therefore, you can identify that when one of the variables increases, the other variable decreases. Hence, you can conclude that the data describes a negative correlation.
3. You need to follow these steps to find the equation of the line of best fit:
- You need to find the average of the x-values by adding them and dividing the Sum by the number of x-values:
[tex]\bar{X}=\frac{27+35+39+32+40+23+18+37+17+23+37+30+23+32+20+30}{16}[/tex][tex]\bar{X}=28.9375[/tex]
- Find the average of the y-values:
[tex]\bar{Y}=\frac{25+19+16+21+15+29+31+15+46+26+17+26+29+19+33+21}{16}[/tex][tex]\bar{Y}=24.25[/tex]
- Find:
[tex]\sum_{i=1}^n(x_i-\bar{X})[/tex]
Where this represents each x-values in the data set:
[tex]x_i[/tex]
You get:
[tex]\sum_{i=1}^n(x_i-\bar{X})=(27-28.9375)+(35-28.9375)+(39-28.9375)+(32-28.9375)+(40-28.9375)+(23-28.9375)+(18-28.9375)+(37-28.9375)+(17-28.9375)+(23-28.9375)+(37-28.9375)+(30-28.9375)+(23-28.9375)+(32-28.9375)+(20-28.9375)+(30-29.9375)[/tex][tex]\sum_{i=1}^n(x_i-\bar{X})=1.0625[/tex]
- Find:
[tex]\sum_{i=1}^n(x_i-\bar{Y})[/tex]
You get:
[tex]\sum_{i=1}^n(x_i-\bar{Y})=(25-24.25)+(19-24.25)+(16-24.25)+(21-24.25)+(15-24.25)+(29-24.25)+(31-24.25)+(15-24.25)+(46-24.25)+(26-24.25)+(17-24.25)+(26-24.25)+(29-24.25)+(19-24.25)+(33-24.25)+(21-24.25)[/tex][tex]\sum_{i=1}^n(x_i-\bar{Y})=-3.25[/tex]
- Find:
[tex]\sum_{i=1}^n(x_i-\bar{X})(y_i-\bar{Y})[/tex]
You get:
[tex]=-857.75[/tex]
- Find:
[tex]\sum_{i=1}^n(x_i-\bar{X})^2[/tex]
You can find it by squaring each Difference of the x-values and the Mean. you get:
[tex]=862.9375[/tex]
- Find the slope of the line
[tex]m=\frac{-857.75}{862.9375}\approx-0.994[/tex]
- Find the y-intercept with this formula:
[tex]b=\bar{Y}-m\bar{X}[/tex][tex]b=24.25-(-0.994)(1.0625)[/tex][tex]b=53.0135[/tex]
Therefore, the line in Slope-Intercept Form:
[tex]y=mx+b[/tex]
is the following:
[tex]y=-0.9940x+53.0135[/tex]
4. If:
[tex]y=30[/tex]
You can predict the vehicle weight by substituting that value into the equation found in Part 3, and solving for "x":
[tex]30=-0.9940x+53.0135[/tex][tex]\frac{30-53.0135}{-0.9949}=x[/tex][tex]x\approx23.1524[/tex]
5. If a vehicle weighs 1500 pounds, then:
[tex]x=1500[/tex]
Then you can determine the expected city MPG of the vehicle by substituting this value into the equation and evaluating:
[tex]y=-0.9940(1500)+53.0135[/tex][tex]y\approx-1437.9865[/tex]
Hence, the answers are:
1.
2. It describes a negative correlation.
3.
[tex]y=-0.9940x+53.0135[/tex]
4.
[tex]x\approx23.1524\text{ \lparen in hundreds of pounds\rparen}[/tex]
5.
[tex]y\approx-1437.9865\text{ \lparen In miles per gallon\rparen}[/tex]
