Chapter 12.2, Problem 43E

(a)

To determine

To construct: a scatterplot to assume horizontal distance travelled and clarified from ramp height.

Answer to Problem 43E

True positive curved correlation

Explanation of Solution

Given:

Distance	Height
1500	1000
1340	828
1328	800
1172	600
800	300

Graph:

  

Direction: Positive, as the scatterplot swings upwards.

Form: Curved, since there is no straight line to the ends.

Strength: Solid, since all points in the same pattern lay very close together.

(b)

To determine

To Explain: The distance-to-height relationship tends to follow an exponential formula, or a power model.

Answer to Problem 43E

Power Model

Explanation of Solution

Given:

Distance	Height
1500	1000
1340	828
1328	800
1172	600
800	300

Graph:

Exponential Model

  

Power Model

  

In Exponential Model the height is in the horizontal dimension, and the distance logarithm is in the y axis and in Power Model the Height logarithm is in the horizontal axis, and the Distance logarithm is in the vertical axis.

The model for the relationship between the variables is Power Model, with the most linear form in the accompanying scatterplot.

(c)

To determine

To Perform: least-square regression on the transformed results, and give your regression line equation and identify some variables.

Answer to Problem 43E

  $\ln \widehat{y}=3.751+0.515\ln x$

Explanation of Solution

Given:

Distance	Height
1500	1000
1340	828
1328	800
1172	600
800	300

Formula used:

  $\begin{array}{l}a=\frac{{\displaystyle \sum Y}.{\displaystyle \sum X^2}-{\displaystyle \sum X}.{\displaystyle \sum XY}}{n{\displaystyle \sum X^2}-{\left({\displaystyle \sum X}\right)}^2}\\ b=\frac{n{\displaystyle \sum XY}-{\displaystyle \sum X}.{\displaystyle \sum Y}}{n{\displaystyle \sum X^2}-{\left({\displaystyle \sum X}\right)}^2}\end{array}$

Calculation:

Value of height and distance is taken as logarithmic

Height (X)	Distance (Y)	XY	XX
6.907755	7.31322	50.517932	47.71708
6.719013	7.200425	48.379749	45.14514
6.684612	7.191429	48.071913	44.68404
6.39693	7.066467	45.203695	40.92071
5.703782	6.684612	38.12757	32.53313
${\displaystyle \sum X}$=32.412092	${\displaystyle \sum Y}$=35.456153	${\displaystyle \sum XY}$=230.30086	${\displaystyle \sum X^2}$=211.0001

  $\begin{array}{l}a=\frac{{\displaystyle \sum Y}.{\displaystyle \sum X^2}-{\displaystyle \sum X}.{\displaystyle \sum XY}}{n{\displaystyle \sum X^2}-{\left({\displaystyle \sum X}\right)}^2}=3.751\\ b=\frac{n{\displaystyle \sum XY}-{\displaystyle \sum X}.{\displaystyle \sum Y}}{n{\displaystyle \sum X^2}-{\left({\displaystyle \sum X}\right)}^2}=0.515\end{array}$

  $\begin{array}{l}y=a+bx\\ y=3.751+0.515x\end{array}$

Therefore, the equation is

  $\ln \widehat{y}=3.751+0.515\ln x$

Here $x$ is the height and $y$ distance.

(d)

To determine

To Predict: a ball would move the horizontal distance if the height of the ramp was 700, using part (a).

Answer to Problem 43E

  $\widehat{y}=1242.41$

Explanation of Solution

Given:

From part (c)

  $\ln \widehat{y}=3.751+0.515\ln x$

Calculation:

Putting the value of $x$ =700 in the equation of part (c)

  $\begin{array}{c}\ln \widehat{y}=3.751+0.515\ln \left(700\right)\\ =7.12481\end{array}$

Taking the exponential

  $\begin{array}{c}\widehat{y}=e^{7.12481}\\ =1242.41\end{array}$