Chapter 3, Problem 28RE

28. Bono Length Research performed at NASA, led by Dr. Emily R. Morey-Holton, measured the lengths of the right humerus and right tibia in 11 rats that were sent to space on Spacelab Life Sciences 2. The following data were collected.

(a) Draw a scatter diagram of the data treating length of the right humerus as the independent variable.

(b) Based on the scatter diagram, do you think that there is a linear relation between the length of the right humerus and the length of the right tibia?

(c) Use a graphing utility to find the line of best fit relating length of the right humerus and length of the right tibia.

(d) Predict the length of the right tibia on a rat whose right humerus is 26.5 millimeters (mm).

To determine

To calculate:

1. Draw a scatter diagram of the given data.
2. Determine the type of the relation.
3. Find the line of best fit.
4. Predict the length of the right tibia on a rat whose right humerus is $\text{26}\text{.5}$ mm.

Answer to Problem 28RE

Solution:

1. The scatter diagram is

   

2. It is not a linear relation. The relation is a polynomial relation.
3. The equation of best fit is $y=-0.1127x^2+7.0702x-70.399$
4. The length of the right tibia on a rat whose right humerus is $\text{26}\text{.5}$ mm is $\text{37}\text{.82}$ mm.

Explanation of Solution

Given:

The given data is

Formula used:

For a quadratic function $y=a{x}^2+bx+c$, if $a$ is positive, then the vertex $\left(\frac{-b}{2a},y\left(\frac{-b}{2a}\right)\right)$ is the minimum point and if $a$ is negative, the vertex $\left(\frac{-b}{2a},y\left(\frac{-b}{2a}\right)\right)$ is the maximum point.

Calculation:

We can draw the scatter diagram using Microsoft Excel.

Thus, on entering the $x$ and the $y$ values on excel, we have to choose the Scatter diagram form the insert option.

Then for getting the equation of best fit, we have to choose the Layout option and then Trendline and then more Trendline option. Then choose the option polynomial and then display the equation.

(a) The scatter diagram for the given data is

(b) From the above diagram, we can see that the relation is not linear as the average rate if change of the given relation is not constant. We can see that there is a point separated from the other points.

(c) The equation of best fit is

$y=-0.1127x^2+7.0702x-70.399$

(d) When $x=26.5$, we get

$y(26.5)=-0.1127{(26.5)}^2+7.0702(26.5)-70.399$

$=37.82$