Chapter 2, Problem 1EDE

To determine

1. Estimated height of female with 12-inch humerus.
2. Plot for ordered pairs (x,y) for both sexes and corresponding relationship.
3. For 1-inch increase in the length of humerus, corresponding increase in heights for both sexes.
4. Determination of linear functions that model the data.
5. Use the linear functions to get estimation of height for given values of length of humerus.

Answer to Problem 1EDE

Solution:

1. 62.8 inch
2. Plot



3. For 1-inch increase in length of humerus corresponding increase in height is 3.1-inches for female and 3-inches for male.
4. The linear functions that model these data for female and male are $f(x)=3.1x+\text{25}\text{.6}$ and $g(x)=3x+\text{29}$ respectively.
5. The range for the height a female when the humerous is between 9.7 to 10.1 is 55.67 to 56.91 inches, and the range for the height a male when the humerous is between 9.7 to 10.1 is 58.1 to 59.3 inches.

Explanation of Solution

a) From the given table we can observe that the estimated height of a female with 12-inch humerous is 62.8 inches.

b) From the given table we get the ordered pairs for female as $\left(8,50.4\right),$

  $\left(9,53.5\right),$

  $\left(10,56.6\right),$

  $\left(11,59.7\right),$

  $\left(12,62.8\right),$ and $\left(13,65.9\right)$ , and for male as $\left(8,53\right),$

  $\left(9,56\right),$

  $\left(10,59\right),$

  $\left(11,62\right),$

  $\left(12,65\right),$

  $\left(13,68\right)$ , and $\left(14,71\right)$ . Now, plotting these points on we get the graph as shown below.

From the graph we can observe that all the points for female lie on a straight line and for male as well the points lie on a straight line. Therefore, we can say that the relation between the humerous and height of male or female is linear as for each 1-inch increase in the humerous the height of male or female increases by constant amount.

c) From the table we can observe that for 1-inch increase in humerous for female corresponds to 3.1 inches increase in the height, and for 1-inch increase in humerous for male corresponds to 3 inches in increase in the height.

d) For female, each 1-inch increase in humerous causes 3.1-inches increase in height so, the linear relation for female can be written as shown below.

  $f(x)=3.1x+c$

Now, Substitute $x=8$ and $y=50.4$ in the above equation and solve for $c$ .

  $\begin{array}{l}f(x)=3.1x+c\\ \Rightarrow 50.4=3.1(8)+c\\ \Rightarrow c=\text{25}\text{.6}\end{array}$

Substitute the value of $c$ in $f$ to get the relation for female.

  $f(x)=3.1x+\text{25}\text{.6}$

Similarly, for male, each 1-inch increase in humerous causes 3-inches increase in height so, the linear relation for male can be written as shown below.

  $g(x)=3x+k$

Now, for the point $\left(8,53.0\right)$ , we have $x=8$ and $g(8)=53$ . Use these values in the equation for $g$ and solve it for $k$ .

  $\begin{array}{l}g(x)=3x+k\\ \Rightarrow 53=3(8)+k\\ \Rightarrow 53=24+k\\ \Rightarrow k=\text{29}\end{array}$

Now, substitute the value of $k$ in the equation for $g$ to get the relation for male.

  $g(x)=3x+\text{29}$

Thus, the required linear functions that model these data for female and male are

  $f(x)=3.1x+\text{25}\text{.6}$ and $g(x)=3x+\text{29}$ respectively.

e) When $x=9.7$ , $f(9.7)=3.1(9.7)+\text{25}\text{.6 = 55}\text{.67}$ and $g(9.7)=3(9.7)+\text{29 = 58}\text{.1}$ .

When $x=10.1$ , $f(10.1)=3.1(10.1)+\text{25}\text{.6 = 56}\text{.91}$ and $g(10.1)=3(10.1)+\text{29 = 59}\text{.3}$ .

Therefore, the range for the height a female when the humerous is between 9.7 to 10.1 is 55.67 to 56.91 inches, and the range for the height a male when the humerous is between 9.7 to 10.1 is 58.1 to 59.3 inches.