Question

a pediatrician wants to determine the relation that may exist between a childs height and head - circumference. she randomly selects 8 children, measures their height and head - circumference, and obtains the data shown in the table. the pediatrician wants to use height to predict head - circumference.

height (inches) head circumference (inches)
27.75 17.4
25 17.1
26.25 17.3
25.75 17
27.5 17.4
26.25 17.2
26 17.1
26.75 17.4

compute the linear correlation coefficient between the height and head - circumference of a child.
r = □ (round to three decimal places as needed.)

Explanation:

Step1: Recall correlation - coefficient formula

The formula for the linear correlation coefficient \(r\) is \(r=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{\sqrt{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\sqrt{n\sum_{i = 1}^{n}y_i^{2}-(\sum_{i = 1}^{n}y_i)^{2}}}\), where \(n\) is the number of data - points, \(x_i\) are the values of the independent variable (height), and \(y_i\) are the values of the dependent variable (head circumference).
Let \(x\) be the height and \(y\) be the head circumference. First, calculate the following sums for \(n = 8\) data - points:
Let \(x_1 = 27.75,x_2 = 25,x_3 = 26.25,x_4 = 25.75,x_5 = 27.5,x_6 = 26.25,x_7 = 26,x_8 = 26.75\) and \(y_1 = 17.4,y_2 = 17.1,y_3 = 17.3,y_4 = 17,y_5 = 17.4,y_6 = 17.2,y_7 = 17.1,y_8 = 17.4\)
\(\sum_{i = 1}^{8}x_i=27.75 + 25+26.25+25.75+27.5+26.25+26+26.75 = 211.25\)
\(\sum_{i = 1}^{8}y_i=17.4 + 17.1+17.3+17+17.4+17.2+17.1+17.4 = 137.9\)
\(\sum_{i = 1}^{8}x_i^{2}=27.75^{2}+25^{2}+26.25^{2}+25.75^{2}+27.5^{2}+26.25^{2}+26^{2}+26.75^{2}\)
\(=770.0625 + 625+689.0625+663.0625+756.25+689.0625+676+715.5625 = 5584.125\)
\(\sum_{i = 1}^{8}y_i^{2}=17.4^{2}+17.1^{2}+17.3^{2}+17^{2}+17.4^{2}+17.2^{2}+17.1^{2}+17.4^{2}\)
\(=302.76+292.41+299.29+289+302.76+295.84+292.41+302.76 = 2377.23\)
\(\sum_{i = 1}^{8}x_iy_i=(27.75\times17.4)+(25\times17.1)+(26.25\times17.3)+(25.75\times17)+(27.5\times17.4)+(26.25\times17.2)+(26\times17.1)+(26.75\times17.4)\)
\(=482.85+427.5+454.125+437.75+478.5+451.5+444.6+465.45 = 3642.275\)

Step2: Substitute into the formula

\[

$$\begin{align*} r&=\frac{8\times3642.275-211.25\times137.9}{\sqrt{8\times5584.125-(211.25)^{2}}\sqrt{8\times2377.23-(137.9)^{2}}}\\ &=\frac{29138.2 - 29131.375}{\sqrt{44673 - 44627.5625}\sqrt{19017.84 - 19016.41}}\\ &=\frac{6.825}{\sqrt{45.4375}\sqrt{1.43}}\\ &=\frac{6.825}{\sqrt{45.4375\times1.43}}\\ &=\frac{6.825}{\sqrt{64.975625}}\\ &=\frac{6.825}{8.06075}\\ &\approx0.847 \end{align*}$$

\]

Answer:

\(0.847\)