Question

a pediatrician wants to determine the relation that may exist between a childs height and head circumference. she randomly selects 8 children from her practice, measures their height and head circumference, and obtains the data shown in the table. complete parts (a) through (e).

height (in.) head circumference (in.)
27.75 17.4
25.75 17.2
26.5 17.2
25.75 16.9
28 17.5
26.5 17.3
25.75 17.1
27 17.4

click here to see the table of critical values for correlation coefficient.

(e) convert the data to centimeters (1 inch = 2.54 cm), and recompute the linear correlation coefficient. what effect did the conversion have on the linear correlation coefficient?

convert the first four data values to centimeters.

height (centimeters) head circumference (centimeters)

(type integers or decimals. do not round. list the terms in the same order as they appear in the original list.)

Explanation:

Step1: Convert height to centimeters

Multiply each height value (in inches) by \(2.54\).
For \(27.75\) inches: \(27.75\times2.54 = 70.485\)
For \(25.75\) inches: \(25.75\times2.54=65.405\)
For \(26.5\) inches: \(26.5\times2.54 = 67.31\)
For \(25.75\) inches: \(25.75\times2.54 = 65.405\)

Step2: Convert head circumference to centimeters

Multiply each head - circumference value (in inches) by \(2.54\).
For \(17.4\) inches: \(17.4\times2.54=44.196\)
For \(17.2\) inches: \(17.2\times2.54 = 43.688\)
For \(17.2\) inches: \(17.2\times2.54=43.688\)
For \(16.9\) inches: \(16.9\times2.54 = 42.926\)

Answer:

Height (centimeters)	Head Circumference (centimeters)
\(65.405\)	\(43.688\)
\(67.31\)	\(43.688\)
\(65.405\)	\(42.926\)

The linear correlation coefficient is unchanged when we convert the units of measurement. This is because the linear correlation coefficient \(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\), and if \(x_{i}^{*}=a x_{i}+b\) and \(y_{i}^{*}=c y_{i}+d\) (in our case \(a = c=2.54\), \(b = d = 0\)), the formula for \(r\) with \(x_{i}^{*}\) and \(y_{i}^{*}\) will simplify to the same value as with \(x_{i}\) and \(y_{i}\) due to the properties of covariance and variance under linear transformations.