Questions: A pediatrician wants to determine the relation that may exist between a child's height and head circumference. She randomly selects 8 children, measures their height and head circumference, and obtains the data shown in the table. Head Height (inches)Circum 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 Determine if a linear relation exists between height and head circumference. (Note that the linear correlation coefficient between the height and head circumference of a child is r=0.842.)

A pediatrician wants to determine the relation that may exist between a child's height and head circumference. She randomly selects 8 children, measures their height and head circumference, and obtains the data shown in the table.

Head
Height (inches)Circum
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

Determine if a linear relation exists between height and head circumference. (Note that the linear correlation coefficient between the height and head circumference of a child is r=0.842.)

Transcript text: A pediatrician wants to determine the relation that may exist between a child's height and head circumference. She randomly selects 8 children, measures their height and head circumference, and obtains the data shown in the table. \begin{tabular}{|c|c|} \hline \multicolumn{2}{|l|}{Head} \\ \hline \multicolumn{2}{|l|}{Height (inches)Circum} \\ \hline 27.75 & 17.4 \\ \hline 25 & 17.1 \\ \hline 26.25 & 17.3 \\ \hline 25.75 & 17 \\ \hline 27.5 & 17.4 \\ \hline 26.25 & 17.2 \\ \hline 26 & 17.1 \\ \hline 26.75 & 17.4 \\ \hline \end{tabular} (d) Determine if a linear relation exists between height and head circumference. (Note that the linear correlation coefficient between the height and head circumference of a child is $r=0.842$.)

Solution

Solution Steps

Step 1: Calculate Covariance

The covariance between height $ X $ and head circumference $ Y $ is calculated as follows:

\[ \text{Cov}(X,Y) = 0.122 \]

Step 2: Calculate Standard Deviations

The standard deviations of the height and head circumference are given by:

\[ \sigma_X = 0.906 \] \[ \sigma_Y = 0.16 \]

Step 3: Calculate Correlation Coefficient

The correlation coefficient $ r $ is calculated using the formula:

\[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \]

Substituting the values:

\[ r = \frac{0.122}{0.906 \times 0.16} = 0.842 \]

Step 4: Compare with Critical Value

The critical value for $ n = 8 $ (number of children) is $ 0.707 $. Since the calculated correlation coefficient $ r = 0.842 $ is greater than the critical value, we conclude that:

A linear relation exists between height and head circumference.