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).

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

the explanatory variable is head circumference and the response variable is height.

(b) draw a scatter diagram. which of the following represents the data?

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

Explanation:

Step1: Calculate the means

Let \(x\) be height and \(y\) be head circumference.
\(\bar{x}=\frac{27.75 + 25.75+26.5+25.75+28+26.5+25.75+27}{8}=\frac{213}{8}=26.625\)
\(\bar{y}=\frac{17.4 + 17.2+17.2+16.9+17.5+17.3+17.1+17.4}{8}=\frac{138}{8}=17.25\)

Step2: Calculate numerator and denominator for \(r\)

The formula for \(r\) is \(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\)

Calculate \((x_{i}-\bar{x})(y_{i}-\bar{y})\) for each \(i\):
\((27.75 - 26.625)(17.4 - 17.25)=1.125\times0.15 = 0.16875\)
\((25.75 - 26.625)(17.2 - 17.25)=(- 0.875)\times(-0.05)=0.04375\)
\((26.5 - 26.625)(17.2 - 17.25)=(-0.125)\times(-0.05) = 0.00625\)
\((25.75 - 26.625)(16.9 - 17.25)=(-0.875)\times(-0.35)=0.30625\)
\((28 - 26.625)(17.5 - 17.25)=1.375\times0.25 = 0.34375\)
\((26.5 - 26.625)(17.3 - 17.25)=(-0.125)\times0.05=-0.00625\)
\((25.75 - 26.625)(17.1 - 17.25)=(-0.875)\times(-0.15)=0.13125\)
\((27 - 26.625)(17.4 - 17.25)=0.375\times0.15 = 0.05625\)
\(\sum_{i = 1}^{8}(x_{i}-\bar{x})(y_{i}-\bar{y})=0.16875 + 0.04375+0.00625+0.30625+0.34375-0.00625+0.13125+0.05625 = 1.05\)

Calculate \((x_{i}-\bar{x})^{2}\):
\((27.75 - 26.625)^{2}=1.265625\)
\((25.75 - 26.625)^{2}=0.765625\)
\((26.5 - 26.625)^{2}=0.015625\)
\((25.75 - 26.625)^{2}=0.765625\)
\((28 - 26.625)^{2}=1.890625\)
\((26.5 - 26.625)^{2}=0.015625\)
\((25.75 - 26.625)^{2}=0.765625\)
\((27 - 26.625)^{2}=0.140625\)
\(\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=1.265625+0.765625 + 0.015625+0.765625+1.890625+0.015625+0.765625+0.140625=5.625\)

Calculate \((y_{i}-\bar{y})^{2}\):
\((17.4 - 17.25)^{2}=0.0225\)
\((17.2 - 17.25)^{2}=0.0025\)
\((17.2 - 17.25)^{2}=0.0025\)
\((16.9 - 17.25)^{2}=0.1225\)
\((17.5 - 17.25)^{2}=0.0625\)
\((17.3 - 17.25)^{2}=0.0025\)
\((17.1 - 17.25)^{2}=0.0225\)
\((17.4 - 17.25)^{2}=0.0225\)
\(\sum_{i = 1}^{8}(y_{i}-\bar{y})^{2}=0.0225+0.0025+0.0025+0.1225+0.0625+0.0025+0.0225+0.0225 = 0.26\)

Step3: Calculate \(r\)

\(r=\frac{1.05}{\sqrt{5.625\times0.26}}=\frac{1.05}{\sqrt{1.4625}}\approx\frac{1.05}{1.209}\approx0.868\)

Answer:

\(r = 0.868\)