Question

the table shows the time a patient spends at the dentist and the amount of the bill. what is the correlation coefficient for the data in the table? bill amount for time spent at the dentist -0.93 -0.27 0.27 0.93

Explanation:

Step1: Calculate the means

Let \(x\) be the time spent (\(x_1 = 1.4,x_2=2.7,x_3 = 0.75,x_4=1.6\)) and \(y\) be the bill amount (\(y_1 = 235,y_2=867,y_3 = 156,y_4=215\)).
The mean of \(x\), \(\bar{x}=\frac{1.4 + 2.7+0.75 + 1.6}{4}=\frac{6.45}{4}=1.6125\)
The mean of \(y\), \(\bar{y}=\frac{235+867+156+215}{4}=\frac{1473}{4}=368.25\)

Step2: Calculate the numerator and denominator of the correlation formula

The formula for the correlation coefficient \(r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i-\bar{x})^2\sum_{i = 1}^{n}(y_i-\bar{y})^2}}\)

\((x_1-\bar{x})(y_1-\bar{y})=(1.4 - 1.6125)(235 - 368.25)=(- 0.2125)\times(-133.25)=28.315625\)
\((x_2-\bar{x})(y_2-\bar{y})=(2.7-1.6125)(867 - 368.25)=(1.0875)\times(498.75)=541.734375\)
\((x_3-\bar{x})(y_3-\bar{y})=(0.75 - 1.6125)(156 - 368.25)=(-0.8625)\times(-212.25)=183.178125\)
\((x_4-\bar{x})(y_4-\bar{y})=(1.6-1.6125)(215 - 368.25)=(-0.0125)\times(-153.25)=1.915625\)
\(\sum_{i = 1}^{4}(x_i-\bar{x})(y_i-\bar{y})=28.315625 + 541.734375+183.178125+1.915625 = 755.14375\)

\((x_1-\bar{x})^2=(1.4 - 1.6125)^2=0.04515625\)
\((x_2-\bar{x})^2=(2.7 - 1.6125)^2=1.18265625\)
\((x_3-\bar{x})^2=(0.75 - 1.6125)^2=0.74390625\)
\((x_4-\bar{x})^2=(1.6 - 1.6125)^2=0.00015625\)
\(\sum_{i = 1}^{4}(x_i-\bar{x})^2=0.04515625+1.18265625 + 0.74390625+0.00015625=1.971875\)

\((y_1-\bar{y})^2=(235 - 368.25)^2=17750.0625\)
\((y_2-\bar{y})^2=(867 - 368.25)^2=198901.1875\)
\((y_3-\bar{y})^2=(156 - 368.25)^2=44949.5625\)
\((y_4-\bar{y})^2=(215 - 368.25)^2=23483.0625\)
\(\sum_{i = 1}^{4}(y_i-\bar{y})^2=17750.0625+198901.1875+44949.5625+23483.0625=285083.875\)

\(r=\frac{755.14375}{\sqrt{1.971875\times285083.875}}\)
\(\sqrt{1.971875\times285083.875}=\sqrt{562197.7734375}\approx749.8\)
\(r=\frac{755.14375}{749.8}\approx0.93\)

Answer:

0.93