Question

the table shows the time a patient spends at the dentist and the amount of the bill. bill amount for time spent at the dentist time spent at the dentist (in hours) 1.4 2.7 0.75 1.6 bill amount $235 $867 $156 $215 what is the correlation coefficient for the data in the table? -0.93 -0.27 0.27 0.93

Explanation:

Step1: Recall correlation coefficient formula

The formula for the correlation coefficient $r$ is $r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$, where $n$ is the number of data - points, $x$ is the independent variable (time spent at the dentist), and $y$ is the dependent variable (bill amount). Let $x_1 = 0.75,x_2=1.4,x_3 = 2.7,x_4=1.6$ and $y_1 = 156,y_2 = 235,y_3=867,y_4 = 215$, and $n = 4$.

Step2: Calculate sums

Calculate $\sum x=0.75 + 1.4+2.7+1.6=6.45$, $\sum y=156 + 235+867+215 = 1473$, $\sum xy=(0.75\times156)+(1.4\times235)+(2.7\times867)+(1.6\times215)=117+329+2340.9+344 = 3130.9$, $\sum x^{2}=0.75^{2}+1.4^{2}+2.7^{2}+1.6^{2}=0.5625 + 1.96+7.29+2.56 = 12.3725$, $\sum y^{2}=156^{2}+235^{2}+867^{2}+215^{2}=24336+55225+751689+46225 = 877475$.

Step3: Substitute into formula

$n(\sum xy)=4\times3130.9 = 12523.6$, $(\sum x)(\sum y)=6.45\times1473 = 9490.85$, $n\sum x^{2}=4\times12.3725 = 49.49$, $(\sum x)^{2}=6.45^{2}=41.6025$, $n\sum y^{2}=4\times877475 = 3509900$, $(\sum y)^{2}=1473^{2}=2169729$. Then $r=\frac{12523.6 - 9490.85}{\sqrt{(49.49 - 41.6025)(3509900 - 2169729)}}=\frac{3032.75}{\sqrt{7.8875\times1340171}}=\frac{3032.75}{\sqrt{10560301.625}}\approx\frac{3032.75}{3249.66}\approx0.93$.

Answer:

0.93