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? -0.93 -0.27 0.27 0.93

Explanation:

Step1: Calculate the means

Let \(x\) be the time spent at the dentist and \(y\) be the bill amount.
\(\bar{x}=\frac{1.4 + 2.7+0.75 + 1.6}{4}=\frac{6.45}{4}=1.6125\)
\(\bar{y}=\frac{235+867 + 156+215}{4}=\frac{1473}{4}=368.25\)

Step2: Calculate numerator and denominator

Numerator: \(\sum_{i = 1}^{4}(x_{i}-\bar{x})(y_{i}-\bar{y})\)
\((1.4 - 1.6125)(235 - 368.25)+(2.7-1.6125)(867 - 368.25)+(0.75 - 1.6125)(156 - 368.25)+(1.6 - 1.6125)(215 - 368.25)\)
\(=(- 0.2125)(-133.25)+(1.0875)(498.75)+(-0.8625)(-212.25)+(-0.0125)(-153.25)\)
\(=28.315625+542.578125 + 183.046875+1.915625=755.85625\)

Denominator: \(\sqrt{\sum_{i = 1}^{4}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{4}(y_{i}-\bar{y})^{2}}\)
\(\sum_{i = 1}^{4}(x_{i}-\bar{x})^{2}=(1.4 - 1.6125)^{2}+(2.7-1.6125)^{2}+(0.75 - 1.6125)^{2}+(1.6 - 1.6125)^{2}\)
\(=(-0.2125)^{2}+(1.0875)^{2}+(-0.8625)^{2}+(-0.0125)^{2}\)
\(=0.04515625+1.18265625+0.74390625 + 0.00015625=1.971875\)

\(\sum_{i = 1}^{4}(y_{i}-\bar{y})^{2}=(235 - 368.25)^{2}+(867 - 368.25)^{2}+(156 - 368.25)^{2}+(215 - 368.25)^{2}\)
\(=(-133.25)^{2}+(498.75)^{2}+(-212.25)^{2}+(-153.25)^{2}\)
\(=17755.5625+248751.5625+45050.0625+23485.5625=335042.75\)

\(\sqrt{1.971875\times335042.75}=\sqrt{660625.390625}=812.8\)

Step3: Calculate correlation coefficient

\(r=\frac{755.85625}{812.8}\approx0.93\)

Answer:

0.93