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
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 for correlation coefficient formula

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

For \((x_1,y_1)=(1.4,235)\): \((1.4 - 1.6125)(235 - 368.25)=(- 0.2125)(-133.25)=28.315625\)
For \((x_2,y_2)=(2.7,867)\): \((2.7-1.6125)(867 - 368.25)=(1.0875)(498.75)=541.734375\)
For \((x_3,y_3)=(0.75,156)\): \((0.75 - 1.6125)(156 - 368.25)=(-0.8625)(-212.25)=183.01875\)
For \((x_4,y_4)=(1.6,215)\): \((1.6 - 1.6125)(215 - 368.25)=(-0.0125)(-153.25)=1.915625\)

\(\sum_{i = 1}^{4}(x_{i}-\bar{x})(y_{i}-\bar{y})=28.315625 + 541.734375+183.01875+1.915625 = 755\)

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

\((y_1-\bar{y})^{2}=(235 - 368.25)^{2}=17750.0625\)
\((y_2-\bar{y})^{2}=(867 - 368.25)^{2}=198901.6875\)
\((y_3-\bar{y})^{2}=(156 - 368.25)^{2}=45049.0625\)
\((y_4-\bar{y})^{2}=(215 - 368.25)^{2}=23483.0625\)
\(\sum_{i = 1}^{4}(y_{i}-\bar{y})^{2}=17750.0625+198901.6875+45049.0625+23483.0625 = 285183\)

\(\sqrt{\sum_{i = 1}^{4}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{4}(y_{i}-\bar{y})^{2}}=\sqrt{1.971875\times285183}\approx\sqrt{562439.9}\approx750\)

\(r=\frac{755}{750}\approx0.93\)

Answer:

0.93