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 time spent at the dentist (in hours) bill amount 1.4 $235 2.7 $867 0.75 $156 1.6 $215

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 numerator and denominator components

For each \(i\):

• \( (x_1-\bar{x})(y_1 - \bar{y})=(1.4 - 1.6125)(235 - 368.25)=(- 0.2125)\times(-133.25)=28.315625\)
• \( (x_2-\bar{x})(y_2 - \bar{y})=(2.7-1.6125)(867 - 368.25)=(1.0875)\times(498.75)=541.640625\)
• \( (x_3-\bar{x})(y_3 - \bar{y})=(0.75 - 1.6125)(156 - 368.25)=(-0.8625)\times(-212.25)=183.05625\)
• \( (x_4-\bar{x})(y_4 - \bar{y})=(1.6 - 1.6125)(215 - 368.25)=(-0.0125)\times(-153.25)=1.915625\)

Sum of \((x_i-\bar{x})(y_i - \bar{y})\): \(28.315625+541.640625 + 183.05625+1.915625=754.928125\)

For the denominator:

• \((x_1-\bar{x})^2=(1.4 - 1.6125)^2=0.04515625\)
• \((x_2-\bar{x})^2=(2.7-1.6125)^2 = 1.18265625\)
• \((x_3-\bar{x})^2=(0.75 - 1.6125)^2=0.74390625\)
• \((x_4-\bar{x})^2=(1.6 - 1.6125)^2=0.00015625\)

Sum of \((x_i-\bar{x})^2\): \(0.04515625+1.18265625+0.74390625 + 0.00015625=1.971875\)

• \((y_1-\bar{y})^2=(235 - 368.25)^2=17750.0625\)
• \((y_2-\bar{y})^2=(867 - 368.25)^2=198901.640625\)
• \((y_3-\bar{y})^2=(156 - 368.25)^2=45049.0625\)
• \((y_4-\bar{y})^2=(215 - 368.25)^2=23483.0625\)

Sum of \((y_i-\bar{y})^2\): \(17750.0625+198901.640625+45049.0625+23483.0625=285183.828125\)

The correlation coefficient \(r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i-\bar{x})^2\sum_{i = 1}^{n}(y_i-\bar{y})^2}}\)
\(r=\frac{754.928125}{\sqrt{1.971875\times285183.828125}}=\frac{754.928125}{\sqrt{562399.902344}}\approx\frac{754.928125}{750}\approx0.93\)

Answer:

\(0.93\)