the table shows the time a patient spends at the dentist and the amount of the bill. bill amount for time…

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) bill amount 1.4 $235 2.7 $867 0.75 $156 1.6 $215 what is the correlation coefficient for the data in the table? -0.93 -0.27 0.27 0.93

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) bill amount 1.4 $235 2.7 $867 0.75 $156 1.6 $215 what is the correlation coefficient for the data in the table? -0.93 -0.27 0.27 0.93

Answer

Explanation:

Step1: Identify Variables

Let ( x ) be time spent (hours): ( x = [1.4, 2.7, 0.75, 1.6] )
Let ( y ) be bill amount ($): ( y = [235, 867, 156, 215] )

Step2: Calculate Means

( \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 )

Step3: Calculate Deviations

( x_i - \bar{x} ): ( -0.2125, 1.0875, -0.8625, -0.0125 )
( y_i - \bar{y} ): ( -133.25, 498.75, -212.25, -153.25 )

Step4: Calculate Products and Squares

( (x_i - \bar{x})(y_i - \bar{y}) ):
( (-0.2125)(-133.25) \approx 28.31 ), ( (1.0875)(498.75) \approx 542.47 ),
( (-0.8625)(-212.25) \approx 183.07 ), ( (-0.0125)(-153.25) \approx 1.92 )
Sum: ( 28.31 + 542.47 + 183.07 + 1.92 \approx 755.77 )

( (x_i - \bar{x})^2 ):
( 0.0452, 1.1826, 0.7439, 0.00016 )
Sum: ( 0.0452 + 1.1826 + 0.7439 + 0.00016 \approx 1.97186 )

( (y_i - \bar{y})^2 ):
( 17755.56, 248753.56, 45050.06, 23485.56 )
Sum: ( 17755.56 + 248753.56 + 45050.06 + 23485.56 \approx 335044.74 )

Step5: Apply Correlation Formula

( r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} )
( r = \frac{755.77}{\sqrt{1.97186 \times 335044.74}} \approx \frac{755.77}{\sqrt{660700}} \approx \frac{755.77}{812.83} \approx 0.93 )

Answer:

0.93