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?
A. -0.93
B. -0.27
C. 0.27
D. 0.93
Answer (1)
Calculate the mean of x values (time spent at the dentist): x ˉ = 1.6125 .
Calculate the mean of y values (bill amount): y ˉ = 368.25 .
Calculate the standard deviations of x and y values: s x ≈ 0.811 and s y ≈ 334.14 .
Calculate the correlation coefficient: r ≈ 0.93 . The final answer is 0.93 .
Explanation
Understanding the Problem We are given a table with the time spent at the dentist and the corresponding bill amount. We need to find the correlation coefficient for the data in the table. The data points are (1.4, 235), (2.7, 867), (0.75, 156), and (1.6, 215).
Formula for Correlation Coefficient Let x represent the time spent at the dentist (in hours) and y represent the bill amount. We will calculate the correlation coefficient using the formula: r = ( n − 1 ) s x s y ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ ) where x ˉ and y ˉ are the means of x and y values, and s x and s y are the standard deviations of x and y values, and n is the number of data points.
Calculating the Means First, calculate the means of x and y values: x ˉ = 4 1.4 + 2.7 + 0.75 + 1.6 = 4 6.45 = 1.6125 y ˉ = 4 235 + 867 + 156 + 215 = 4 1473 = 368.25
Calculating Standard Deviations Next, calculate the standard deviations of x and y values: s x = n − 1 ∑ ( x i − x ˉ ) 2 = 4 − 1 ( 1.4 − 1.6125 ) 2 + ( 2.7 − 1.6125 ) 2 + ( 0.75 − 1.6125 ) 2 + ( 1.6 − 1.6125 ) 2 s x = 3 ( − 0.2125 ) 2 + ( 1.0875 ) 2 + ( − 0.8625 ) 2 + ( − 0.0125 ) 2 = 3 0.04515625 + 1.18265625 + 0.74390625 + 0.00015625 = 3 1.971875 ≈ 0.811 s y = n − 1 ∑ ( y i − y ˉ ) 2 = 4 − 1 ( 235 − 368.25 ) 2 + ( 867 − 368.25 ) 2 + ( 156 − 368.25 ) 2 + ( 215 − 368.25 ) 2 s y = 3 ( − 133.25 ) 2 + ( 498.75 ) 2 + ( − 212.25 ) 2 + ( − 153.25 ) 2 = 3 17755.5625 + 248751.5625 + 45050.0625 + 23485.5625 = 3 335042.75 ≈ 334.14
Calculating the Correlation Coefficient Now, calculate the correlation coefficient: r = ( n − 1 ) s x s y ∑ ( x i − x ˉ ) ( y i − y ˉ ) = ( 4 − 1 ) ( 0.811 ) ( 334.14 ) ( 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 ) r = 3 ( 0.811 ) ( 334.14 ) ( − 0.2125 ) ( − 133.25 ) + ( 1.0875 ) ( 498.75 ) + ( − 0.8625 ) ( − 212.25 ) + ( − 0.0125 ) ( − 153.25 ) r = 812.4582 28.315625 + 542.484375 + 183.053125 + 1.915625 = 812.4582 755.76875 ≈ 0.93
Final Answer The correlation coefficient is approximately 0.93. This indicates a strong positive correlation between the time spent at the dentist and the bill amount.
Examples
Understanding correlation coefficients is very useful in real life. For example, a marketing team can analyze the correlation between advertising spending and sales revenue to determine the effectiveness of their campaigns. Similarly, doctors can study the correlation between lifestyle factors and the risk of developing certain diseases to provide better health advice. In finance, investors can use correlation coefficients to understand the relationship between different assets and build diversified portfolios.
