IDNLearn.com provides a platform for sharing and gaining valuable knowledge. Get timely and accurate answers to your questions from our dedicated community of experts who are here to help you.

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

\begin{tabular}{|c|c|}
\hline
\begin{tabular}{c}
Time spent at the \\
dentist (in hours)
\end{tabular} & \begin{tabular}{c}
Bill \\
amount
\end{tabular} \\
\hline 1.4 & [tex]$\$[/tex]235[tex]$ \\
\hline 2.7 & $[/tex]\[tex]$867$[/tex] \\
\hline 0.75 & [tex]$\$[/tex]156[tex]$ \\
\hline 1.6 & $[/tex]\[tex]$215$[/tex] \\
\hline
\end{tabular}

What is the correlation coefficient for the data in the table?

A. [tex]$-0.93$[/tex]
B. [tex]$-0.27$[/tex]
C. 0.27
D. 0.93


Sagot :

To determine the correlation coefficient for the given data, we need to follow these steps:

1. Identify the variables:
- Time spent at the dentist (in hours): [tex]\( [1.4, 2.7, 0.75, 1.6] \)[/tex]
- Bill amount (in dollars): [tex]\( [235, 867, 156, 215] \)[/tex]

2. Calculate the means of the variables:
- Calculate the mean time spent at the dentist:
[tex]\[\bar{x} = \frac{1.4 + 2.7 + 0.75 + 1.6}{4} = \frac{6.45}{4} = 1.6125\][/tex]
- Calculate the mean of the bill amounts:
[tex]\[\bar{y} = \frac{235 + 867 + 156 + 215}{4} = \frac{1473}{4} = 368.25\][/tex]

3. Subtract the means from each data point:
- Adjusted time spent values: [tex]\( [1.4 - 1.6125, 2.7 - 1.6125, 0.75 - 1.6125, 1.6 - 1.6125] = [-0.2125, 1.0875, -0.8625, -0.0125] \)[/tex]
- Adjusted bill amounts: [tex]\( [235 - 368.25, 867 - 368.25, 156 - 368.25, 215 - 368.25] = [-133.25, 498.75, -212.25, -153.25] \)[/tex]

4. Calculate the products of the paired deviations:
[tex]\[ \begin{align*} (-0.2125 \times -133.25) &= 28.31875 \\ (1.0875 \times 498.75) &= 542.615625 \\ (-0.8625 \times -212.25) &= 183.31875 \\ (-0.0125 \times -153.25) &= 1.915625 \\ \end{align*} \][/tex]

5. Sum the products of deviations:
[tex]\[ \sum{product} = 28.31875 + 542.615625 + 183.31875 + 1.915625 = 756.16875 \][/tex]

6. Calculate the squared deviations and their sums:
- Sum the squared deviations for time spent:
[tex]\[ \begin{align*} (-0.2125)^2 &= 0.0451 \\ (1.0875)^2 &= 1.1822 \\ (-0.8625)^2 &= 0.7444 \\ (-0.0125)^2 &= 0.0002 \\ \end{align*} \][/tex]

[tex]\[ \sum{(x_i - \bar{x})^2} = 0.0451 + 1.1822 + 0.7444 + 0.0002 = 1.9719 \][/tex]

- Sum the squared deviations for the bill amounts:
[tex]\[ \begin{align*} (-133.25)^2 &= 17754.5625 \\ (498.75)^2 &= 248750.625 \\ (-212.25)^2 &= 45050.0625 \\ (-153.25)^2 &= 23485.5625 \\ \end{align*} \][/tex]

[tex]\[ \sum{(y_i - \bar{y})^2} = 17754.5625 + 248750.625 + 45050.0625 + 23485.5625 = 334040.8125 \][/tex]

7. Calculate the correlation coefficient:
- The formula for the correlation coefficient ([tex]\(r\)[/tex]) is:
[tex]\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}} \][/tex]

[tex]\[ r = \frac{756.16875}{\sqrt{1.9719 \cdot 334040.8125}} \][/tex]

- Simplifying the denominator:
[tex]\[ \sqrt{1.9719 \times 334040.8125} \approx \sqrt{658471.37} \approx 811.48 \][/tex]

- Therefore:
[tex]\[ r = \frac{756.16875}{811.48} \approx 0.93 \][/tex]

The correlation coefficient for the data given in the table is [tex]\(0.93\)[/tex]. Hence, the correct answer is [tex]\(0.93\)[/tex].