Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
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].
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].
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
