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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].
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