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To determine the correlation coefficient for the given data, we can follow these steps:
1. Write down the data in pairs:
- (8, 12)
- (12, 40)
- (6, 15)
- (20, 20)
2. Calculate the means of each variable:
- Mean number of flowers: [tex]\(\bar{x} = \frac{8 + 12 + 6 + 20}{4} = \frac{46}{4} = 11.5\)[/tex]
- Mean total cost: [tex]\(\bar{y} = \frac{12 + 40 + 15 + 20}{4} = \frac{87}{4} = 21.75\)[/tex]
3. Calculate the covariance:
- First, compute the deviation of each value from the mean:
- Deviations for number of flowers: [tex]\(8 - 11.5 = -3.5\)[/tex], [tex]\(12 - 11.5 = 0.5\)[/tex], [tex]\(6 - 11.5 = -5.5\)[/tex], [tex]\(20 - 11.5 = 8.5\)[/tex]
- Deviations for total cost: [tex]\(12 - 21.75 = -9.75\)[/tex], [tex]\(40 - 21.75 = 18.25\)[/tex], [tex]\(15 - 21.75 = -6.75\)[/tex], [tex]\(20 - 21.75 = -1.75\)[/tex]
- Compute the product of these deviations:
- [tex]\((-3.5 \times -9.75) = 34.125\)[/tex]
- [tex]\((0.5 \times 18.25) = 9.125\)[/tex]
- [tex]\((-5.5 \times -6.75) = 37.125\)[/tex]
- [tex]\((8.5 \times -1.75) = -14.875\)[/tex]
- Sum these products: [tex]\(34.125 + 9.125 + 37.125 - 14.875 = 65.5\)[/tex]
- Estimate the sample covariance: Cov(X, Y) = [tex]\(\frac{65.5}{4-1} = \frac{65.5}{3} = 21.8333\)[/tex]
4. Calculate the standard deviations:
- Standard deviation for the number of flowers:
- [tex]\((8 - 11.5)^2 = 12.25\)[/tex]
- [tex]\((12 - 11.5)^2 = 0.25\)[/tex]
- [tex]\((6 - 11.5)^2 = 30.25\)[/tex]
- [tex]\((20 - 11.5)^2 = 72.25\)[/tex]
- Sum: [tex]\(12.25 + 0.25 + 30.25 + 72.25 = 115\)[/tex]
- Variance: [tex]\(\frac{115}{4-1} = \frac{115}{3} = 38.333\)[/tex]
- Standard deviation: [tex]\(\sqrt{38.333} \approx 6.19\)[/tex]
- Standard deviation for total cost:
- [tex]\((12 - 21.75)^2 = 95.0625\)[/tex]
- [tex]\((40 - 21.75)^2 = 333.0625\)[/tex]
- [tex]\((15 - 21.75)^2 = 45.5625\)[/tex]
- [tex]\((20 - 21.75)^2 = 3.0625\)[/tex]
- Sum: [tex]\(95.0625 + 333.0625 + 45.5625 + 3.0625 = 476.75\)[/tex]
- Variance: [tex]\(\frac{476.75}{3} = 158.9167\)[/tex]
- Standard deviation: [tex]\(\sqrt{158.9167} \approx 12.61\)[/tex]
5. Calculate the correlation coefficient:
- Correlation coefficient, [tex]\(r\)[/tex], is given by:
[tex]\[ r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} = \frac{21.8333}{6.19 \times 12.61} \approx \frac{21.8333}{78.0599} \approx 0.2797 \][/tex]
6. Select the correct answer:
- Among the given choices:
- [tex]\( -0.57 \)[/tex]
- [tex]\( -0.28 \)[/tex]
- [tex]\( 0.28 \)[/tex]
- [tex]\( 0.57 \)[/tex]
- The value closest to [tex]\(0.2797\)[/tex] is [tex]\(0.28\)[/tex].
Thus, the correlation coefficient for the data in the table is approximately [tex]\(0.28\)[/tex]. The correct answer is therefore:
[tex]\[ \boxed{0.28} \][/tex]
1. Write down the data in pairs:
- (8, 12)
- (12, 40)
- (6, 15)
- (20, 20)
2. Calculate the means of each variable:
- Mean number of flowers: [tex]\(\bar{x} = \frac{8 + 12 + 6 + 20}{4} = \frac{46}{4} = 11.5\)[/tex]
- Mean total cost: [tex]\(\bar{y} = \frac{12 + 40 + 15 + 20}{4} = \frac{87}{4} = 21.75\)[/tex]
3. Calculate the covariance:
- First, compute the deviation of each value from the mean:
- Deviations for number of flowers: [tex]\(8 - 11.5 = -3.5\)[/tex], [tex]\(12 - 11.5 = 0.5\)[/tex], [tex]\(6 - 11.5 = -5.5\)[/tex], [tex]\(20 - 11.5 = 8.5\)[/tex]
- Deviations for total cost: [tex]\(12 - 21.75 = -9.75\)[/tex], [tex]\(40 - 21.75 = 18.25\)[/tex], [tex]\(15 - 21.75 = -6.75\)[/tex], [tex]\(20 - 21.75 = -1.75\)[/tex]
- Compute the product of these deviations:
- [tex]\((-3.5 \times -9.75) = 34.125\)[/tex]
- [tex]\((0.5 \times 18.25) = 9.125\)[/tex]
- [tex]\((-5.5 \times -6.75) = 37.125\)[/tex]
- [tex]\((8.5 \times -1.75) = -14.875\)[/tex]
- Sum these products: [tex]\(34.125 + 9.125 + 37.125 - 14.875 = 65.5\)[/tex]
- Estimate the sample covariance: Cov(X, Y) = [tex]\(\frac{65.5}{4-1} = \frac{65.5}{3} = 21.8333\)[/tex]
4. Calculate the standard deviations:
- Standard deviation for the number of flowers:
- [tex]\((8 - 11.5)^2 = 12.25\)[/tex]
- [tex]\((12 - 11.5)^2 = 0.25\)[/tex]
- [tex]\((6 - 11.5)^2 = 30.25\)[/tex]
- [tex]\((20 - 11.5)^2 = 72.25\)[/tex]
- Sum: [tex]\(12.25 + 0.25 + 30.25 + 72.25 = 115\)[/tex]
- Variance: [tex]\(\frac{115}{4-1} = \frac{115}{3} = 38.333\)[/tex]
- Standard deviation: [tex]\(\sqrt{38.333} \approx 6.19\)[/tex]
- Standard deviation for total cost:
- [tex]\((12 - 21.75)^2 = 95.0625\)[/tex]
- [tex]\((40 - 21.75)^2 = 333.0625\)[/tex]
- [tex]\((15 - 21.75)^2 = 45.5625\)[/tex]
- [tex]\((20 - 21.75)^2 = 3.0625\)[/tex]
- Sum: [tex]\(95.0625 + 333.0625 + 45.5625 + 3.0625 = 476.75\)[/tex]
- Variance: [tex]\(\frac{476.75}{3} = 158.9167\)[/tex]
- Standard deviation: [tex]\(\sqrt{158.9167} \approx 12.61\)[/tex]
5. Calculate the correlation coefficient:
- Correlation coefficient, [tex]\(r\)[/tex], is given by:
[tex]\[ r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} = \frac{21.8333}{6.19 \times 12.61} \approx \frac{21.8333}{78.0599} \approx 0.2797 \][/tex]
6. Select the correct answer:
- Among the given choices:
- [tex]\( -0.57 \)[/tex]
- [tex]\( -0.28 \)[/tex]
- [tex]\( 0.28 \)[/tex]
- [tex]\( 0.57 \)[/tex]
- The value closest to [tex]\(0.2797\)[/tex] is [tex]\(0.28\)[/tex].
Thus, the correlation coefficient for the data in the table is approximately [tex]\(0.28\)[/tex]. The correct answer is therefore:
[tex]\[ \boxed{0.28} \][/tex]
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