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To calculate the correlation coefficient [tex]\( r \)[/tex] for the given sets of data [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we need to follow these steps:
1. Understand the Data: We have two sets of data:
- [tex]\( x = [1, 2, 2.3, 3, 3.3, 4] \)[/tex]
- [tex]\( y = [1.7, 1.2, 3, 1.9, 2.6, 3.1] \)[/tex]
2. Calculate the Means: Calculate the mean of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
[tex]\[ \bar{x} = \frac{1 + 2 + 2.3 + 3 + 3.3 + 4}{6} \approx 2.6 \][/tex]
[tex]\[ \bar{y} = \frac{1.7 + 1.2 + 3 + 1.9 + 2.6 + 3.1}{6} \approx 2.25 \][/tex]
3. Subtract the Mean from Each Value: Create two new sets of data by subtracting the mean from each data point.
[tex]\[ x' = \left[1 - 2.6, 2 - 2.6, 2.3 - 2.6, 3 - 2.6, 3.3 - 2.6, 4 - 2.6\right] = [-1.6, -0.6, -0.3, 0.4, 0.7, 1.4] \][/tex]
[tex]\[ y' = \left[1.7 - 2.25, 1.2 - 2.25, 3 - 2.25, 1.9 - 2.25, 2.6 - 2.25, 3.1 - 2.25\right] = [-0.55, -1.05, 0.75, -0.35, 0.35, 0.85] \][/tex]
4. Multiply Corresponding Values: Multiply corresponding values of [tex]\( x' \)[/tex] and [tex]\( y' \)[/tex].
[tex]\[ \left[-1.6 \times -0.55, -0.6 \times -1.05, -0.3 \times 0.75, 0.4 \times -0.35, 0.7 \times 0.35, 1.4 \times 0.85\right] = [0.88, 0.63, -0.225, -0.14, 0.245, 1.19] \][/tex]
5. Calculate Sum of Products:
[tex]\[ \sum = 0.88 + 0.63 - 0.225 - 0.14 + 0.245 + 1.19 = 2.58 \][/tex]
6. Calculate Sum of Squares:
[tex]\[ \sum (x'^2) = (-1.6)^2 + (-0.6)^2 + (-0.3)^2 + (0.4)^2 + (0.7)^2 + (1.4)^2 = 2.56 + 0.36 + 0.09 + 0.16 + 0.49 + 1.96 = 5.62 \][/tex]
[tex]\[ \sum (y'^2) = (-0.55)^2 + (-1.05)^2 + (0.75)^2 + (-0.35)^2 + (0.35)^2 + (0.85)^2 = 0.3025 + 1.1025 + 0.5625 + 0.1225 + 0.1225 + 0.7225 = 2.935 \][/tex]
7. Calculate the Correlation Coefficient:
[tex]\[ r = \frac{\sum (x' \times y')}{\sqrt{\sum (x'^2) \times \sum (y'^2)}} = \frac{2.58}{\sqrt{5.62 \times 2.935}} = \frac{2.58}{\sqrt{16.497}} = \frac{2.58}{4.06} \approx 0.635254 \][/tex]
8. Round the Correlation Coefficient: Round [tex]\( r \)[/tex] to the nearest hundredth.
[tex]\[ r \approx 0.64 \][/tex]
So the correct answer is [tex]\( \boxed{0.64} \)[/tex].
1. Understand the Data: We have two sets of data:
- [tex]\( x = [1, 2, 2.3, 3, 3.3, 4] \)[/tex]
- [tex]\( y = [1.7, 1.2, 3, 1.9, 2.6, 3.1] \)[/tex]
2. Calculate the Means: Calculate the mean of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
[tex]\[ \bar{x} = \frac{1 + 2 + 2.3 + 3 + 3.3 + 4}{6} \approx 2.6 \][/tex]
[tex]\[ \bar{y} = \frac{1.7 + 1.2 + 3 + 1.9 + 2.6 + 3.1}{6} \approx 2.25 \][/tex]
3. Subtract the Mean from Each Value: Create two new sets of data by subtracting the mean from each data point.
[tex]\[ x' = \left[1 - 2.6, 2 - 2.6, 2.3 - 2.6, 3 - 2.6, 3.3 - 2.6, 4 - 2.6\right] = [-1.6, -0.6, -0.3, 0.4, 0.7, 1.4] \][/tex]
[tex]\[ y' = \left[1.7 - 2.25, 1.2 - 2.25, 3 - 2.25, 1.9 - 2.25, 2.6 - 2.25, 3.1 - 2.25\right] = [-0.55, -1.05, 0.75, -0.35, 0.35, 0.85] \][/tex]
4. Multiply Corresponding Values: Multiply corresponding values of [tex]\( x' \)[/tex] and [tex]\( y' \)[/tex].
[tex]\[ \left[-1.6 \times -0.55, -0.6 \times -1.05, -0.3 \times 0.75, 0.4 \times -0.35, 0.7 \times 0.35, 1.4 \times 0.85\right] = [0.88, 0.63, -0.225, -0.14, 0.245, 1.19] \][/tex]
5. Calculate Sum of Products:
[tex]\[ \sum = 0.88 + 0.63 - 0.225 - 0.14 + 0.245 + 1.19 = 2.58 \][/tex]
6. Calculate Sum of Squares:
[tex]\[ \sum (x'^2) = (-1.6)^2 + (-0.6)^2 + (-0.3)^2 + (0.4)^2 + (0.7)^2 + (1.4)^2 = 2.56 + 0.36 + 0.09 + 0.16 + 0.49 + 1.96 = 5.62 \][/tex]
[tex]\[ \sum (y'^2) = (-0.55)^2 + (-1.05)^2 + (0.75)^2 + (-0.35)^2 + (0.35)^2 + (0.85)^2 = 0.3025 + 1.1025 + 0.5625 + 0.1225 + 0.1225 + 0.7225 = 2.935 \][/tex]
7. Calculate the Correlation Coefficient:
[tex]\[ r = \frac{\sum (x' \times y')}{\sqrt{\sum (x'^2) \times \sum (y'^2)}} = \frac{2.58}{\sqrt{5.62 \times 2.935}} = \frac{2.58}{\sqrt{16.497}} = \frac{2.58}{4.06} \approx 0.635254 \][/tex]
8. Round the Correlation Coefficient: Round [tex]\( r \)[/tex] to the nearest hundredth.
[tex]\[ r \approx 0.64 \][/tex]
So the correct answer is [tex]\( \boxed{0.64} \)[/tex].
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