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To find the linear correlation coefficient [tex]\(r\)[/tex] for the given data, follow these steps:
1. List the provided information:
- Hours: [tex]\( [5, 10, 4, 6, 10, 9] \)[/tex]
- Scores: [tex]\( [64, 86, 69, 86, 59, 87] \)[/tex]
2. Calculate the mean of the hours and scores:
[tex]\[ \text{Mean of hours} = \frac{5 + 10 + 4 + 6 + 10 + 9}{6} = \frac{44}{6} = 7.3333 \approx 7.33 \][/tex]
[tex]\[ \text{Mean of scores} = \frac{64 + 86 + 69 + 86 + 59 + 87}{6} = \frac{451}{6} \approx 75.17 \][/tex]
3. Calculate the numerator for the correlation coefficient formula:
[tex]\[ \text{Numerator} = \sum_{i=1}^{6} (x_i - \bar{x})(y_i - \bar{y}) \][/tex]
Where [tex]\( x_i \)[/tex] represents the hours and [tex]\( y_i \)[/tex] represents the scores.
Performing these subtractions and multiplications:
[tex]\[ ((5 - 7.33) \times (64 - 75.17)) + ((10 - 7.33) \times (86 - 75.17)) + ((4 - 7.33) \times (69 - 75.17)) + ((6 - 7.33) \times (86 - 75.17)) + ((10 - 7.33) \times (59 - 75.17)) + ((9 - 7.33) \times (87 - 75.17)) \][/tex]
[tex]\[ = (-2.33 \times -11.17) + (2.67 \times 10.83) + (-3.33 \times -6.17) + (-1.33 \times 10.83) + (2.67 \times -16.17) + (1.67 \times 11.83) \][/tex]
[tex]\[ \approx 26 + 28.90 + 20.57 - 14.40 - 43.22 + 19.85 \][/tex]
[tex]\[ = 37.67 \][/tex]
4. Calculate the denominator for the correlation coefficient formula:
[tex]\[ \text{Denominator} = \sqrt{\left(\sum_{i=1}^{6} (x_i - \bar{x})^2\right) \times \left(\sum_{i=1}^{6} (y_i - \bar{y})^2\right)} \][/tex]
Calculating these terms individually:
[tex]\[ \sum_{i=1}^{6} (x_i - \bar{x})^2 = (5 - 7.33)^2 + (10 - 7.33)^2 + (4 - 7.33)^2 + (6 - 7.33)^2 + (10 - 7.33)^2 + (9 - 7.33)^2 \][/tex]
[tex]\[ = 5.44 + 7.11 + 11.09 + 1.77 + 7.11 + 2.78 = 35.30 \][/tex]
[tex]\[ \sum_{i=1}^{6} (y_i - \bar{y})^2 = (64 - 75.17)^2 + (86 - 75.17)^2 + (69 - 75.17)^2 + (86 - 75.17)^2 + (59 - 75.17)^2 + (87 - 75.17)^2 \][/tex]
[tex]\[ = 124.10 + 117.35 + 38.00 + 117.35 + 261.32 + 139.88 = 797.99 \][/tex]
Then the denominator:
[tex]\[ \text{Denominator} = \sqrt{35.30 \times 797.99} \approx 168.00 \][/tex]
5. Finally, calculate the correlation coefficient [tex]\(r\)[/tex]:
[tex]\[ r = \frac{\text{Numerator}}{\text{Denominator}} = \frac{37.67}{168.00} \approx 0.224 \][/tex]
Thus, the value of the linear correlation coefficient [tex]\(r\)[/tex] is approximately 0.224. Therefore, the correct answer is:
[tex]\[ \boxed{0.224} \][/tex]
1. List the provided information:
- Hours: [tex]\( [5, 10, 4, 6, 10, 9] \)[/tex]
- Scores: [tex]\( [64, 86, 69, 86, 59, 87] \)[/tex]
2. Calculate the mean of the hours and scores:
[tex]\[ \text{Mean of hours} = \frac{5 + 10 + 4 + 6 + 10 + 9}{6} = \frac{44}{6} = 7.3333 \approx 7.33 \][/tex]
[tex]\[ \text{Mean of scores} = \frac{64 + 86 + 69 + 86 + 59 + 87}{6} = \frac{451}{6} \approx 75.17 \][/tex]
3. Calculate the numerator for the correlation coefficient formula:
[tex]\[ \text{Numerator} = \sum_{i=1}^{6} (x_i - \bar{x})(y_i - \bar{y}) \][/tex]
Where [tex]\( x_i \)[/tex] represents the hours and [tex]\( y_i \)[/tex] represents the scores.
Performing these subtractions and multiplications:
[tex]\[ ((5 - 7.33) \times (64 - 75.17)) + ((10 - 7.33) \times (86 - 75.17)) + ((4 - 7.33) \times (69 - 75.17)) + ((6 - 7.33) \times (86 - 75.17)) + ((10 - 7.33) \times (59 - 75.17)) + ((9 - 7.33) \times (87 - 75.17)) \][/tex]
[tex]\[ = (-2.33 \times -11.17) + (2.67 \times 10.83) + (-3.33 \times -6.17) + (-1.33 \times 10.83) + (2.67 \times -16.17) + (1.67 \times 11.83) \][/tex]
[tex]\[ \approx 26 + 28.90 + 20.57 - 14.40 - 43.22 + 19.85 \][/tex]
[tex]\[ = 37.67 \][/tex]
4. Calculate the denominator for the correlation coefficient formula:
[tex]\[ \text{Denominator} = \sqrt{\left(\sum_{i=1}^{6} (x_i - \bar{x})^2\right) \times \left(\sum_{i=1}^{6} (y_i - \bar{y})^2\right)} \][/tex]
Calculating these terms individually:
[tex]\[ \sum_{i=1}^{6} (x_i - \bar{x})^2 = (5 - 7.33)^2 + (10 - 7.33)^2 + (4 - 7.33)^2 + (6 - 7.33)^2 + (10 - 7.33)^2 + (9 - 7.33)^2 \][/tex]
[tex]\[ = 5.44 + 7.11 + 11.09 + 1.77 + 7.11 + 2.78 = 35.30 \][/tex]
[tex]\[ \sum_{i=1}^{6} (y_i - \bar{y})^2 = (64 - 75.17)^2 + (86 - 75.17)^2 + (69 - 75.17)^2 + (86 - 75.17)^2 + (59 - 75.17)^2 + (87 - 75.17)^2 \][/tex]
[tex]\[ = 124.10 + 117.35 + 38.00 + 117.35 + 261.32 + 139.88 = 797.99 \][/tex]
Then the denominator:
[tex]\[ \text{Denominator} = \sqrt{35.30 \times 797.99} \approx 168.00 \][/tex]
5. Finally, calculate the correlation coefficient [tex]\(r\)[/tex]:
[tex]\[ r = \frac{\text{Numerator}}{\text{Denominator}} = \frac{37.67}{168.00} \approx 0.224 \][/tex]
Thus, the value of the linear correlation coefficient [tex]\(r\)[/tex] is approximately 0.224. Therefore, the correct answer is:
[tex]\[ \boxed{0.224} \][/tex]
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