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Sagot :
To solve this problem step by step, let's first calculate the relevant sums based on the given data for hours spent surfing online and the corresponding test scores:
Given:
- Hours surfing online (x): [20, 15, 10, 18, 22, 28, 20, 12, 15, 25]
- Test scores (y): [79, 87, 95, 81, 70, 50, 73, 94, 90, 62]
(a) Relevant sums:
1. [tex]\(\sum x = 20 + 15 + 10 + 18 + 22 + 28 + 20 + 12 + 15 + 25 = 185\)[/tex]
2. [tex]\(\sum y = 79 + 87 + 95 + 81 + 70 + 50 + 73 + 94 + 90 + 62 = 781\)[/tex]
3. [tex]\(\sum x^2 = 20^2 + 15^2 + 10^2 + 18^2 + 22^2 + 28^2 + 20^2 + 12^2 + 15^2 + 25^2 = 3711\)[/tex]
4. [tex]\(\sum y^2 = 79^2 + 87^2 + 95^2 + 81^2 + 70^2 + 50^2 + 73^2 + 94^2 + 90^2 + 62^2 = 62905\)[/tex]
5. [tex]\(\sum xy = 20 \cdot 79 + 15 \cdot 87 + 10 \cdot 95 + 18 \cdot 81 + 22 \cdot 70 + 28 \cdot 50 + 20 \cdot 73 + 12 \cdot 94 + 15 \cdot 90 + 25 \cdot 62 = 13721\)[/tex]
So, the sums are:
[tex]\[ \sum x = 185, \quad \sum y = 781, \quad \sum x^2 = 3711, \quad \sum y^2 = 62905, \quad \sum xy = 13721 \][/tex]
(b) Correlation coefficient:
To find the correlation coefficient [tex]\(r\)[/tex], we use the formula:
[tex]\[ r = \frac{n \sum xy - (\sum x)(\sum y)}{\sqrt{(n \sum x^2 - (\sum x)^2)(n \sum y^2 - (\sum y)^2)}} \][/tex]
Here, [tex]\(n\)[/tex] is the number of data points, which is 10.
Plugging in the values:
1. Calculate the numerator:
[tex]\[ n \sum xy - (\sum x)(\sum y) = 10 \cdot 13721 - 185 \cdot 781 = 137210 - 144485 = -7275 \][/tex]
2. Calculate the denominator:
[tex]\[ \sqrt{(n \sum x^2 - (\sum x)^2)(n \sum y^2 - (\sum y)^2)} = \sqrt{(10 \cdot 3711 - 185^2)(10 \cdot 62905 - 781^2)} \][/tex]
Breaking it down further:
[tex]\[ n \sum x^2 - (\sum x)^2 = 10 \cdot 3711 - 185^2 = 37110 - 34225 = 2885 \][/tex]
[tex]\[ n \sum y^2 - (\sum y)^2 = 10 \cdot 62905 - 781^2 = 629050 - 609961 = 19089 \][/tex]
So, the denominator is:
[tex]\[ \sqrt{2885 \cdot 19089} \approx \sqrt{55012765} \approx 7417.01 \][/tex]
3. Finally, calculate the correlation coefficient:
[tex]\[ r = \frac{-7275}{7417.01} \approx -0.980 \][/tex]
Thus, the relevant sums and the correlation coefficient are:
(a)
[tex]\[ \sum x = 185, \sum y = 781, \sum x^2 = 3711, \sum y^2 = 62905, \sum xy = 13721 \][/tex]
(b) The correlation coefficient:
[tex]\[ r \approx -0.980 \][/tex]
Given:
- Hours surfing online (x): [20, 15, 10, 18, 22, 28, 20, 12, 15, 25]
- Test scores (y): [79, 87, 95, 81, 70, 50, 73, 94, 90, 62]
(a) Relevant sums:
1. [tex]\(\sum x = 20 + 15 + 10 + 18 + 22 + 28 + 20 + 12 + 15 + 25 = 185\)[/tex]
2. [tex]\(\sum y = 79 + 87 + 95 + 81 + 70 + 50 + 73 + 94 + 90 + 62 = 781\)[/tex]
3. [tex]\(\sum x^2 = 20^2 + 15^2 + 10^2 + 18^2 + 22^2 + 28^2 + 20^2 + 12^2 + 15^2 + 25^2 = 3711\)[/tex]
4. [tex]\(\sum y^2 = 79^2 + 87^2 + 95^2 + 81^2 + 70^2 + 50^2 + 73^2 + 94^2 + 90^2 + 62^2 = 62905\)[/tex]
5. [tex]\(\sum xy = 20 \cdot 79 + 15 \cdot 87 + 10 \cdot 95 + 18 \cdot 81 + 22 \cdot 70 + 28 \cdot 50 + 20 \cdot 73 + 12 \cdot 94 + 15 \cdot 90 + 25 \cdot 62 = 13721\)[/tex]
So, the sums are:
[tex]\[ \sum x = 185, \quad \sum y = 781, \quad \sum x^2 = 3711, \quad \sum y^2 = 62905, \quad \sum xy = 13721 \][/tex]
(b) Correlation coefficient:
To find the correlation coefficient [tex]\(r\)[/tex], we use the formula:
[tex]\[ r = \frac{n \sum xy - (\sum x)(\sum y)}{\sqrt{(n \sum x^2 - (\sum x)^2)(n \sum y^2 - (\sum y)^2)}} \][/tex]
Here, [tex]\(n\)[/tex] is the number of data points, which is 10.
Plugging in the values:
1. Calculate the numerator:
[tex]\[ n \sum xy - (\sum x)(\sum y) = 10 \cdot 13721 - 185 \cdot 781 = 137210 - 144485 = -7275 \][/tex]
2. Calculate the denominator:
[tex]\[ \sqrt{(n \sum x^2 - (\sum x)^2)(n \sum y^2 - (\sum y)^2)} = \sqrt{(10 \cdot 3711 - 185^2)(10 \cdot 62905 - 781^2)} \][/tex]
Breaking it down further:
[tex]\[ n \sum x^2 - (\sum x)^2 = 10 \cdot 3711 - 185^2 = 37110 - 34225 = 2885 \][/tex]
[tex]\[ n \sum y^2 - (\sum y)^2 = 10 \cdot 62905 - 781^2 = 629050 - 609961 = 19089 \][/tex]
So, the denominator is:
[tex]\[ \sqrt{2885 \cdot 19089} \approx \sqrt{55012765} \approx 7417.01 \][/tex]
3. Finally, calculate the correlation coefficient:
[tex]\[ r = \frac{-7275}{7417.01} \approx -0.980 \][/tex]
Thus, the relevant sums and the correlation coefficient are:
(a)
[tex]\[ \sum x = 185, \sum y = 781, \sum x^2 = 3711, \sum y^2 = 62905, \sum xy = 13721 \][/tex]
(b) The correlation coefficient:
[tex]\[ r \approx -0.980 \][/tex]
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