Dive into the world of knowledge and get your queries resolved at IDNLearn.com. Whether your question is simple or complex, our community is here to provide detailed and trustworthy answers quickly and effectively.
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]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com has the solutions you’re looking for. Thanks for visiting, and see you next time for more reliable information.
