Connect with a community of experts and enthusiasts on IDNLearn.com. Discover prompt and accurate answers from our experts, ensuring you get the information you need quickly.

The table below shows the number of hours students surf online and their test scores.

\begin{tabular}{|c|c|}
\hline
Surfing Online (hrs) & Test Scores \\
\hline
20 & 79 \\
\hline
15 & 87 \\
\hline
10 & 95 \\
\hline
18 & 81 \\
\hline
22 & 70 \\
\hline
28 & 50 \\
\hline
20 & 73 \\
\hline
12 & 94 \\
\hline
15 & 90 \\
\hline
25 & 62 \\
\hline
\end{tabular}

Find the relevant sums:
(a) [tex]\(\sum x = 185\)[/tex], [tex]\(\sum y = 781\)[/tex], [tex]\(\sum x^2 = 3711\)[/tex], [tex]\(\sum y^2 = 62910\)[/tex], [tex]\(\sum xy = 13721\)[/tex].

(b) The correlation coefficient using the formula:

[tex]\[ r = \frac{n \sum xy - \sum x \sum y}{\sqrt{n \sum x^2 - (\sum x)^2} \sqrt{n \sum y^2 - (\sum y)^2}} \][/tex]


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]