Find the correlation coefficient, r, of the data described below.
Vijay thinks that text messaging is causing him to talk less on the phone. For one month, he examined his text message and call logs with his closest friends.
For each friend, Vijay checked the number of text messages he sent to that friend, x, and the number of minutes they spoke on the phone, y.
Text messages sent	Minutes on the phone
4	33
7	14
11	9
32	50
54	3
Round your answer to the nearest thousandth.

To find the correlation coefficient, \( r \), we can use the formula:

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

where:
- \( n \) is the number of data points,
- \( x \) and \( y \) are the variables (text messages and minutes on the phone, respectively),
- \( \sum (xy) \) is the sum of the product of each pair of \( x \) and \( y \),
- \( \sum x \) is the sum of the \( x \) values,
- \( \sum y \) is the sum of the \( y \) values,
- \( \sum (x^2) \) is the sum of the squares of the \( x \) values,
- \( \sum (y^2) \) is the sum of the squares of the \( y \) values.

Let's calculate these values step by step:

Given dataTo find the correlation coefficient, \( r \), we can use the formula:

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

where:
- \( n \) is the number of data points,
- \( x \) and \( y \) are the variables (text messages and minutes on the phone, respectively),
- \( \sum (xy) \) is the sum of the product of each pair of \( x \) and \( y \),
- \( \sum x \) is the sum of the \( x \) values,
- \( \sum y \) is the sum of the \( y \) values,
- \( \sum (x^2) \) is the sum of the squares of the \( x \) values,
- \( \sum (y^2) \) is the sum of the squares of the \( y \) values.

Let's calculate these values step by step:

Given 
\[
\begin{align*}
x &: 4, 7, 11, 32, 54 \\
y &: 33, 14, 9, 50, 3
\end{align*}
\]

Calculations:
\[
\begin{align*}
\sum x &= 4 + 7 + 11 + 32 + 54, \\
\sum y &= 33 + 14 + 9 + 50 + 3, \\
\sum (xy) &= (4 \cdot 33) + (7 \cdot 14) + (11 \cdot 9) + (32 \cdot 50) + (54 \cdot 3), \\
\sum (x^2) &= 4^2 + 7^2 + 11^2 + 32^2 + 54^2, \\
\sum (y^2) &= 33^2 + 14^2 + 9^2 + 50^2 + 3^2.
\end{align*}
\]

Let’s compute these values and find \( r \).The correlation coefficient, \( r \), of the data is approximately \(-0.161\) when rounded to the nearest thousandth. This indicates a very weak negative correlation between the number of text messages sent and the minutes spent on the phone.

Would you like more details, or do you have any questions?

Here are some related questions that might interest you:

1. How do you interpret the strength and direction of a correlation coefficient?
2. What are the limitations of using the correlation coefficient in data analysis?
3. How can we use a scatter plot to visually assess the correlation between two variables?
4. What is the difference between correlation and causation?
5. How does outlier data affect the calculation of the correlation coefficient?

**Tip:** A correlation coefficient near zero suggests a weak relationship between the variables, but it doesn't necessarily imply there's no relationship at all—it could be nonlinear!

Find the correlation coefficient, r, of the data described below. Vijay thinks that text messaging is causing him to talk less on the phone. For one month, he examined his text message and call logs with his closest friends. For each friend, Vijay checked the number of text messages he sent to that friend, x, and the number of minutes they spoke on the phone, y. Text messages sent: 4, 7, 11, 32, 54 Minutes on the phone: 33, 14, 9, 50, 3. Round your answer to the nearest thousandth.

Find the correlation coefficient, r, between the number of text messages sent and minutes on the phone. This problem involves statistics and linear relationships. It provides step-by-step calculations of the correlation coefficient formula, using the given Text messages (4, 7, 11, 32, 54) and Minutes on the phone (33, 14, 9, 50, 3). Suitable for students in Grades 10-12.

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