Compute​ r, the correlation​ coefficient, using the following data.
x
7
1
3
8
2
5
 
y
5
9
9
3
9
7


To calculate the correlation coefficient \( r \) between the two variables \( x \) and \( y \), we will use the formula for Pearson's correlation coefficient:

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

Where:
- \( n \) is the number of data points.
- \( \sum x \) is the sum of the \( x \)-values.
- \( \sum y \) is the sum of the \( y \)-values.
- \( \sum xy \) is the sum of the product of corresponding \( x \) and \( y \) 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.

### Step-by-step Calculation:

Given the data:

\[
x = [7, 1, 3, 8, 2, 5]
\]
\[
y = [5, 9, 9, 3, 9, 7]
\]

Let's calculate the necessary sums:

1. **Sum of \( x \):**
\[
\sum x = 7 + 1 + 3 + 8 + 2 + 5 = 26
\]

2. **Sum of \( y \):**
\[
\sum y = 5 + 9 + 9 + 3 + 9 + 7 = 42
\]

3. **Sum of \( x^2 \):**
\[
\sum x^2 = 7^2 + 1^2 + 3^2 + 8^2 + 2^2 + 5^2 = 49 + 1 + 9 + 64 + 4 + 25 = 152
\]

4. **Sum of \( y^2 \):**
\[
\sum y^2 = 5^2 + 9^2 + 9^2 + 3^2 + 9^2 + 7^2 = 25 + 81 + 81 + 9 + 81 + 49 = 326
\]

5. **Sum of \( xy \):**
\[
\sum xy = (7 \times 5) + (1 \times 9) + (3 \times 9) + (8 \times 3) + (2 \times 9) + (5 \times 7) = 35 + 9 + 27 + 24 + 18 + 35 = 148
\]

Now, we can plug these values into the formula:

\[
r = \frac{6(148) - (26)(42)}{\sqrt{[6(152) - (26)^2][6(326) - (42)^2]}}
\]

\[
r = \frac{888 - 1092}{\sqrt{[912 - 676][1956 - 1764]}}
\]

\[
r = \frac{-204}{\sqrt{[236][192]}}
\]

\[
r = \frac{-204}{\sqrt{45216}} = \frac{-204}{212.56}
\]

\[
r \approx -0.96
\]

### Final Answer:
The correlation coefficient \( r \) is approximately **-0.96**, which indicates a very strong negative linear relationship between \( x \) and \( y \).

---

Would you like a more detailed breakdown or have any questions about the process?

Here are some related questions you might find interesting:
1. How does the value of the correlation coefficient relate to the strength and direction of the relationship between two variables?
2. What would happen to the correlation coefficient if the data points were reversed (i.e., switching \( x \) and \( y \))?
3. How do you interpret a correlation coefficient of -1 or 1?
4. What are other ways to assess the relationship between two variables besides the correlation coefficient?
5. Can the correlation coefficient be used for non-linear relationships?

**Tip**: If you get a very high or very low correlation coefficient (close to 1 or -1), double-check the data for outliers or patterns that might be influencing the result too strongly!

Compute​ r, the correlation​ coefficient, using the following data. x: 7, 1, 3, 8, 2, 5; y: 5, 9, 9, 3, 9, 7.

Learn how to compute the Pearson correlation coefficient r using a given set of data. This step-by-step solution explains the use of the formula for calculating correlation, with data points for x and y provided. Perfect for high school and college-level students.

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