Calculate the correlation coefficient for the following data. Round your answer to the nearest thousandth.

xy

22

40.340.3

55

46.246.2

66

65.665.6

99

56.256.2

1313

89.389.3

1515

72.572.5

1818

98.698.6

To calculate the Pearson correlation coefficient for the given data, we will use the formula:

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

Let's first summarize the given data and then calculate each component of the formula.

Given dataTo calculate the Pearson correlation coefficient for the given data, we will use the formula:

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

Let's first summarize the given data and then calculate each component of the formula.

Given 

| \(x\) | \(y\)  |
|:----:|:-----:|
|  2   |  2    |
|  4   |  5    |
|  6   |  6    |
|  9   |  9    |
|  13  |  13   |
|  15  |  15   |
|  18  |  18   |

1. Calculate the necessary sums:

\[
\begin{align*}
\sum x & = 2 + 4 + 6 + 9 + 13 + 15 + 18 = 67 \\
\sum y & = 2 + 5 + 6 + 9 + 13 + 15 + 18 = 68 \\
\sum (xy) & = (2 \cdot 2) + (4 \cdot 5) + (6 \cdot 6) + (9 \cdot 9) + (13 \cdot 13) + (15 \cdot 15) + (18 \cdot 18) \\
         & = 4 + 20 + 36 + 81 + 169 + 225 + 324 = 859 \\
\sum (x^2) & = 2^2 + 4^2 + 6^2 + 9^2 + 13^2 + 15^2 + 18^2 \\
          & = 4 + 16 + 36 + 81 + 169 + 225 + 324 = 855 \\
\sum (y^2) & = 2^2 + 5^2 + 6^2 + 9^2 + 13^2 + 15^2 + 18^2 \\
          & = 4 + 25 + 36 + 81 + 169 + 225 + 324 = 864
\end{align*}
\]

2. Plug these values into the formula:

\[
r = \frac{7 \cdot 859 - 67 \cdot 68}{\sqrt{[7 \cdot 855 - 67^2][7 \cdot 864 - 68^2]}}
\]

3. Simplify the numerator and denominator:

\[
\begin{align*}
\text{Numerator} & : 7 \cdot 859 - 67 \cdot 68 = 6013 - 4556 = 1457 \\
\text{Denominator} & : \sqrt{[7 \cdot 855 - 67^2][7 \cdot 864 - 68^2]} \\
                  & = \sqrt{[5985 - 4489][6048 - 4624]} \\
                  & = \sqrt{1496 \cdot 1424} \\
                  & = \sqrt{2130304} \\
                  & = 1460.41
\end{align*}
\]

4. Calculate the correlation coefficient:

\[
r = \frac{1457}{1460.41} \approx 0.998
\]

Thus, the Pearson correlation coefficient for the given data, rounded to the nearest thousandth, is \( r \approx 0.998 \).

Would you like more details or have any questions?

Here are 5 questions you might ask:
1. What do the calculated sums represent in the Pearson correlation formula?
2. How do you interpret the value of the Pearson correlation coefficient?
3. What steps are involved in simplifying the correlation coefficient formula?
4. Can you explain the impact of outliers on the Pearson correlation coefficient?
5. What are some alternative methods for measuring correlation?

**Tip:** When calculating the Pearson correlation coefficient, it is important to ensure that both variables are on an interval or ratio scale and that the relationship between the variables is linear.

Learn how to calculate the Pearson correlation coefficient for a set of data points using the formula. This problem involves calculating sums of products and squares, essential for understanding statistical relationships. Suitable for college-level students and beyond.

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