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To calculate the correlation coefficient for the given data, we will use the formula for the Pearson 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,
- \( x \) and \( y \) are the individual data points,
- \( \sum x \), \( \sum y \), \( \sum x^2 \), and \( \sum y^2 \) are the sums of \( x \), \( y \), squares of \( x \), and squares of \( y \), respectively,
- \( \sum xy \) is the sum of the products of corresponding \( x \) and \( y \) values.

Let's start by calculating the necessary values step-by-step from the provided table:

|  x  |  y  |  \( x \times y \)  |  \( x^2 \)  |  \( y^2 \)  |
|-----|-----|--------------------|-------------|-------------|
|  1  |  3  |  3                 |  1          |  9          |
|  3  |  8  |  24                |  9          |  64         |
|  5  | 11  |  55                |  25         |  121        |
|  7  | 12  |  84                |  49         |  144        |
|  9  | 16  |  144               |  81         |  256        |
| 11  | 21  |  231               |  121        |  441        |
| 13  | 28  |  364               |  169        |  784        |

Now, we compute the sums:

- \( \sum x = 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 \)
- \( \sum y = 3 + 8 + 11 + 12 + 16 + 21 + 28 = 99 \)
- \( \sum xy = 3 + 24 + 55 + 84 + 144 + 231 + 364 = 905 \)
- \( \sum x^2 = 1 + 9 + 25 + 49 + 81 + 121 + 169 = 455 \)
- \( \sum y^2 = 9 + 64 + 121 + 144 + 256 + 441 + 784 = 1819 \)

We can now substitute these values into the Pearson correlation formula:

\[
r = \frac{7 \times 905 - 49 \times 99}{\sqrt{(7 \times 455 - 49^2)(7 \times 1819 - 99^2)}}
\]

Let's compute this step-by-step: math

# Values from the table
n = 7
sum_x = 49
sum_y = 99
sum_xy = 905
sum_x_squared = 455
sum_y_squared = 1819

# Calculate the correlation coefficient using the formula
numerator = n * sum_xy - sum_x * sum_y
denominator = math.sqrt((n * sum_x_squared - sum_x ** 2) * (n * sum_y_squared - sum_y ** 2))

# Pearson correlation coefficient
r = numerator / denominator
rThe Pearson correlation coefficient for the given data is approximately **0.979**. This indicates a very strong positive linear relationship between the variables \( x \) and \( y \).

Would you like further details or clarification on any part of the calculation?

Here are 5 related questions that could help expand on this topic:
1. How can you interpret the value of the correlation coefficient in terms of the strength and direction of the relationship?
2. What other methods can be used to assess the linear relationship between two variables?
3. How does the correlation coefficient relate to the regression line in a scatter plot?
4. What are the limitations of using the Pearson correlation coefficient for data analysis?
5. How can you perform a hypothesis test for the correlation coefficient?

**Tip:** The correlation coefficient is sensitive to outliers. A small number of extreme values can heavily influence its value, so it's always useful to visualize the data (using a scatter plot) to check for such outliers before making conclusions.

Find the correlation coefficient for the data displayed in the table below: x: {1, 3, 5, 7, 9, 11, 13}, y: {3, 8, 11, 12, 16, 21, 28}.

Step-by-step solution to calculate the Pearson correlation coefficient for x: {1, 3, 5, 7, 9, 11, 13} and y: {3, 8, 11, 12, 16, 21, 28}. Includes key statistics concepts and formulas for understanding linear relationships between variables.

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