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solve this with each step with explanation

To find the coefficient of correlation for the given data, we can use Pearson's correlation coefficient formula. Let's go through each step in detail.

### Given Data
The data given in the table is:

| X  | Y  |
|----|----|
| 10 | 18 |
| 14 | 12 |
| 18 | 24 |
| 22 | 6  |
| 26 | 30 |
| 30 | 36 |

### Pearson's Correlation Coefficient Formula
The Pearson correlation coefficient \( r \) is calculated using the formula:

\[
r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n\sum X^2 - (\sum X)^2][n\sum Y^2 - (\sum Y)^2]}}
\]

Where:
- \( n \) is the number of data points.
- \( \sum XY \) is the sum of the product of each pair of \( X \) and \( Y \).
- \( \sum X \) and \( \sum Y \) are the sums of \( X \) and \( Y \) values respectively.
- \( \sum X^2 \) and \( \sum Y^2 \) are the sums of the squares of \( X \) and \( Y \) values respectively.

Let's calculate each part step by step.

### Step 1: Calculate the Sums

1. **Calculate \(\sum X\) and \(\sum Y\):**

   \[
   \sum X = 10 + 14 + 18 + 22 + 26 + 30 = 120
   \]

   \[
   \sum Y = 18 + 12 + 24 + 6 + 30 + 36 = 126
   \]

2. **Calculate \(\sum XY\):**

   \[
   \sum XY = (10 \times 18) + (14 \times 12) + (18 \times 24) + (22 \times 6) + (26 \times 30) + (30 \times 36)
   \]

   \[
   = 180 + 168 + 432 + 132 + 780 + 1080 = 2772
   \]

3. **Calculate \(\sum X^2\) and \(\sum Y^2\):**

   \[
   \sum X^2 = 10^2 + 14^2 + 18^2 + 22^2 + 26^2 + 30^2
   \]

   \[
   = 100 + 196 + 324 + 484 + 676 + 900 = 2680
   \]

   \[
   \sum Y^2 = 18^2 + 12^2 + 24^2 + 6^2 + 30^2 + 36^2
   \]

   \[
   = 324 + 144 + 576 + 36 + 900 + 1296 = 3276
   \]

### Step 2: Calculate the Correlation Coefficient

Substitute the calculated sums into the Pearson correlation formula:

\[
r = \frac{6 \times 2772 - 120 \times 126}{\sqrt{(6 \times 2680 - 120^2)(6 \times 3276 - 126^2)}}
\]

\[
r = \frac{16632 - 15120}{\sqrt{(16080 - 14400)(19656 - 15876)}}
\]

\[
r = \frac{1512}{\sqrt{1680 \times 3780}}
\]

\[
r = \frac{1512}{\sqrt{6350400}}
\]

\[
r = \frac{1512}{2520}
\]

\[
r \approx 0.6
\]

### Conclusion

The Pearson correlation coefficient \( r \approx 0.6 \) indicates a moderate positive correlation between \( X \) and \( Y \).

---

Would you like any further explanation or details on any step? Here are some additional questions you might have:

1. What does the correlation coefficient tell us about the relationship between \( X \) and \( Y \)?
2. How can we interpret the result in terms of statistical significance?
3. What are the limitations of using Pearson's correlation coefficient?
4. How would the correlation change if we add more data points?
5. Can you explain how to calculate a correlation coefficient using a different method?

**Tip:** When analyzing correlations, always visualize your data with a scatter plot to see if the linear correlation is a suitable model.

Learn how to calculate Pearson's correlation coefficient step-by-step with detailed explanations and numerical examples. Understand the statistical significance of correlation coefficients and their interpretation in real-world data analysis scenarios.

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