Based on the data shown below, calculate the correlation coefficient (rounded to three decimal places)
| x | y |
|----|-------|
| 4 | 36.2 |
| 5 | 35.78 |
| 6 | 33.56 |
| 7 | 31.34 |
| 8 | 27.32 |
| 9 | 24.5 |
| 10 | 22.18 |
| 11 | 20.76 |
| 12 | 16.34 |
| 13 | 14.82 |
| 14 | 13.7 |


To calculate the **correlation coefficient (\(r\))**, we 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:
- \(x\) and \(y\) are the variables.
- \(n\) is the number of data points.
- \(\sum{x}\), \(\sum{y}\), \(\sum{xy}\), \(\sum{x^2}\), and \(\sum{y^2}\) are calculated based on the data.

---

### **Step 1: Compute Necessary Sums**

Let’s calculate:
1. \(\sum{x}\)
2. \(\sum{y}\)
3. \(\sum{xy}\)
4. \(\sum{x^2}\)
5. \(\sum{y^2}\)

| \(x\) | \(y\)    | \(x^2\) | \(y^2\)     | \(xy\)       |
|------|----------|---------|-------------|--------------|
| 4    | 36.2     | 16      | 1310.44     | 144.8        |
| 5    | 35.78    | 25      | 1280.2084   | 178.9        |
| 6    | 33.56    | 36      | 1126.8736   | 201.36       |
| 7    | 31.34    | 49      | 982.5956    | 219.38       |
| 8    | 27.32    | 64      | 746.1424    | 218.56       |
| 9    | 24.5     | 81      | 600.25      | 220.5        |
| 10   | 22.18    | 100     | 492.7524    | 221.8        |
| 11   | 20.76    | 121     | 431.5776    | 228.36       |
| 12   | 16.34    | 144     | 267.6356    | 196.08       |
| 13   | 14.82    | 169     | 219.5024    | 192.66       |
| 14   | 13.7     | 196     | 187.69      | 191.8        |

Now sum each column:

\[
\sum{x} = 99, \quad \sum{y} = 276.2, \quad \sum{x^2} = 1001, \quad \sum{y^2} = 7645.268, \quad \sum{xy} = 2214.16
\]

---

### **Step 2: Plug Values into the Correlation Formula**

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

Substitute:
- \(n = 11\),
- \(\sum{xy} = 2214.16\),
- \(\sum{x} = 99\),
- \(\sum{y} = 276.2\),
- \(\sum{x^2} = 1001\),
- \(\sum{y^2} = 7645.268\).

#### Numerator:
\[
n\sum{(xy)} - \sum{x}\sum{y} = 11(2214.16) - (99)(276.2) = 24355.76 - 27343.8 = -988.04
\]

#### Denominator:
1. First term: \(n\sum{x^2} - (\sum{x})^2\):
\[
11(1001) - (99)^2 = 11011 - 9801 = 1210
\]

2. Second term: \(n\sum{y^2} - (\sum{y})^2\):
\[
11(7645.268) - (276.2)^2 = 84097.948 - 76280.44 = 7817.508
\]

3. Denominator product:
\[
\sqrt{1210 \cdot 7817.508} = \sqrt{9463160.68} = 3076.05
\]

#### Correlation Coefficient:
\[
r = \frac{-988.04}{3076.05} = -0.321
\]

---

### **Final Answer**
The correlation coefficient is:
\[
\boxed{-0.321}
\]

This indicates a weak negative correlation between \(x\) and \(y\). Let me know if you'd like further clarification!

---

### Related Questions
1. What does the sign of the correlation coefficient indicate about the relationship between two variables?  
2. How would outliers affect the value of the correlation coefficient?  
3. What are some real-world examples of weak negative correlations?  
4. Can the correlation coefficient determine causation? Why or why not?  
5. How can we test the significance of a correlation coefficient?  

**Tip**: Always visualize data (e.g., scatter plots) when interpreting correlation coefficients to spot patterns or outliers!

Based on the data shown below, calculate the correlation coefficient (rounded to three decimal places) | x | y | |----|-------| | 4 | 36.2 | | 5 | 35.78 | | 6 | 33.56 | | 7 | 31.34 | | 8 | 27.32 | | 9 | 24.5 | | 10 | 22.18 | | 11 | 20.76 | | 12 | 16.34 | | 13 | 14.82 | | 14 | 13.7 |

Learn how to calculate the correlation coefficient for a given data set. Using the provided values for x and y, this detailed solution walks through all steps to compute Pearson's correlation coefficient, showing the necessary sums and final result of -0.321, indicating a weak negative correlation. Suitable for students in Grades 9-12.

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