use Pearsons correlation coefficient to solve for r . n=40, Σxy= 67102.6,Σx= 9662, Σy= 7456,Σ x^2=-14609.9 and Σy^2= 31580.6

To solve for Pearson's correlation coefficient (\(r\)) using the given data, we can use the following formula:

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

Given:
- \( n = 40 \)
- \( \Sigma{xy} = 67102.6 \)
- \( \Sigma{x} = 9662 \)
- \( \Sigma{y} = 7456 \)
- \( \Sigma{x^2} = -14609.9 \) (this value seems odd; typically \(\Sigma{x^2}\) is positive)
- \( \Sigma{y^2} = 31580.6 \)

First, we need to compute the numerator and the denominator separately.

### Numerator
\[ n\Sigma{xy} - \Sigma{x}\Sigma{y} \]

\[ 40 \times 67102.6 - 9662 \times 7456 \]

### Denominator
\[ \sqrt{(n\Sigma{x^2} - (\Sigma{x})^2)(n\Sigma{y^2} - (\Sigma{y})^2)} \]

\[ \sqrt{(40 \times \Sigma{x^2} - (\Sigma{x})^2)(40 \times \Sigma{y^2} - (\Sigma{y})^2)} \]

Now, let's compute the numerator:

\[ 40 \times 67102.6 = 2684104 \]
\[ 9662 \times 7456 = 72028352 \]

\[ \text{Numerator} = 2684104 - 72028352 = -69344248 \]

### Denominator
\[ \text{First term} = 40 \times (-14609.9) - 9662^2 \]

\[ 40 \times (-14609.9) = -584396 \]
\[ 9662^2 = 93354844 \]

\[ 40 \times (-14609.9) - 9662^2 = -584396 - 93354844 = -93939240 \]

\[ \text{Second term} = 40 \times 31580.6 - 7456^2 \]

\[ 40 \times 31580.6 = 1263224 \]
\[ 7456^2 = 55636416 \]

\[ 40 \times 31580.6 - 7456^2 = 1263224 - 55636416 = -54373192 \]

\[ \text{Denominator} = \sqrt{(-93939240)(-54373192)} \]

\[ \sqrt{5110021251423680} \approx 71484038.98 \]

### Pearson's correlation coefficient
\[ r = \frac{-69344248}{71484038.98} \approx -0.9701 \]

Therefore, the Pearson's correlation coefficient \(r\) is approximately \(-0.9701\).

Would you like further details or have any questions?

Here are 8 related questions to expand your understanding:
1. What is the significance of the sign of the Pearson correlation coefficient?
2. How can outliers affect the Pearson correlation coefficient?
3. What is the range of values that Pearson's correlation coefficient can take?
4. How is Pearson's correlation coefficient different from Spearman's rank correlation coefficient?
5. What assumptions must be met to use Pearson's correlation coefficient?
6. How can you test the statistical significance of Pearson's correlation coefficient?
7. What are some potential limitations of using Pearson's correlation coefficient?
8. Can Pearson's correlation coefficient be used with categorical data?

**Tip:** Always visualize your data with a scatter plot before calculating Pearson's correlation coefficient to understand the relationship better.

Learn how to calculate Pearson's correlation coefficient (r) with the provided data of n=40, Σxy=67102.6, Σx=9662, Σy=7456, Σx^2=-14609.9, and Σy^2=31580.6. Understand the steps involved and the significance of the result. Suitable for advanced college-level students studying statistics.

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