https://file.aiquickdraw.com/mathbot/file/1725775532697uywus7pp.jpg

The equation in the image appears to represent the formula for Pearson's correlation coefficient, which is often used to measure the strength and direction of a linear relationship between two variables \( X \) and \( Y \). The formula given is:

\[
r_{XY} = \frac{95 \times (74) - (1 \times 13)}{\sqrt{(95 \times 1^2 - (1)^2) \times (95 \times 13^2 - 13^2)}}
\]

Let's break down and simplify this step-by-step.

### Step 1: Simplify the numerator

\[
\text{Numerator} = 95 \times 74 - (1 \times 13) = 7030 - 13 = 7017
\]

### Step 2: Simplify the terms inside the square root in the denominator

1. For the first part:
   
   \[
   95 \times 1^2 - 1^2 = 95 - 1 = 94
   \]

2. For the second part:

   \[
   95 \times 13^2 - 13^2 = 95 \times 169 - 169 = 16055 - 169 = 15886
   \]

### Step 3: Compute the denominator

\[
\text{Denominator} = \sqrt{94 \times 15886}
\]

Calculating \( 94 \times 15886 \):

\[
94 \times 15886 = 1493244
\]

Now, take the square root:

\[
\sqrt{1493244} \approx 1222.97
\]

### Step 4: Combine to find \( r_{XY} \)

\[
r_{XY} = \frac{7017}{1222.97} \approx 5.74
\]

Therefore, the value of \( r_{XY} \approx 5.74 \).

Do you have any questions or need further details?

### Related Questions
1. What is Pearson's correlation coefficient and how is it used?
2. How do changes in data affect the value of the correlation coefficient?
3. What are the limits and interpretations of correlation coefficients?
4. Can you calculate the correlation coefficient using different datasets?
5. How does a correlation coefficient differ from covariance?

### Tip
Always check if the values make logical sense, especially when calculating measures like correlation coefficients; values outside of -1 to 1 might indicate an error in calculations.

Calculate Pearson's correlation coefficient using the formula in the image.

This page provides a detailed step-by-step guide to calculating Pearson's correlation coefficient for two variables. Using real data and breaking down the formula, learn how to find the correlation value using statistical principles.

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