https://file.aiquickdraw.com/mathbot/file/1726466301925kcj5atjh.jpg

To find the correlation coefficient between \( X \) and \( Y \), we can use the formula for the Pearson correlation coefficient \( r \):

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

However, the simplified form when we have the mean and standard deviations given is:

\[
r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{(n-1) \cdot s_X \cdot s_Y}
\]

Given data from the problem:
- Number of pairs (\( n \)) = 15
- Arithmetic mean of \( X \) (\( \bar{X} \)) = 25
- Arithmetic mean of \( Y \) (\( \bar{Y} \)) = 18
- Standard deviation of \( X \) (\( s_X \)) = 3
- Standard deviation of \( Y \) (\( s_Y \)) = 3
- Sum of the products of \( X \) and \( Y \) (\( \sum XY \)) = 6870

To find \( r \), we need \( \sum X \) and \( \sum Y \). Since \( \bar{X} \) and \( \bar{Y} \) are given:
\[
\sum X = n \times \bar{X} = 15 \times 25 = 375
\]
\[
\sum Y = n \times \bar{Y} = 15 \times 18 = 270
\]

Now, we can use the correlation formula:
\[
r = \frac{n \sum XY - (\sum X)(\sum Y)}{\sqrt{[n \sum X^2 - (\sum X)^2][n \sum Y^2 - (\sum Y)^2]}}
\]

But for this case, we can use the simplified form since the standard deviations and means are given:
\[
r = \frac{\sum XY - n \bar{X} \bar{Y}}{(n-1) \cdot s_X \cdot s_Y}
\]

Calculating \( \sum XY - n \bar{X} \bar{Y} \):
\[
\sum XY - n \bar{X} \bar{Y} = 6870 - 15 \times 25 \times 18 = 6870 - 6750 = 120
\]

Now, substituting the values:
\[
r = \frac{120}{(15-1) \cdot 3 \cdot 3} = \frac{120}{14 \times 9} = \frac{120}{126} = \frac{20}{21} \approx 0.952
\]

So, the correlation coefficient \( r \) is approximately **0.952**.

Would you like more details on this calculation, or do you have any other questions?

Here are 5 related questions to further expand your understanding:
1. What does a correlation coefficient of 0.952 indicate about the relationship between \( X \) and \( Y \)?
2. How would the correlation coefficient change if the standard deviations of \( X \) and \( Y \) were different?
3. What are the limitations of using the Pearson correlation coefficient?
4. How would you interpret a negative correlation coefficient?
5. How can we calculate the covariance from the given data?

**Tip:** Always check the correlation coefficient's value to understand the strength and direction of the relationship between two variables.

Find the correlation coefficient between X and Y given the following information: Number of pairs of X and Y series = 15, Arithmetic mean of X = 25, Arithmetic mean of Y = 18, Standard deviation of X = 3, Standard deviation of Y = 3, Sum of products of X and Y (ΣXY) = 6870.

Learn how to find the correlation coefficient between two variables, X and Y, given their means, standard deviations, and sum of products. This step-by-step solution uses the Pearson correlation coefficient formula, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation