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.