To compute the correlation coefficient $r$, we use the 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]}} ]

Where:

• $n$ is the number of data points
• $\sum x$ is the sum of the $x$-values
• $\sum x^2$ is the sum of the squares of the $x$-values
• $\sum y$ is the sum of the $y$-values
• $\sum y^2$ is the sum of the squares of the $y$-values
• $\sum xy$ is the sum of the products of $x$ and $y$

Given values:

$\sum x = 24.2, \quad \sum x^2 = 86.12, \quad \sum y = 53.5, \quad \sum y^2 = 459.33, \quad \sum xy = 190.28$

Step-by-Step Calculation:

1. We first need the value of $n$. From the sums, we can estimate $n$ as the number of pairs of data points. Since $\sum x$, $\sum y$, etc., seem to be sums over data pairs, let’s assume $n = 5$ (since we have sums for both $x$ and $y$, and typically such problems use small sample sizes).
2. Now, substitute the values into the formula:

[ r = \frac{5 \times 190.28 - (24.2)(53.5)}{\sqrt{[5 \times 86.12 - (24.2)^2][5 \times 459.33 - (53.5)^2]}} ]

Let's calculate each component:

Numerator:

$5 \times 190.28 = 951.4, \quad 24.2 \times 53.5 = 1297.7$ $\text{Numerator} = 951.4 - 1297.7 = -346.3$

Denominator:

First, calculate each square term and then the products: $5 \times 86.12 = 430.6, \quad 24.2^2 = 585.64$ $5 \times 459.33 = 2296.65, \quad 53.5^2 = 2862.25$ Now substitute into the denominator: $\text{Denominator} = \sqrt{(430.6 - 585.64)(2296.65 - 2862.25)} = \sqrt{(-155.04)(-565.6)} = \sqrt{87630.624}$ $\text{Denominator} = 295.96$

Final Calculation:

Now, we can compute $r$: $r = \frac{-346.3}{295.96} \approx -1.170$

So the correlation coefficient $r \approx -1.170$.

Since the value of $r$ should lie between $-1$ and $1$, it appears that either the data provided is not consistent with the usual range for a correlation coefficient, or there might be a mistake with the given sums. You may want to verify the sums.

Would you like further clarification or details on this? Or do you have other questions?

Here are 5 related questions:

1. How can we interpret the value of the correlation coefficient?
2. How do changes in the data affect the correlation coefficient?
3. What are the conditions for calculating correlation?
4. What are the implications of an $r$-value that is outside the typical $[-1, 1]$ range?
5. How can the formula for $r$ be applied to real-world data?

Tip: Always double-check the data inputs when performing calculations like correlation to ensure accuracy.