Let's break down the solution step-by-step for each part of the question:
### Part (a): Sketch a Scatter Plot
To create a scatter plot, we plot each pair [tex]\((X, Y)\)[/tex] on a coordinate system where [tex]\(X\)[/tex] values are on the horizontal axis and [tex]\(Y\)[/tex] values are on the vertical axis. Here are the pairs to plot:
- [tex]\((7, 6)\)[/tex]
- [tex]\((9, 6)\)[/tex]
- [tex]\((6, 3)\)[/tex]
- [tex]\((12, 5)\)[/tex]
- [tex]\((9, 6)\)[/tex]
- [tex]\((5, 4)\)[/tex]
Though I cannot directly draw a scatter plot here, you can visualize the points on a graph:
1. Draw your horizontal axis (X-axis) and label it with values from 5 to 12.
2. Draw your vertical axis (Y-axis) and label it with values from 3 to 6.
3. Plot the points on the graph accordingly.
### Part (b): Describe the Correlation
Looking at the scatter plot:
- Direction: You may observe a positive correlation, meaning as [tex]\(X\)[/tex] increases, [tex]\(Y\)[/tex] also tends to increase. However, it's not perfectly linear.
- Strength: The points are somewhat scattered but there is a noticeable pattern that suggests moderate positive linear association. The correlation is not extremely strong but it's not weak either.
### Part (c): Compute the Pearson Correlation
To calculate the Pearson correlation coefficient [tex]\( r \)[/tex], use the formula:
[tex]\[ r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n\sum X^2 - (\sum X)^2][n\sum Y^2 - (\sum Y)^2]}} \][/tex]
Where [tex]\( n \)[/tex] is the number of data points (here [tex]\( n = 6 \)[/tex]).
Given the values:
[tex]\[ X = [7, 9, 6, 12, 9, 5] \][/tex]
[tex]\[ Y = [6, 6, 3, 5, 6, 4] \][/tex]
We'll sum the necessary components:
- [tex]\( \sum X = 7 + 9 + 6 + 12 + 9 + 5 = 48 \)[/tex]
- [tex]\( \sum Y = 6 + 6 + 3 + 5 + 6 + 4 = 30 \)[/tex]
- [tex]\( \sum XY = (76) + (96) + (63) + (125) + (96) + (54) = 42 + 54 + 18 + 60 + 54 + 20 = 248 \)[/tex]
- [tex]\( \sum X^2 = 7^2 + 9^2 + 6^2 + 12^2 + 9^2 + 5^2 = 49 + 81 + 36 + 144 + 81 + 25 = 416 \)[/tex]
- [tex]\( \sum Y^2 = 6^2 + 6^2 + 3^2 + 5^2 + 6^2 + 4^2 = 36 + 36 + 9 + 25 + 36 + 16 = 158 \)[/tex]
Now plug these values into the Pearson correlation formula:
[tex]\[ r = \frac{6(248) - (48)(30)}{\sqrt{[6(416) - (48)^2][6(158) - (30)^2]}} \][/tex]
[tex]\[ r = \frac{1488 - 1440}{\sqrt{[2496 - 2304][948 - 900]}} \][/tex]
[tex]\[ r = \frac{48}{\sqrt{192 \cdot 48}} \][/tex]
[tex]\[ r = \frac{48}{\sqrt{9216}} \][/tex]
[tex]\[ r = \frac{48}{96} \][/tex]
[tex]\[ r = 0.5 \][/tex]
So, the Pearson correlation coefficient is approximately [tex]\(0.5\)[/tex], which indicates a moderate positive correlation between [tex]\(X\)[/tex] and [tex]\(Y\)[/tex].
