### Part (i): Scatter Diagram of the Data
To create a scatter diagram, we need to plot the years of experience (X) on the horizontal axis and the number of rejects (Y) on the vertical axis.
Scatter Plot:
- Horizontal Axis (X-axis): Years of experience
- Vertical Axis (Y-axis): Number of rejects
Given data:
| Years of Experience (X) | 4 | 5 | 7 | 9 | 10 | 11 | 12 | 14 |
|-------------------------|----|----|----|----|----|----|----|----|
| Number of Rejects (Y) | 21 | 22 | 15 | 18 | 14 | 14 | 11 | 13 |
To draw the scatter diagram, plot each pair [tex]\((X, Y)\)[/tex] as points on a graph:
- Point (4, 21)
- Point (5, 22)
- Point (7, 15)
- Point (9, 18)
- Point (10, 14)
- Point (11, 14)
- Point (12, 11)
- Point (14, 13)
This scatter plot visually shows the relationship between years of experience and the number of rejects.
### Part (ii): Calculate the Product Moment Correlation Coefficient
The formula for the Pearson product-moment correlation coefficient [tex]\( r \)[/tex] is:
[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
- [tex]\(X\)[/tex] and [tex]\(Y\)[/tex] are the variables
Step-by-Step Calculation:
1. Calculate the sums and sums of squares:
Given data:
[tex]\[
\begin{align*}
\sum{X} &= 4 + 5 + 7 + 9 + 10 + 11 + 12 + 14 = 72 \\
\sum{Y} &= 21 + 22 + 15 + 18 + 14 + 14 + 11 + 13 = 128 \\
\sum{XY} &= (4 \cdot 21) + (5 \cdot 22) + (7 \cdot 15) + (9 \cdot 18) + (10 \cdot 14) + (11 \cdot 14) + (12 \cdot 11) + (14 \cdot 13) \\
&= 84 + 110 + 105 + 162 + 140 + 154 + 132 + 182 = 1069 \\
\sum{X^2} &= 4^2 + 5^2 + 7^2 + 9^2 + 10^2 + 11^2 + 12^2 + 14^2 \\
&= 16 + 25 + 49 + 81 + 100 + 121 + 144 + 196 = 732 \\
\sum{Y^2} &= 21^2 + 22^2 + 15^2 + 18^2 + 14^2 + 14^2 + 11^2 + 13^2 \\
&= 441 + 484 + 225 + 324 + 196 + 196 + 121 + 169 = 2156
\end{align*}
\][/tex]
2. Substitute these values into the correlation coefficient formula:
[tex]\[
\begin{align*}
r &= \frac{8 \cdot 1069 - 72 \cdot 128}{\sqrt{[8 \cdot 732 - 72^2][8 \cdot 2156 - 128^2]}} \\
&= \frac{8552 - 9216}{\sqrt{[5856 - 5184][17248 - 16384]}} \\
&= \frac{-664}{\sqrt{672 \cdot 864}} \\
&= \frac{-664}{\sqrt{580608}} \\
&= \frac{-664}{761.95} \\
&= -0.871
\end{align*}
\][/tex]
Conclusion:
The product moment correlation coefficient [tex]\( r \)[/tex] is approximately [tex]\(-0.871\)[/tex]. This indicates a strong negative correlation between the years of experience and the number of rejects; as years of experience increase, the number of rejects tends to decrease.
