91Ó°ÊÓ

A scatter diagram is a useful tool to visualize the relationship between two variables. In this exercise, you need to plot the pairs \( (X, Y) \) on a Cartesian coordinate system. This helps in initially assessing any potential correlation.
For example, using values \( X = (0, 1.2, 1.8, 2.3, 3.5, 4.1) \) and \( Y = (2, 6.4, 7, 7.3, 4.5, 2.4) \), plot the points accordingly:
* \( (0, 2) \)
* \( (1.2, 6.4) \)
* \( (1.8, 7) \)
* \( (2.3, 7.3) \)
* \( (3.5, 4.5) \)
* \( (4.1, 2.4) \)
This scatter plot gives a preliminary visual indication of the relationship between \( X \) and \( Y \). Whether the points fall close to a line can suggest whether the variables are correlated.

The rank correlation coefficient, specifically Spearman's \( r_s \), measures the strength and direction of association between two ranked variables.
To compute this, follow these steps:
1. Rank both sets of data. For the values \( X = (0, 1.2, 1.8, 2.3, 3.5, 4.1) \) and \( Y = (2, 6.4, 7, 7.3, 4.5, 2.4) \), assign ranks. If there are ties, assign the average rank.
2. Calculate the differences between the pairs of ranks, denoted as \( d_i \).
3. Square these differences and sum them, \( \frac{6 \times \text{sum of squared differences}}{n(n^2-1)} \).
4. Plug this into the formula: \[ r_s = 1 - \frac{6 \times \text{sum of} \ d_i^2}{n(n^2-1)} \]
This provides a value between -1 and 1, indicating the degree of rank correlation:
* \( r_s = 1 \): perfectly positive correlation
* \( r_s = -1 \): perfectly negative correlation
* \( r_s = 0 \): no correlation.

Statistical significance helps determine if the observed correlation is genuine or occurred by chance.
At a significance level \( \alpha = 0.05 \), we need to compare the calculated Spearman's \( r_s \) to a critical value from the Spearman's rank correlation table. This step determines whether we reject or accept the null hypothesis that the variables are uncorrelated.
For \ n = 6 \, the critical value at \ \alpha = 0.05 \ is typically found in a standard table. If \( r_s \) exceeds this critical value, you conclude that there is a statistically significant association at the \ 0.05 \ level, indicating that \ X \ and \ Y \ are significantly related. If not, we do not have enough evidence to claim such an association.
This process provides a rigorous statistical foundation to support or refute the observed correlation.