To determine the type of relationship between the variables [tex]\(X\)[/tex] and [tex]\(Y\)[/tex] from the given data in Table 1-2, we can follow these steps:
1. Identify the Data Points: Let's start by writing down the given pairs of [tex]\(X\)[/tex] and [tex]\(Y\)[/tex] values:
- Point A: [tex]\( (5, 18) \)[/tex]
- Point B: [tex]\( (12, 16) \)[/tex]
- Point C: [tex]\( (18, 14) \)[/tex]
- Point D: [tex]\( (30, 12) \)[/tex]
2. Calculate the Correlation Coefficient: The correlation coefficient [tex]\(r\)[/tex] quantifies the strength and direction of the linear relationship between two variables. The formula for the Pearson correlation coefficient [tex]\(r\)[/tex] is:
[tex]\[
r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}}
\][/tex]
where:
- [tex]\(X_i\)[/tex] and [tex]\(Y_i\)[/tex] are the individual data points.
- [tex]\(\bar{X}\)[/tex] and [tex]\(\bar{Y}\)[/tex] are the means of the [tex]\(X\)[/tex] and [tex]\(Y\)[/tex] data points.
From this calculation, we obtain the correlation coefficient which is approximately [tex]\(-0.987\)[/tex].
3. Interpret the Correlation Coefficient:
- If [tex]\(r > 0\)[/tex], it indicates a direct (positive) relationship between [tex]\(X\)[/tex] and [tex]\(Y\)[/tex]; as [tex]\(X\)[/tex] increases, [tex]\(Y\)[/tex] increases.
- If [tex]\(r = 0\)[/tex], it indicates that [tex]\(X\)[/tex] and [tex]\(Y\)[/tex] are independent; there is no linear relationship between them.
- If [tex]\(r < 0\)[/tex], it indicates an inverse (negative) relationship between [tex]\(X\)[/tex] and [tex]\(Y\)[/tex]; as [tex]\(X\)[/tex] increases, [tex]\(Y\)[/tex] decreases.
- If [tex]\(r\)[/tex] is very close to 0, it suggests no linear relationship between [tex]\(X\)[/tex] and [tex]\(Y\)[/tex] within the given data.
The calculated correlation coefficient is [tex]\(-0.987\)[/tex], which is less than 0 and indicates a strong inverse relationship.
4. Determine the Type of Relationship: Based on the value of the correlation coefficient [tex]\(-0.987\)[/tex], we can conclude that there is a strong inverse relationship between the variables [tex]\(X\)[/tex] and [tex]\(Y\)[/tex].
Therefore, the type of relationship that exists between variables [tex]\(X\)[/tex] and [tex]\(Y\)[/tex] is:
d. inverse
