To analyze the association between the given \(x\) and \(y\) values, we can look at the relationship between the two sets of data:
Here's the data from the table:
\[ \begin{array}{|c|c|} \hline x & y \ \hline 25 & 375 \ 30 & 225 \ 80 & 120 \ 150 & 75 \ 250 & 50 \ 350 & 100 \ 425 & 175 \ 450 & 275 \ 500 & 400 \ \hline \end{array} \]
Steps to Determine the Association:
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Plot the Points: We would plot the points on a graph where the x-values are on the horizontal axis and the y-values are on the vertical axis.
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Visual Inspection: As we plot the points, we observe how they trend with respect to each other.
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Analyze the Trend:
- For smaller values of \(x\) (from 25 to about 250), \(y\) decreases significantly from 375 to 50.
- As we increase \(x\) from 250 to 500, \(y\) starts to increase again but does not return to higher values than it started.
Type of Association:
- Negative Linear Association: The initial trend shows that as \(x\) increases, \(y\) initially decreases, suggesting a negative relationship. Although it doesn't remain consistently negative throughout (as it begins to rise again towards the extreme values), the overall behavior suggests that there is a dominating negative trend in the early segment of the scatterplot.
Given this analysis, the type of association that best characterizes the relationship between \(x\) and \(y\) is:
Negative Linear Association.