To determine which scatterplot shows a linear association, we need to look for a pattern in the data points that suggests a consistent relationship between \(x\) and \(y\). A linear association indicates that as \(x\) increases, \(y\) either tends to increase or decrease in a consistent manner.
Let’s analyze the \(x\) and \(y\) values from each of the data sets (A, B, C, and D):
A
- \(x: 5, 8, 2, 4, 6, 12, 10, 1, 3, 7\)
- \(y: 550, 800, 250, 500, 500, 400, 650, 150, 400, 750\)
B
- \(x: 5, 8, 2, 4, 6, 12, 10, 1, 3, 7\)
- \(y: 550, 800, 250, 300, 700, 1000, 650, 300, 100, 700\)
C
- \(x: 5, 8, 2, 4, 6, 12, 10, 1, 3, 7\)
- \(y: 800, 700, 350, 450, 500, 1000, 900, 250, 400, 750\)
D
- \(x: 5, 8, 2, 4, 6, 12, 10, 1, 3, 7\)
- \(y: 550, 800, 350, 450, 700, 1000, 900, 250, 400, 750\)
Observing the Data:
- Plot A shows a somewhat consistent relationship, but the values fluctuate significantly, causing a non-linear trend.
- Plot B shows substantial variation in \(y\) values that do not present a consistent upward or downward trend, indicating a non-linear association.
- Plot C shows varying \(y\) values with no clear trend either up or down.
- Plot D shows a clearer linear tendency, where \(y\) increases generally as \(x\) increases, especially towards the larger \(x\) values.
Conclusion:
Among the data sets, plot D appears to demonstrate the most linear association, as it reflects a trend where increases in \(x\) more consistently lead to increases in \(y\).
Thus, the scatterplot that shows a linear association is D.
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