To determine which scatterplot shows a linear association, we would typically look for a pattern in the points plotted on a graph where \(x\) values are on the horizontal axis and \(y\) values are on the vertical axis. A linear association means that the points will closely follow a straight line (either increasing or decreasing).
Let's analyze the provided data sets one by one, based on their \(x\) and \(y\) values.
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First Dataset:
- \(x\): 5, 8, 2, 4, 6, 12, 10, 1, 3, 7
- \(y\): 800, 700, 350, 450, 500, 1000, 900, 250, 400, 750
This dataset appears to fluctuate without a clear upward or downward trend. Thus, it does not exhibit a linear association.
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Second Dataset:
- \(x\): 5, 8, 2, 4, 6, 12, 10, 1, 3, 7
- \(y\): 550, 800, 350, 450, 700, 1000, 900, 250, 400, 750
Similar to the first dataset, there is considerable fluctuation in the \(y\) values without a clear linear trend.
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Third Dataset:
- \(x\): 5, 8, 2, 4, 6, 12, 10, 1, 3, 7
- \(y\): 550, 800, 250, 500, 500, 400, 650, 150, 400, 750
This dataset also shows irregularities in the data points, lacking a discernible linear trend.
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Fourth Dataset:
- \(x\): 5, 8, 2, 4, 6, 12, 10, 1, 3, 7
- \(y\): 550, 800, 250, 300, 700, 1000, 650, 300, 100, 700
This dataset categorically shows a more defined pattern. The values of \(y\) seem to increase with the increase in \(x\) until a certain point and then fluctuate a bit towards the end, but there are segments that mimic a linear increase.
After evaluating all the datasets, if we rely solely on the appearances suggested by the numbers without graphing, the fourth dataset appears to show the closest behavior to a linear association due to its relatively consistent increase in many parts, even if it is not perfectly linear throughout.
In summary, based on analysis:
- The fourth dataset shows the most linear association.