Part (A)
The input variable is the shoe size because it is along the horizontal axis, or x axis. It's not clear what the units for the shoe size are, but it's possible the units are in inches.
The output variable is the y value, which in this case is the height. The units for the height is in inches.
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Part (B)
We have a positive trend. We determine this visually by noting the regression line, or trend line, is going upward as we move from left to right.
Algebraically, the equation for the trend line has a positive slope, which leads to a positive trend.
A positive trend is where x and y increase together. As shoe size goes up, height goes up as well.
While not all the points are on the line, there is a tendency for the points to go uphill as we move from left to right.
As for the strength of the correlation, that part is subjective. If we knew the correlation coefficient r, then we could be more certain; however, we don't know this value. So instead we have to eyeball the given graph and give an estimate. Based on the graph, I'd say there's a moderate positive correlation going on. It's somewhere in the middle of weak and strong positive correlation because the data points are somewhat scattered randomly, but they trend upward nonetheless.
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Part (C)
The rate of change is the slope. The slope is the coefficient of the independent variable x. In this problem, the slope is 1.728
So the rate of change is 1.728 inches per shoe size.
In other words, as the shoe size increases by 1 unit (1 inch perhaps?), the height will increase by roughly 1.728 inches.
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Extra info:
The equation
Height = 51.46 + 1.728(Shoe Size)
is the same as
y = 51.46 + 1.728x
where x is the shoe size and y is the height
We can rearrange that into
y = 1.728x + 51.46
So that it matches with the familiar y = mx+b slope intercept form. This is optional.
