Question

can the scatter plot be fit exactly or only approximately by a linear function? how do you know?

x | y
1 | 4
3 | 8
5 | 9
7 | 14
9 | 16

choose the correct answer below.
a. the scatter plot can only be fit approximately by a linear function because no straight line will fit all points.

b. the scatter plot can be fit exactly by a linear function because the first differences of the outputs are equal.

c. the scatter plot can be fit exactly by a linear function because the first differences of the outputs are nearly constant.

d. the scatter plot can only be fit approximately by a linear function because a straight line will pass through all points.

Explanation:

Step1: Calculate x differences

$\Delta x_1=3-1=2$, $\Delta x_2=5-3=2$, $\Delta x_3=7-5=2$, $\Delta x_4=9-7=2$

Step2: Calculate y first differences

$\Delta y_1=8-4=4$, $\Delta y_2=9-8=1$, $\Delta y_3=14-9=5$, $\Delta y_4=16-14=2$

Step3: Analyze linear fit condition

For exact linear fit, $\frac{\Delta y}{\Delta x}$ must be constant. Here, $\frac{4}{2}=2$, $\frac{1}{2}=0.5$, $\frac{5}{2}=2.5$, $\frac{2}{2}=1$, which are not equal. So no straight line fits all points.

Answer:

A. The scatter plot can only be fit approximately by a linear function because no straight line will fit all points.