Question

write
explain the relationship
between the terms trend line
and model.
remember
a trend line is a straight line that is as close to as many
points as possible but does not have to go through any of
the points on the scatterplot.
a trend line can be used to make predictions about
bivariate data through interpolation and extrapolation.
practice

1. estimate the equation of the line of best fit for each graph.

a.
b.
(0,1) (4,9)
x₁ y₁ (x₂ y₂)
(9 - 1)/(4 - 0) = 8/4 = 2
b = 1
+ b

Explanation:

Part (a)

Step1: Identify two points on the line

From the graph (and the handwritten notes), we can use the points \((0, 1)\) and \((4, 9)\).

Step2: Calculate the slope (\(m\))

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Substituting \(x_1 = 0,y_1 = 1,x_2 = 4,y_2 = 9\), we get \(m=\frac{9 - 1}{4 - 0}=\frac{8}{4}=2\).

Step3: Determine the y - intercept (\(b\))

The y - intercept is the value of \(y\) when \(x = 0\). From the point \((0,1)\), when \(x = 0\), \(y=1\), so \(b = 1\).

Step4: Write the equation of the line

The equation of a line in slope - intercept form is \(y=mx + b\). Substituting \(m = 2\) and \(b = 1\), we get \(y = 2x+1\).

Part (b)

Step1: Select two points on the trend line

We can estimate two points on the trend line. Let's assume we pick \((0,15)\) (since when \(x = 0\), the trend line seems to cross the y - axis around \(y = 15\)) and \((24,8)\) (by looking at the end - point of the scatter plot and the trend line).

Step2: Calculate the slope (\(m\))

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), with \(x_1=0,y_1 = 15,x_2 = 24,y_2=8\), we have \(m=\frac{8 - 15}{24 - 0}=\frac{- 7}{24}\approx- 0.29\) (or we can use other points. If we pick \((5,12)\) and \((15,9)\), \(m=\frac{9 - 12}{15 - 5}=\frac{-3}{10}=-0.3\)).

Step3: Determine the y - intercept (\(b\))

When \(x = 0\), from the estimated point \((0,15)\), the y - intercept \(b\approx15\).

Step4: Write the equation of the line

Using the slope - intercept form \(y=mx + b\). If we take \(m=-0.3\) and \(b = 15\), the equation is \(y=-0.3x + 15\) (or with a more accurate slope calculation, but this is an estimate).

Answer:

s:
a. \(y = 2x+1\)
b. \(y=-0.3x + 15\) (or a similar estimated equation based on the trend line)