Math / Data & Statistics

Question4. A random sample of people was asked to report the age and distance driven of their first car. A line was fit to the data to model the relationship.
Write an equation of the line of best fit in $\mathbf{y}=\mathbf{m x + b}$ : Steps:
1. Identify $y$ -intercept (where the line crosses the y -axis) y -intercept, $\mathrm{b}=(\mathrm{O}$ , $\qquad$ )
2. Identify another point on the line (it has to be on the line)
2nd point ( $\qquad$ , $\qquad$ )
3. Find slope, $m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ between these two points $\mathrm{m}=$ $\qquad$ $=$
4. Substitute $m$ and $b$ values into $y=m x+b$ $y=$ $\qquad$

Studdy Solution

STEP 1

What is this asking? We need to find the equation of the line that best fits the data shown in the scatter plot, basically figuring out a formula that relates the age of a person's first car to how much they drove it. Watch out! Make sure to read the values from the graph carefully!
The axes might be tricky.

STEP 2

1. Find the y-intercept
2. Find the slope
3. Build the equation

STEP 3

The y-intercept is where the line crosses the y-axis.
Looking at the graph, the line crosses the y-axis at about $\textbf{16}$ on the "Age (years)" axis.
So, our y-intercept, which we call $\textbf{b}$ , is $\textbf{16}$ .

STEP 4

We already have one point, the y-intercept: $(0, 16)$ .
Now we need another point that's clearly on the line.
Let's pick the point where "Kilometers driven" is $\textbf{40}$ (thousand) and "Age" is $\textbf{12}$ .
So our second point is $(40, 12)$ .

STEP 5

Remember, the slope, which we call $\textbf{m}$ , is the change in $\textbf{y}$ divided by the change in $\textbf{x}$ .
We can use the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Let's plug in our points.
Our first point is $(0, 16)$ , so $x_1 = 0$ and $y_1 = 16$ .
Our second point is $(40, 12)$ , so $x_2 = 40$ and $y_2 = 12$ . $m = \frac{12 - 16}{40 - 0}$ $m = \frac{-4}{40}$ $m = -\frac{1}{10}$ So our slope, $\textbf{m}$ , is $\textbf{-1/10}$ or $\textbf{-0.1}$ .
This makes sense because the line is going downwards, meaning as kilometers driven increases, the age decreases!

STEP 6

We know the equation of a line is $y = mx + b$ .
We found that $\textbf{b = 16}$ and $\textbf{m = -1/10}$ .
Let's plug these values into our equation: $y = -\frac{1}{10}x + 16$ And there we have it!

STEP 7

The equation of the line of best fit is $y = -\frac{1}{10}x + 16$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the y-intercept
2. Find the slope
3. Build the equation

STEP 3

The y-intercept is where the line crosses the y-axis.
Looking at the graph, the line crosses the y-axis at about $\textbf{16}$ on the "Age (years)" axis.
So, our y-intercept, which we call $\textbf{b}$ , is $\textbf{16}$ .

STEP 4

We already have one point, the y-intercept: $(0, 16)$ .
Now we need another point that's clearly on the line.
Let's pick the point where "Kilometers driven" is $\textbf{40}$ (thousand) and "Age" is $\textbf{12}$ .
So our second point is $(40, 12)$ .

STEP 5

STEP 6

We know the equation of a line is $y = mx + b$ .
We found that $\textbf{b = 16}$ and $\textbf{m = -1/10}$ .
Let's plug these values into our equation: $y = -\frac{1}{10}x + 16$ And there we have it!

STEP 7

The equation of the line of best fit is $y = -\frac{1}{10}x + 16$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the y-intercept2. Find the slope3. Build the equation

STEP 3

The y-intercept is where the line crosses the y-axis.Looking at the graph, the line crosses the y-axis at about 16\textbf{16}16 on the "Age (years)" axis.So, our **y-intercept**, which we call b\textbf{b}b, is 16\textbf{16}16.

STEP 4

We already have one point, the y-intercept: (0,16)(0, 16)(0,16).Now we need another point that's clearly on the line.Let's pick the point where "Kilometers driven" is 40\textbf{40}40 (thousand) and "Age" is 12\textbf{12}12.So our second point is (40,12)(40, 12)(40,12).

STEP 5

STEP 6

We know the equation of a line is y=mx+by = mx + by=mx+b.We found that b = 16\textbf{b = 16}b = 16 and m = -1/10\textbf{m = -1/10}m = -1/10.Let's plug these values into our equation: y=−110x+16 y = -\frac{1}{10}x + 16 y=−101​x+16 And there we have it!

STEP 7

The equation of the line of best fit is y=−110x+16y = -\frac{1}{10}x + 16y=−101​x+16.

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1. Find the y-intercept
2. Find the slope
3. Build the equation

The y-intercept is where the line crosses the y-axis.
Looking at the graph, the line crosses the y-axis at about $\textbf{16}$ on the "Age (years)" axis.
So, our y-intercept, which we call $\textbf{b}$ , is $\textbf{16}$ .

We already have one point, the y-intercept: $(0, 16)$ .
Now we need another point that's clearly on the line.
Let's pick the point where "Kilometers driven" is $\textbf{40}$ (thousand) and "Age" is $\textbf{12}$ .
So our second point is $(40, 12)$ .

We know the equation of a line is $y = mx + b$ .
We found that $\textbf{b = 16}$ and $\textbf{m = -1/10}$ .
Let's plug these values into our equation: $y = -\frac{1}{10}x + 16$ And there we have it!

The equation of the line of best fit is $y = -\frac{1}{10}x + 16$ .