Question

the scatter plot shows the number of years of experience, x, and the amount charged per hour, y, for each of 23 dog sitters in california. use the scatter plot to answer the parts below. (note that you can use the graphing tools to help you approximate the line.)
(a) write an approximate equation of the line of best fit. round the coefficients to the nearest hundredth.
( y = square )
(b) using your equation from part (a), predict the amount charged per hour by a dog sitter with 14 years of experience. round your prediction to the nearest hundredth.

Explanation:

To solve this problem, we need to determine the line of best fit for the scatter plot and then use it to make a prediction. Here's the step-by-step process:

Part (a): Equation of the Line of Best Fit

To find the equation of the line of best fit, we can use the two-point form or estimate the slope and y-intercept from the scatter plot. Let's assume we estimate two points on the line of best fit. For example, let's take the points (2, 6) and (20, 20) (these are approximate points from the scatter plot).

Step 1: Calculate the Slope (\(m\))

The slope \(m\) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points (2, 6) and (20, 20):
\[
m = \frac{20 - 6}{20 - 2} = \frac{14}{18} \approx 0.78
\]

Step 2: Calculate the Y-Intercept (\(b\))

Using the point-slope form \(y - y_1 = m(x - x_1)\) with the point (2, 6):
\[
y - 6 = 0.78(x - 2)
\]
Simplify to slope-intercept form (\(y = mx + b\)):
\[
y = 0.78x - 1.56 + 6
\]
\[
y = 0.78x + 4.44
\]

So, the approximate equation of the line of best fit is \(y = 0.78x + 4.44\) (rounded to the nearest hundredth).

Part (b): Prediction for 14 Years of Experience

To predict the amount charged per hour for a dog sitter with 14 years of experience, substitute \(x = 14\) into the equation from part (a):

Step 1: Substitute \(x = 14\) into the Equation

\[
y = 0.78(14) + 4.44
\]

Step 2: Calculate the Value

\[
y = 10.92 + 4.44 = 15.36
\]

Final Answers

(a) The equation of the line of best fit is \(y = 0.78x + 4.44\) (rounded to the nearest hundredth).
(b) The predicted amount charged per hour for a dog sitter with 14 years of experience is \(\$15.36\) (rounded to the nearest hundredth).

Answer:

To solve this problem, we need to determine the line of best fit for the scatter plot and then use it to make a prediction. Here's the step-by-step process:

Part (a): Equation of the Line of Best Fit

To find the equation of the line of best fit, we can use the two-point form or estimate the slope and y-intercept from the scatter plot. Let's assume we estimate two points on the line of best fit. For example, let's take the points (2, 6) and (20, 20) (these are approximate points from the scatter plot).

Step 1: Calculate the Slope (\(m\))

The slope \(m\) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points (2, 6) and (20, 20):
\[
m = \frac{20 - 6}{20 - 2} = \frac{14}{18} \approx 0.78
\]

Step 2: Calculate the Y-Intercept (\(b\))

Using the point-slope form \(y - y_1 = m(x - x_1)\) with the point (2, 6):
\[
y - 6 = 0.78(x - 2)
\]
Simplify to slope-intercept form (\(y = mx + b\)):
\[
y = 0.78x - 1.56 + 6
\]
\[
y = 0.78x + 4.44
\]

So, the approximate equation of the line of best fit is \(y = 0.78x + 4.44\) (rounded to the nearest hundredth).

Part (b): Prediction for 14 Years of Experience

To predict the amount charged per hour for a dog sitter with 14 years of experience, substitute \(x = 14\) into the equation from part (a):

Step 1: Substitute \(x = 14\) into the Equation

\[
y = 0.78(14) + 4.44
\]

Step 2: Calculate the Value

\[
y = 10.92 + 4.44 = 15.36
\]

Final Answers

(a) The equation of the line of best fit is \(y = 0.78x + 4.44\) (rounded to the nearest hundredth).
(b) The predicted amount charged per hour for a dog sitter with 14 years of experience is \(\$15.36\) (rounded to the nearest hundredth).