Question

2-5 study guide and intervention scatter plots and lines of regression
scatter plots and prediction equations a set of data points graphed as ordered pairs in a coordinate plane is called a scatter plot. a scatter plot can be used to determine if there is a relationship among the data. a line of fit is a line that closely approximates a set of data graphed in a scatter plot. the equation of a line of fit is called a prediction equation because it can be used to predict values not given in the data set.
example: storage costs according to a certain prediction equation, the cost of 200 square feet of storage space is $60. the cost of 325 square feet of storage space is $160.
a. find the slope of the prediction equation. what does it represent?
since the cost depends upon the square footage, let x represent the amount of storage space in square feet and y represent the cost in dollars. the slope can be found using the formula m = (y₂ - y₁)/(x₂ - x₁). so, m = (160 - 60)/(325 - 200)=100/125 = 0.8
the slope of the prediction equation is 0.8. this means that the price of storage increases 80¢ for each one - square - foot increase in storage space.
b. find a prediction equation.
using the slope and one of the points on the line, you can use the point - slope form to find a prediction equation.
y - y₁ = m(x - x₁) point - slope form
y - 60 = 0.8(x - 200) (x₁, y₁)=(200, 60), m = 0.8
y - 60 = 0.8x - 160 distributive property
y = 0.8x - 100 add 60 to both sides
a prediction equation is y = 0.8x - 100.
exercises

1. salaries the table below shows the years of experience for eight technicians at lewis techomatic and the hourly rate of pay each technician earns.

experience (years)	9	4	3	1	10	6	12	8
hourly rate of pay	$17	$10	$10	$7	$19	$12	$20	$15

a. draw a scatter plot to show how years of experience are related to hourly rate of pay. draw a line of fit and describe the correlation.
b. write a prediction equation to show how years of experience (x) are related to hourly rate of pay (y).
c. use the function to predict the hourly rate of pay for 15 years of experience.

Explanation:

Step1: Draw scatter - plot and line of fit

1. On the x - axis, mark years of experience (ranging from 1 - 12 as per data). On the y - axis, mark hourly rate of pay (ranging from 0 - 24).
2. Plot the points: (9,17), (4,10), (3,10), (1,7), (10,19), (6,12), (12,20), (8,15).
3. Draw a line of fit that approximately passes through the middle of the points. The correlation appears to be positive as the line has a positive slope, meaning as years of experience increase, the hourly rate of pay also tends to increase.

Step2: Find the slope

Let $(x_1,y_1)=(1,7)$ and $(x_2,y_2)=(12,20)$.
Using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$, we have $m=\frac{20 - 7}{12 - 1}=\frac{13}{11}\approx1.18$.

Step3: Find the prediction equation

Using the point - slope form $y - y_1=m(x - x_1)$ with $(x_1,y_1)=(1,7)$ and $m = \frac{13}{11}$.
$y-7=\frac{13}{11}(x - 1)$
$y-7=\frac{13}{11}x-\frac{13}{11}$
$y=\frac{13}{11}x-\frac{13}{11}+7$
$y=\frac{13}{11}x-\frac{13}{11}+\frac{77}{11}$
$y=\frac{13}{11}x+\frac{64}{11}\approx1.18x + 5.82$

Step4: Predict the hourly rate for 15 years of experience

Substitute $x = 15$ into the equation $y=\frac{13}{11}x+\frac{64}{11}$.
$y=\frac{13\times15}{11}+\frac{64}{11}=\frac{195 + 64}{11}=\frac{259}{11}\approx23.55$

Answer:

a. After drawing the scatter - plot and line of fit, there is a positive correlation.
b. The prediction equation is $y=\frac{13}{11}x+\frac{64}{11}\approx1.18x + 5.82$.
c. The predicted hourly rate of pay for 15 years of experience is approximately $\$23.55$.