Question

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

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

Plot the points \((9,17)\), \((4,10)\), \((3,10)\), \((1,7)\), \((10,19)\), \((6,12)\), \((12,20)\), \((8,15)\) on the given coordinate system where the x - axis represents years of experience and the y - axis represents hourly rate of pay. Then draw a line of fit by eye or using a statistical software (for simplicity, by eye here). The line seems to have a positive slope, indicating a positive correlation between years of experience and hourly rate of pay.

Step2: Find the prediction equation

Let the prediction equation be of the form \(y = mx + b\) (linear regression). We can use the least - squares method or estimate the slope \(m\) and y - intercept \(b\) by eye. First, find two points on the line of fit, say \((x_1,y_1)\) and \((x_2,y_2)\). Let's assume the line passes through \((1,7)\) and \((10,19)\).
The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{19 - 7}{10 - 1}=\frac{12}{9}=\frac{4}{3}\).
Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(1,7)\) and \(m = \frac{4}{3}\), we have \(y-7=\frac{4}{3}(x - 1)\).
Expanding, \(y-7=\frac{4}{3}x-\frac{4}{3}\), so \(y=\frac{4}{3}x+7-\frac{4}{3}=\frac{4}{3}x+\frac{21 - 4}{3}=\frac{4}{3}x+\frac{17}{3}\approx1.33x + 5.67\).

Step3: Make a prediction

Substitute \(x = 15\) into the equation \(y=\frac{4}{3}x+\frac{17}{3}\).
\(y=\frac{4}{3}\times15+\frac{17}{3}=\frac{60 + 17}{3}=\frac{77}{3}\approx25.67\).

Answer:

a. After drawing the scatter - plot and line of fit, we observe a positive correlation between years of experience and hourly rate of pay.
b. The prediction equation is \(y=\frac{4}{3}x+\frac{17}{3}\) (or approximately \(y = 1.33x+5.67\)).
c. The predicted hourly rate of pay for 15 years of experience is approximately \(\$25.67\).