a Plot the data set. What kind of association does it appear to be?
B
b The residual is the difference between the actual value and the model's value.
C
c The correlation coefficient is obtained by performing a linear regression on a graphing calculator. It's labeled r.
A
aScatterplot:
Model: y=1.266x+10.633.
B
bSketch:
Is the linear model appropriate? Yes, it is.
C
cCorrelation Coefficient: r=0.998
What does it tell you? We have a strong positive correlation.
Practice makes perfect
a The hourly wage of a worker depends on, among other factors, the experience of the worker. Therefore, the experience in years is the independent variable, while the hourly wage is the dependent variable. With this information we can draw our scatter plot.
To describe this association we need to perform a linear regression using a graphing calculator. First, we have to enter the values into lists. Push STAT, choose Edit, and then enter the values in the first two columns.
To perform the linear regression of the data set, push STAT, scroll to the right to view the CALC options, and then choose the fourth option in the list, LinReg.
The association between experience and hourly wage can be modeled by y=1.266x+10.633. Therefore, with no experience, a technology worker starts at about $10.60 an hour, and for each year of experience the pay increases by 1.26 dollars.
b To sketch the residual plot we first have to calculate the residuals, which are the difference between the actual value and the predicted value.
Residual= Actual value-Predicted value
Let's determine the predicted value by substituting the experience in years in our model and simplifying.
x
1.266x+10.633
y
1
1.266( 1)+10.633
≈ 11.9
2
1.266( 2)+10.633
≈ 13.2
3
1.266( 3)+10.633
≈ 14.4
4
1.266( 4)+10.633
≈ 15.7
5
1.266( 5)+10.633
≈ 17.0
6
1.266( 5)+10.633
≈ 18.2
7
1.266( 7)+10.633
≈ 19.5
8
1.266( 8)+10.633
≈ 20.8
9
1.266( 9)+10.633
≈ 22
10
1.266( 10)+10.633
≈ 23.3
When we know the predicted value, we can calculate the residual by subtracting it from the actual value.
x
Actual
Predicted
Residual
1
12.00
11.9
0.1
2
13.25
13.2
0.05
3
14
14.4
- 0.4
4
16
15.7
0.3
5
17
17
0
6
18
18.2
- 0.2
7
19.5
19.5
0
8
21
20.8
0.2
9
22
22
0
10
23.25
23.3
- 0.05
Now we can sketch the residual plot.
As we can see, the residuals are evenly dispersed around the x-axis. This tells us our model runs in the middle of the data points, which means the linear model is appropriate.
c From Part A we know that the correlation coefficient is r≈ 0.998. The closer the correlation coefficient is to either 1 or - 1, the better predictive power the model has. A coefficient close to 1 means we have a strong positive correlation.
