c The residual is the actual value minus the predicted value.
D
d Are the values evenly dispersed around the x-axis?
E
e Substitute 16.8 in place of x to calculate the predicted value.
F
f Is the length of the paint greater or lower than the predicted length if the residual is positive?
A
aCalculator plot:
Sketch:
B
b y=0.248x+1.3
C
cResidual plot:
D
dIs a linear model appropriate? Yes
E
eResidual: 0.5
F
f See solution.
Practice makes perfect
a To graph the data, we first have to enter the values into lists. Push STAT, choose Edit, and then enter the values in the first two columns.
Having entered the values, we can plot them by pushing 2nd and Y=. Then we will choose one of the plots in the list. Make sure you turn the plot ON, choose the type to be a scatterplot, and assign L1 and L2 as XList and Ylist. Finally, you can pick whichever type of mark you want.
Pushing the GRAPH button will tell the calculator to plot the dataset. If you are using a standard viewing window, you will need to change the settings so that you can see all of the data points.
Let's also graph the data on paper.
b To view the linear regression analysis of the dataset, we press STAT, scroll right to view the CALC options, and finally choose the fourth option in the list, LinReg.
Now we can write the line of best fit.
y=0.248x+1.3
Let's add it to our scatterplot.
c Since the residual is the actual value minus the predicted value, we first want to find all of the predicted values.
When we know the predicted values, we can calculate the residuals by subtracting them from the actual values.
c|c|c|c
x & Actual & Predicted & Residual
3.8 & 2.3 & 2.24 & 0.06
4.5 & 2.3 & 2.41 & -0.09
5.2 & 2.7 & 2.58 & 0.12
6.3 & 2.8 & 2.86 & - 0.06
7.2 & 3.0 & 3.08 & - 0.08
8.5 & 3.4 & 3.40 & 0
9.3 & 3.6 & 3.60 & 0
9.8 & 3.8 & 3.72 & 0.08
10.7 & 4.1 & 3.94 & 0.16
12.6 & 4.3 & 4.41 & - 0.11
13.7 & 4.7 & 4.68 & - 0.02
Now we can draw our residual plot.
d Examining the residual plot in Part C, we notice that the residuals are evenly dispersed around the x-axis. This means the linear model runs through the middle of the observations, which tells us that it's an appropriate model.
e To determine the residual, we have to substitute 16.8 instead of x in the linear regression.
y=0.248x+1.3By substituting the length of paint instead of x, we can determine how much the pencil should weigh.
The pencil should weigh about 5.5 grams. Now we can calculate the residual.
Residual=6-5.5=0.5
f Since the residual is the actual values minus the predicted values, a positive residual must mean that the weight of the pencil was greater than the predicted weight given the length of paint.
