a A high correlation coefficient means a strong correlation.
B
b Does the passage of time cause texting?
C
c Residuals are calculated by subtracting the models value from the observations value.
D
d Do you know what evenly dispersed residuals look like?
A
aLine of best fit: y=513.5x-298
Correlation coefficient is 0.993 so it's a good model.
B
b It's not a causal relationship.
Explanation: See solution.
C
cResiduals: 25.5, - 128, 117.5, 50, - 63.5
Scatter plot:
It's a good fit.
D
d See solution.
Practice makes perfect
a To write an equation of the regression line, we first have to enter our ordered pairs into lists in our calculator. We can do this if we press the STAT button and then choose Edit.
Once the values have been entered, a regression can be performed by pressing the STAT button again, followed by the Right Arrow key to select CALC in the menu. This menu lists the various regressions that are available. If we choose LinReg, the calculator performs a linear regression test.
The regression line can be written as
y=513.5x-298.
We can also see that the correlation coefficient is 0.993 which is very strong. This means there is a strong correlation between years passed and the rise in text messages.
b A casual relationship would imply that the passage of time actually causes the number of text messages to increase. But this cannot be the case. It's the existence of humans and a larger population owning cellphones year after year, that causes it. If humans cease to exist, so will texting.
c The residuals show us how much the observations deviate from the estimated ones according to our line of best fit. Therefore, by subtracting the model value from the observation value, we get the residual.
Year
Observation
Observation-(513.5x-298)
Residual
1
241
241-(513.5* 1-298)
25.5
2
601
601-(513.5* 2-298)
- 128
3
1360
1360-(513.5* 3-298)
117.5
4
1860
1860-(513.5* 4-298)
50
5
2206
2206-(513.5* 5-298)
- 63.5
Let's also plot the residuals using our graphing calculator. To do this, you have to go to your STAT PLOT menu and choose the square plots. Use L1 for Xlist and RESID for Ylist. To graph the residuals, press 2nd followed by STAT and then choose RESID from the list of names.
The residuals seem to be evenly distributed around the x-axis, which suggests that the regression fits the model well. Note that we changed the window settings to - 1≤ x≤ 6 and - 500 ≤ y ≤ 500 to fit the data.
d Determining if a model fits its data using residuals is very much up for interpretation. What do evenly distributed residuals look like? Would you know it when you saw it? The good thing about using a correlation coefficient, is that our judgement is reduced to a number between 0 and 1. If the number is closer to 1, it's a good model. If it's closer to 0 , it's not a good model.
