Here, the polynomial has a degree of 1. Therefore, this is a linear equation.
y=24.0-0.1972 x
b To determine which model is the best predictor, we should first consider the residual plots. If a scatter plot is a good fit the residual plot is random, which means there is no discernible pattern.
Regression equation 1: We see that the residual plot is random. We can not make out a pattern.
Regression equation 2: Again, the residual plot is random. There is no discernible pattern.
Regression equation 3: For this regression, we do have a pattern. The residual is first mostly above the x-axis. Then it dips below the x-axis and rises above it again.
This means we must choose between Regression 1 and 2. Remember that the scatterplot shows how fast each member runs given how much they ran during the previous week. Since more training generally leads to better run times, the scatterplot should decrease the further to the right we go. This means the quadratic regression is not a good fit, because it eventually increases. We should choose the exponential equation.
c By substituting x=10, 20, and 30 into Regression 1, we can predict the run times for athletes who train 10, 20, and 30 km a week.
x
16.0+10.0(0.95)^x
y
10
16.0+10.0(0.95)^(10)
21.98
20
16.0+10.0(0.95)^(20)
19.58
30
16.0+10.0(0.95)^(30)
18.14
d As with all studies, we have to think about cause and effect. Does running a long distance the previous week really ensure that a runner can run 5k in less than 20 minutes? Not necessarily, because the data does not consider that runners who ran more than 18km per week have probably been doing so for a longer time, and are therefore better.
