Question

a student is interested in the depth of the water off the end of the local pier. starting at midnight, he measures the depth of the water every three hours for an entire day and records the results in the table. the equation of the least - squares regression line is $hat{y}=13.0 - 0.259x$, where $hat{y}$ is the depth of the water and $x$ is the number of hours past midnight. which shows the residual plot?

Explanation:

Step1: Calculate the predicted values

For \(x = 0\): \(\hat{y}=13.0-0.259\times0 = 13.0\)
For \(x = 3\): \(\hat{y}=13.0-0.259\times3=13.0 - 0.777=12.223\)
For \(x = 6\): \(\hat{y}=13.0-0.259\times6=13.0-1.554 = 11.446\)
For \(x = 9\): \(\hat{y}=13.0-0.259\times9=13.0 - 2.331=10.669\)
For \(x = 12\): \(\hat{y}=13.0-0.259\times12=13.0-3.108 = 9.892\)
For \(x = 15\): \(\hat{y}=13.0-0.259\times15=13.0-3.885 = 9.115\)
For \(x = 18\): \(\hat{y}=13.0-0.259\times18=13.0-4.662 = 8.338\)
For \(x = 21\): \(\hat{y}=13.0-0.259\times21=13.0-5.439 = 7.561\)
For \(x = 24\): \(\hat{y}=13.0-0.259\times24=13.0-6.216 = 6.784\)

Step2: Calculate the residuals

Residual formula: \(e=y-\hat{y}\)
For \(x = 0\): \(e = 12.5244-13.0=- 0.4756\)
For \(x = 3\): \(e=12.9987 - 12.223=0.7757\)
For \(x = 6\): \(e=12.4255-11.446 = 0.9795\)
For \(x = 9\): \(e=11.0050 - 10.669=0.336\)
For \(x = 12\): \(e=9.2334-9.892=-0.6586\)
For \(x = 15\): \(e=7.7296 - 9.115=-1.3854\)
For \(x = 18\): \(e=7.0189-8.338=-1.3191\)
For \(x = 21\): \(e=7.3496 - 7.561=-0.2114\)
For \(x = 24\): \(e=8.6062-6.784 = 1.8222\)

Step3: Analyze the residual plot

The \(x\) - axis is the number of hours after midnight (\(x\)) and the \(y\) - axis is the residual (\(e\)). We plot the points \((0,-0.4756),(3,0.7757),(6,0.9795),(9,0.336),(12,-0.6586),(15,-1.3854),(18,-1.3191),(21,-0.2114),(24,1.8222)\)

The given residual plot in the problem (the one with the points) is the correct one as it matches the calculated residuals' general pattern (negative, positive, positive, positive, negative, negative, negative, negative, positive) when considering the order of \(x\) values.

Answer:

The residual plot provided in the problem (the one with the green - colored points plotted against hours after midnight on the \(x\) - axis and residual on the \(y\) - axis) is the correct one.