(a) Graph airplane count (
) versus helicopter count (
), and draw in the estimated
(b) Carry out two
-tests regarding the slope
: the usual test of
and also a
= 1. Which test seems more relevant to this study?
(c) Run the following code in
new=data.frame(heli=seq(28, 88, 0.5))
pred.PI = predict(lm1, new, level=0.95, interval="prediction")
pred.CI = predict(lm1, new, level=0.95, interval="confidence")
n=10; x.mean=mean(heli); sxx=sum((heli-x.mean)^2);
half.width=summary(lm1)$sigma*sqrt(2*qf(0.95, 2, n-2))*
plot(c(28, 88), range(air, pred.PI, pred.CI, band.CI), type="n")
points(heli, air, xlab="Manatee Counts from Helicopter", ylab="From Airplane")
lines(new$heli, pred.PI[,2], lty=1, col="red")
lines(new$heli, pred.PI[,3], lty=1, col="red")
lines(new$heli, pred.CI[,2], lty=2, col="blue")
lines(new$heli, pred.CI[,3], lty=2, col="blue")
lines(new$heli, band.CI[,1], lty=3, col="green")
lines(new$heli, band.CI[,2], lty=3, col="green")
Explain what are plotted in the figure (you don’t need to include this figure in
your homework). Explain why the blue intervals are shorter than the red intervals.
Explain why the blue intervals are shorter than the green intervals.
(d) If the helicopter count really were accurate, and airplane observers counted no
imaginary manatees (although they might miss some real ones), the relation between
these two counts should be a regression through the origin (because when
we should have
= 0 too). Conduct a regression of airplane count on helicopter
count by excluding the intercept, and graph the result. Is the slope in this graph
significantly different from 1?
14. Old Faithful Geyser Data. The data set
gives information about eruptions of
the Old Faithful geyser in Yellowstone National Park, Wyoming, USA. Variables are
: the eruption time in minutes;
: the waiting time to the next eruption in minutes.
Fit a linear regression model to predict
(a) What’re the estimated slope and intercept?
(b) Give an interpretation of the estimated slope. Construct a 99% CI for the slope.
(c) Report the residual standard error and the