In American football, there was a scandal where the New England Patriots had inflated their balls rather poorly ahead of a decisive game against the Indianapolis Colts. Our local football coach informs us that balls behave differently when they are flaccid than when they are hard, and he wants to demonstrate this to us with a ball that has been unused for a few months. He drops it from a height of \(227 \mathrm{~cm}\), and measures its bounce after having inflated it with, respectively, \(0,1,2,3\), and 4 pumps.

The coach tells us the uncertainty in his measurements was \(\sigma_{0}=4 \mathrm{~cm}\). He was confident that this was an accurate assessment, so we assume \(n_{0}=\infty\), and use the Normal approximation \(t_{(m, s, \infty)}=\phi_{(m, s)}(x)\).

(a) Find the linear regression line \(y=\alpha+\beta(x-\bar{x})\).
In the rest of this exercise, we will estimate the regression line \(y(x)=a_{*}+b(x-\bar{x})\).

(b) Find the posterior distribution of \(b\).

(c) Find the posterior distribution of \(a_{*}\).

(d) Find the probability distribution of the next observation of the rebound height, given a total of 20 pumpings.

(e) Find a \(90 \%\) credible interval for \(Y_{+}\), given \(x_{+}=20\) pumpings.

(f) Why is the linear regression a good model for the rebound height as a function of the number of times the ball has been pumped? Why is the linear regression a bad model?

Transcribed Image Text:

Pumps, x Rebound height, y (cm) 0 1 95 111 23 115 3 122 4 129