Regional daily demand for gasoline in the summer driving months is assumed to be the outcome of a \(N\left(\mu, \sigma^{2}ight)\) random variable. Assume you have 40 iid daily observations on daily demand for gasoline with quantity demanded measured in millions of gallons, and \(\bar{x}=43\) and \(s^{2}=2\).

(a) Show that \(N\left(\mu, \sigma^{2}ight)\) is a location-scale parameter family of PDFs (recall Theorem 10.13).

(b) Define a pivotal quantity for \(\mu\), and use it to define a .95 level confidence interval for \(\mu\). Also, define a . 95 lower confidence bound for \(\mu\).

(c) Define a pivotal quantity for \(\sigma^{2}\), and use it to define a .95 level confidence interval for \(\sigma^{2}\).

(d) A colleague claims that mean daily gasoline demand is only 37 million gallons. Is your answer to

(b) consistent with this claim? Explain.