In a test of corrosion resistance, a sample of \(60 \mathrm{In}\) coloy steel specimens were immersed in acidified brine for four hours, after which each specimen had developed a number of corrosive pits. The maximum pit depth was measured for each specimen. The mean depth was \(850 \mu \mathrm{m}\). The population standard deviation is \(153 \mu \mathrm{m}\). The specification is that the population mean depth \(\mu\) is less than \(900 \mu \mathrm{m}\).

a. Find the \(P\)-value for testing \(H_{0}: \mu \geq 900\) versus \(H_{1}: \mu<900\).

b. Do you believe it is plausible that the mean depth is at least \(900 \mu \mathrm{m}\), or are you convinced that it is less than \(900 \mu \mathrm{m}\) ? Explain.