Normal Probabilities and Unknown Parameters

Write \(X\sim N(\mu,\sigma^2)\), with \(\sigma>0\). Calculator normal-probability commands normally require standard deviation \(\sigma\), not variance \(\sigma^2\). Use the lower and upper bounds for interval probabilities; inverse-normal inputs must match the chosen tail.

For known \(\mu=50\) and \(\sigma=8\), \(P(X<58)=P(Z<1)\approx0.8413\). A right-tail probability \(0.10\) corresponds to left-tail probability \(0.90\).

For unknown parameters, first find the corresponding standard normal quantiles using technology. If \(P(X<42)=0.158655\) and \(P(X<66)=0.841345\), the quantiles are approximately \(-1\) and \(1\).

\(\begin{aligned}42&=\mu-\sigma\\66&=\mu+\sigma\\\mu&=54,\quad\sigma=12\end{aligned}\)

The variance is \(144\). Substitute the parameters back into both probability conditions; keep unrounded calculator values until the final answer.