7. The random variable \(X\) is normally distributed with mean 79 and variance 144 .
Find
- \(\mathrm { P } ( X < 70 )\),
- \(\mathrm { P } ( 64 < X < 96 )\).
It is known that \(\mathrm { P } ( 79 - a \leq X \leq 79 + b ) = 0.6463\). This information is shown in the figure below.
\includegraphics[max width=\textwidth, alt={}, center]{df898ff4-c3ef-400c-b4f7-f4df3757941d-6_581_983_818_590}
Given that \(\mathrm { P } ( X \geq 79 + b ) = 2 \mathrm { P } ( X \leq 79 - a )\), - show that the area of the shaded region is 0.1179 .
- Find the value of \(b\).