Let $\bar { X }$ be the sample mean obtained by randomly sampling 9 items from a population following the normal distribution $\mathrm { N } \left( 0,4 ^ { 2 } \right)$, and let $\bar { Y }$ be the sample mean obtained by randomly sampling 16 items from a population following the normal distribution $\mathrm { N } \left( 3,2 ^ { 2 } \right)$. What is the value of the constant $a$ satisfying $\mathrm { P } ( \bar { X } \geq 1 ) = \mathrm { P } ( \bar { Y } \leq a )$? [3 points] (1) $\frac { 19 } { 8 }$ (2) $\frac { 5 } { 2 }$ (3) $\frac { 21 } { 8 }$ (4) $\frac { 11 } { 4 }$ (5) $\frac { 23 } { 8 }$
Let $\bar { X }$ be the sample mean obtained by randomly sampling 9 items from a population following the normal distribution $\mathrm { N } \left( 0,4 ^ { 2 } \right)$, and let $\bar { Y }$ be the sample mean obtained by randomly sampling 16 items from a population following the normal distribution $\mathrm { N } \left( 3,2 ^ { 2 } \right)$.\\
What is the value of the constant $a$ satisfying $\mathrm { P } ( \bar { X } \geq 1 ) = \mathrm { P } ( \bar { Y } \leq a )$? [3 points]\\
(1) $\frac { 19 } { 8 }$\\
(2) $\frac { 5 } { 2 }$\\
(3) $\frac { 21 } { 8 }$\\
(4) $\frac { 11 } { 4 }$\\
(5) $\frac { 23 } { 8 }$