The known values are,

\(\displaystyle\mu={192.3}\),

\(\displaystyle\sigma={66.5}\),

\(\displaystyle{n}={15}\)

The probability that a sample size \(\displaystyle{n}={15}\) is randomly selected with a mean less than 185.4 is,

\(\displaystyle{P}{\left({M}{<}{185.4}\right)}={P}{\left({\frac{{{M}-\mu}}{{\frac{\sigma}{\sqrt{{{n}}}}}}}{<}{\frac{{{185.4}-{192.3}}}{{\frac{{66.5}}{\sqrt{{{15}}}}}}}\right)}\)</span>

\(\displaystyle={P}{\left({z}{<}-{0.402}\right)}\)</span>

\(\displaystyle={\left(={N}{O}{R}{M}{D}{I}{S}{T}{\left(-{0.402}\right)}\right)}{\left({b}{e}{g}\in{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}{U}{\sin{{g}}}\ {t}{h}{e}\ {E}{x}{c}{e}{l}\ {f}{\quad\text{or}\quad}\mu{l}{a}\backslash{\left(={N}{O}{R}{M}{D}{I}{S}{T}{\left({z}\right)}\right)}{e}{n}{d}{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}\right)}\)

\(\displaystyle={0.343842}\approx{0.3438}\)

Therefore, the required probability is, 0.3438

\(\displaystyle\mu={192.3}\),

\(\displaystyle\sigma={66.5}\),

\(\displaystyle{n}={15}\)

The probability that a sample size \(\displaystyle{n}={15}\) is randomly selected with a mean less than 185.4 is,

\(\displaystyle{P}{\left({M}{<}{185.4}\right)}={P}{\left({\frac{{{M}-\mu}}{{\frac{\sigma}{\sqrt{{{n}}}}}}}{<}{\frac{{{185.4}-{192.3}}}{{\frac{{66.5}}{\sqrt{{{15}}}}}}}\right)}\)</span>

\(\displaystyle={P}{\left({z}{<}-{0.402}\right)}\)</span>

\(\displaystyle={\left(={N}{O}{R}{M}{D}{I}{S}{T}{\left(-{0.402}\right)}\right)}{\left({b}{e}{g}\in{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}{U}{\sin{{g}}}\ {t}{h}{e}\ {E}{x}{c}{e}{l}\ {f}{\quad\text{or}\quad}\mu{l}{a}\backslash{\left(={N}{O}{R}{M}{D}{I}{S}{T}{\left({z}\right)}\right)}{e}{n}{d}{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}\right)}\)

\(\displaystyle={0.343842}\approx{0.3438}\)

Therefore, the required probability is, 0.3438