\(\displaystyle\overline{{{X}}}={\frac{{{1}}}{{{n}}}}{\sum_{{{i}={1}}}^{{n}}}{X}_{{i}}=\)

Mean value.

Therefore, the mean of \(\displaystyle\overline{{{X}}}\) is 116.3,

standard deviation is \(\displaystyle{\frac{{{27.5}}}{{\sqrt{{{n}}}}}}={\frac{{{27.5}}}{{\sqrt{{{249}}}}}}\), as \(\displaystyle{n}={249}\).

The probability of a mean greater than 117.3 is

\(\displaystyle{P}{\left[\overline{{{X}}}{>}{117.3}\right]}\)

\(\displaystyle={P}{\left[{\frac{{\overline{{{X}}}-{116.3}}}{{\frac{{27.5}}{\sqrt{{{249}}}}}}}{>}{\frac{{{117.3}-{116.3}}}{{\frac{{27.5}}{\sqrt{{{249}}}}}}}\right]}\)

\(\displaystyle={P}{\left[{Z}{>}{\frac{{\sqrt{{{249}}}}}{{{27.5}}}}\right]},{Z}={\frac{{\overline{{{X}}}-{116.3}}}{{\frac{{27.5}}{\sqrt{{{249}}}}}}}\sim\) Normal(0,1)

\(\displaystyle={1}-{P}{\left[{Z}\leq{\frac{{\sqrt{{{249}}}}}{{{27.5}}}}\right]}\)

\(\displaystyle={1}-\phi{\left({0.574}\right)}\)

\(\displaystyle={1}-{0.717}={0.283}\)

The value of \(\displaystyle\phi{\left({0.574}\right)}\) is taken from the Normal distribution table.

Therefore, the probability of the mean is greater than 117.3 is 0.283.

Mean value.

Therefore, the mean of \(\displaystyle\overline{{{X}}}\) is 116.3,

standard deviation is \(\displaystyle{\frac{{{27.5}}}{{\sqrt{{{n}}}}}}={\frac{{{27.5}}}{{\sqrt{{{249}}}}}}\), as \(\displaystyle{n}={249}\).

The probability of a mean greater than 117.3 is

\(\displaystyle{P}{\left[\overline{{{X}}}{>}{117.3}\right]}\)

\(\displaystyle={P}{\left[{\frac{{\overline{{{X}}}-{116.3}}}{{\frac{{27.5}}{\sqrt{{{249}}}}}}}{>}{\frac{{{117.3}-{116.3}}}{{\frac{{27.5}}{\sqrt{{{249}}}}}}}\right]}\)

\(\displaystyle={P}{\left[{Z}{>}{\frac{{\sqrt{{{249}}}}}{{{27.5}}}}\right]},{Z}={\frac{{\overline{{{X}}}-{116.3}}}{{\frac{{27.5}}{\sqrt{{{249}}}}}}}\sim\) Normal(0,1)

\(\displaystyle={1}-{P}{\left[{Z}\leq{\frac{{\sqrt{{{249}}}}}{{{27.5}}}}\right]}\)

\(\displaystyle={1}-\phi{\left({0.574}\right)}\)

\(\displaystyle={1}-{0.717}={0.283}\)

The value of \(\displaystyle\phi{\left({0.574}\right)}\) is taken from the Normal distribution table.

Therefore, the probability of the mean is greater than 117.3 is 0.283.