\(\displaystyle{X}_{{{1}}},{X}_{{{2}}},\ldots{X}_{{{134}}}\) be a random sample from Normal distribution with \(\displaystyle\mu={154.5}\) and \(\displaystyle\sigma={96.1}\).

Then the sample mean

\(\displaystyle{M}={\frac{{{1}}}{{{134}}}}{\sum_{{{i}={1}}}^{{{134}}}}{X}_{{{i}}}\) follows Normal distribution with mean 154.5 and standard deviation \(\displaystyle{\frac{{{96.1}}}{{\sqrt{{{134}}}}}}\).

Hence, the probability that sample mean greater than 167.

\(\displaystyle{P}{\left({M}{>}{167}\right)}\)

\(\displaystyle={1}-{P}{\left({M}\leq{167}\right)}\)

\(\displaystyle={1}-{P}{\left({\frac{{{M}-{154.5}}}{{\frac{{96.1}}{\sqrt{{{134}}}}}}}\leq{\frac{{{167}-{154.5}}}{{\frac{{96.1}}{\sqrt{{{134}}}}}}}\right)}\)

\(\displaystyle={1}-{P}{\left({Z}\leq{1.506}\right)},{Z}={\left({\frac{{{M}-{154.5}}}{{\frac{{96.1}}{\sqrt{{{134}}}}}}}\right)}\) follows Normal (0,1)

\(\displaystyle={1}-\phi{\left({1.506}\right)},\phi{\left({1.506}\right)}\) calculated from Normal distribution table.

\(\displaystyle={1}-{0.934}={0.066}\)

Therefore, the probability that a sample of size \(\displaystyle{n}={134}\) is randomly selected with a mean greater than 167 is 0.066.

Then the sample mean

\(\displaystyle{M}={\frac{{{1}}}{{{134}}}}{\sum_{{{i}={1}}}^{{{134}}}}{X}_{{{i}}}\) follows Normal distribution with mean 154.5 and standard deviation \(\displaystyle{\frac{{{96.1}}}{{\sqrt{{{134}}}}}}\).

Hence, the probability that sample mean greater than 167.

\(\displaystyle{P}{\left({M}{>}{167}\right)}\)

\(\displaystyle={1}-{P}{\left({M}\leq{167}\right)}\)

\(\displaystyle={1}-{P}{\left({\frac{{{M}-{154.5}}}{{\frac{{96.1}}{\sqrt{{{134}}}}}}}\leq{\frac{{{167}-{154.5}}}{{\frac{{96.1}}{\sqrt{{{134}}}}}}}\right)}\)

\(\displaystyle={1}-{P}{\left({Z}\leq{1.506}\right)},{Z}={\left({\frac{{{M}-{154.5}}}{{\frac{{96.1}}{\sqrt{{{134}}}}}}}\right)}\) follows Normal (0,1)

\(\displaystyle={1}-\phi{\left({1.506}\right)},\phi{\left({1.506}\right)}\) calculated from Normal distribution table.

\(\displaystyle={1}-{0.934}={0.066}\)

Therefore, the probability that a sample of size \(\displaystyle{n}={134}\) is randomly selected with a mean greater than 167 is 0.066.