Since we want the sampling distribution of X, then we have the following:
[tex]\begin{gathered} n=25 \\ \mu_{\bar{X}}=\mu=73 \\ \sigma_{\bar{X}}=\frac{\sigma}{\sqrt[]{n}}=\frac{12.5}{\sqrt[]{25}}=2.5 \end{gathered}[/tex]
then, we can calculate the probability by normalizing the variable X hat:
[tex]P(\bar{X}<77)=P(\frac{\bar{X}-\mu_X}{\sigma_X}<\frac{77-73}{2.5})=P(Z<1.6)=0.9452[/tex]
therefore, the probability is 0.9452
