Step 1

From the provided information,

Mean \(\displaystyle{\left(\mu\right)}={13.7}\)

Standard deviation \(\displaystyle{\left(\sigma\right)}={22}\)

Let X be a random variable which represents the score.

\(\displaystyle{X}\sim{N}{\left({13.7},{22}\right)}\)

Sample size \(\displaystyle{\left({n}\right)}={78}\)

Step 2

The required probability that a single randomly selected value is less than 11.5

can be obtained as:

\(\displaystyle{P}{\left({X},{11.5}\right)}={P}{\left({\frac{{{x}-\mu}}{{\sigma}}}{<}{\frac{{{11.5}-{13.7}}}{{{22}}}}\right)}\)

\(\displaystyle={P}{\left({Z}{<}-{0.1}\right)}={0.4602}\) (Using standard normal table)

Thus, the required probability is 0.4602.