Let the mean and standard deviation of the normal distribution be denoted by \(\mu\). Since the mean is equal to the standard deviation, we have \(\sigma = \mu\).

We need to find \(P(X < 1.5\mu)\).

Standardizing the variable, we use the formula:

\(z = \frac{X - \mu}{\sigma}\)

Substituting \(X = 1.5\mu\), \(\mu = \sigma\), we get:

\(z = \frac{1.5\mu - \mu}{\mu} = \frac{0.5\mu}{\mu} = 0.5\)

Thus, \(P(X < 1.5\mu) = P(z < 0.5)\).

Using standard normal distribution tables or a calculator, \(P(z < 0.5) = 0.692\).