Confidence Intervals · Challenge problem

A student has a random sample of only $n = 5$ values from a normal population with unknown $\sigma$, and builds the interval $\bar x \pm 1.96\,\dfrac{s}{\sqrt n}$, using the normal critical value instead of the correct $t^*$. What is the actual probability that this interval contains $\mu$? You may use the fact that for the $t$ distribution with $4$ degrees of freedom, $P(|T| \le t) = \dfrac{t(t^2 + 6)}{(t^2 + 4)^{3/2}}$. Give your answer to three decimal places.