Johnson's (1978) adjustment for skewness.
\begin{equation}
t_{1} =
\left [ (\bar{x} - \mu) +
\frac{\mu^3}{6\sigma^2N} +
\frac{\mu^3}{3\sigma^4}
(\bar{x} - \mu)^2\right ]
\left[\frac{s^2}{N}
\right]^{-\frac{1}{2}}
\end{equation}
Rearranging.
\begin{equation}
t_{1} =
(\bar{x} - \mu)
\frac{\sqrt{N}}{s}+
\frac{\mu^3}{6\sigma^2N}\frac{\sqrt{N}}{s}+
\frac{\mu^3}{3\sigma^4}
(\bar{x} - \mu)^2\frac{\sqrt{N}}{s}
\end{equation}
Simplifying terms.
\begin{equation}
t_{1} =
(\bar{x} - \mu)
\frac{\sqrt{N}}{s}+
\frac{\mu^3}{6\sigma^2s\sqrt{N}}+
\frac{\mu^3}{3\sigma^4}
(\bar{x} - \mu)^2\frac{\sqrt{N}}{s}
\end{equation}
Formula for t-statistic.
\begin{equation}
t = (\bar{x} - \mu)\frac{\sqrt{N}}{\sigma}
\hspace{20pt} \approx \hspace{20pt}
(\bar{x} - \mu)^2=\frac{t^2s^2}{N}
\end{equation}
Substituting.
\begin{equation}
t_{1} =
t +
\frac{\mu^3}{6\sigma^2s\sqrt{N}}+
\frac{\mu^3}{3\sigma^4}
\frac{t^2s^2}{N}
\frac{\sqrt{N}}{s}
\end{equation}
Simplifying
\begin{equation}
t_{1} =
t +
\frac{\mu^3}{6\sigma^2s\sqrt{N}}+
\frac{\mu^3}{3\sigma^4}
\frac{t^2s}{\sqrt{N}}
\end{equation}
\begin{equation}
t_{1} =
t +
\frac{\mu^3}{6\sigma^3\sqrt{N}}+
\frac{\mu^3t^2}{3\sigma^3\sqrt{N}}
\end{equation}
Therefore, formula at Skewness-adjusted t-statistic
\begin{equation} t_1 = t +
\frac{g}{6\sqrt{n}}+
\frac{gt^2}{3\sqrt{n}}
\hspace{20pt} where \hspace{20pt}
g = \frac{\hat{\mu_3}}{s^3}
\hspace{10pt} and \hspace{10pt}
g \approx \frac{{\mu_3}}{\sigma^3}
\end{equation}
