Find the critical value $z_{\alpha /2}$ needed to construct a confidence interval with level $86\mathrm{\%}$.
Round the answer to two decimal places.

Step 1 ：Given a confidence level of $86\mathrm{\%}$, this leaves $14\mathrm{\%}$ of the data in the tails of the distribution. This $14\mathrm{\%}$ is split evenly between the two tails, so we have $7\mathrm{\%}$ in each tail.

Step 2 ：The value of $\alpha$ is the area in the tails, so $\alpha =0.14$. Therefore, $\alpha /2=0.07$.

Step 3 ：We want to find the $z$-score that leaves an area of $0.07$ in the upper tail of the standard normal distribution.

Step 4 ：We can use the standard normal distribution table or a calculator with a function for the standard normal distribution to find this $z$-score.

Step 5 ：Looking up $1-0.07=0.93$ in the standard normal distribution table, we find a $z$-score of approximately $1.48$.

Step 6 ：Therefore, the critical value $z_{\alpha /2}$ needed to construct a confidence interval with level $86\mathrm{\%}$ is approximately $1.48$.

Step 7 ：Checking our result, we see that a $z$-score of $1.48$ does indeed leave an area of $0.07$ in the upper tail of the standard normal distribution, so our result meets the requirements of the problem.

Step 8 ：So, the critical value $z_{\alpha /2}$ needed to construct a confidence interval with level $86\mathrm{\%}$ is approximately $1.48$ (rounded to two decimal places).

Step 9 ：$\boxed{{\displaystyle 1.48}}$ is the final answer.