Find the critical value that corresponds to a confidence level of 80%.

Refer to the figure shown below.
It shows a portion of the standard normal distribution table with z-scores listed vertically on the left, with fractional (or decimal) portions on the top row. 

For each z-score, the portion of the area under the normal distribution curve is provided.

At the 80% confidence level, 40% of the total area lies to either side of the mean, shown shaded on the right side of the figure.

The closest area to 0.4 is 0.3997, and it is boxed within a black rectangle. This area corresponds to a z-score of 1.2 + 0.08 = 1.28.
This is the critical value of the z-score, denoted by z*, at the 80% confidence level.

The critical value is z* = 1.28.