What is 'z' for an 88% confidence interval?

Feb 21, 2017

From software: $z = 1.554774 .$
From table lookup: $z \approx 1.56 .$

Explanation:

If we seek an 88% confidence interval, that means we only want a 12% chance that our interval does not contain the true value. Assuming a two-sided test, that means we want a 6% chance attributed to each tail of the $Z$-distribution. Thus, we seek the ${z}_{\alpha / 2}$ value of ${z}_{0.06}$.

This $z$ value at $\alpha / 2 = 0.06$ is the coordinate of the $Z$-curve that has 6% of the distribution's area to its right, and thus 94% of the area to its left. We find this $z$-value by reverse-lookup in a $z$-table.

Find the closest value in the table to 0.9400 as you can, then see what its row and column is. From observation, we see that 0.9394 and 0.9406 are in the table with $z$-values of 1.55 and 1.56 respectively, and so to err on the side of caution, we'll choose the value that gives us a wider interval, $z = 1.56 .$

Note: We could also get an answer from software like R, by typing the command $\text{qt(0.94, Inf)}$, which would give us a more precise value of 1.554774.