Find the z score such that the area under the standard normal curve to the left is 0.42

Answer 1

As always, there are several options to solve this. I'll show one with minimal technology and one with reasonably available technology.

The minimal technology approach:

This uses the standard normal table. One difficulty with this is that there are multiple versions out there. But it can be beneficial to try it this way, because this really forces you to get a visual representation of what is going on here. One thing that you hopefully notice right away is that 0.42 is less than 0.5, and so the right boundary for this area will be to the left of the mean of 0. This ought to tell you that the z-score will be negative. Some standard normal tables have an extra page for negative z-scores, and some don't. The truth is, you don't really need one that has negative z-scores if you have a good intuitive grasp of the normal curve, particularly its symmetry.

The standard normal table without negative z-scores only deals with the right half of the normal curve and, therefore, with cumulative areas (to the left of the z-score) that are 0.5 or greater. Well, if we can't find 0.42 on our chart, we can do the next best thing. We can find its mirror image on the right side of the curve. We can either subtract 0.42 from 1, or we can see how far left of 0.5 it is, and add that difference onto 0.5. Either way, we find that its counterpart on the right is 0.58. What this means is that, from negative infinity up to that z-score on the right side of the normal curve, the area is 0.58.

Then we can look on the normal table, find the area that is closest to 0.58, and then we'll have our corresponding z-score (which will be positive, but we'll make it negative so that it will match its left counterpart). The closest z-score we can find is -0.20.

Note: It may be tempting to complain that this isn't very many decimal places. Just bear in mind that these calculations were made by people before calculators existed. And for calculation difficulty, these functions are absolute stinkers! The people who did the calculations for the normal table deserve a medal!

The tech method

The TI-84 calculator has a VARS button, and on top of it is the command DISTR, which you get by pushing the 2ND button first. This brings down a menu of a whole host of random variable pdfs and cdfs. Since you know the area and are wanting the z-score, you would select invNorm(. Then for the area, you'll put 0.42. You want the mean to be 0, and the standard deviation to be 1, which are the default values that will be there if you haven't changed them. Then scroll down to Paste, and hit ENTER until you get an answer. The calculator can give you more decimal places than the table, so what you'll see is -0.2018934725.

Bonus Fact: Initially, I used neither method. Some time ago, I made a cardstock slide rule, to which I added several scales that it didn't have, including what is effectively a normal table on a slide rule.