Timothy K. answered • 09/18/19

Former Adjunct Math and Statistics Professor

It sounds like your teacher is asking you to understand what the z-score is representing.

You should start by drawing the bell curve and giving a rough estimate of where you would find the z-score that has 12% of area to the left of it (the lowest 12%). Since the median (the middle value that has 50% below and 50% above it) is at zero on a normal distribution, we can guess that it will be below zero. My picture would have a vertical line toward the left side of the curve with the area to the left filled in. I would also suggest labeling that are with a 12% to show that you understand what we are looking at.

Then we need to find the z-score that would give us that area. There are two main way to do this. One is to use a calculator/app that can give us the z-score if we give it the area under the curve we are looking for. The other is to use a table that is probably in the back of your textbook or possible in an additional resource that your instructor gave you.

If you have a TI graphing calculator, the "invNorm(" under the DISTR key ([2nd] and [VARS]) would be where you need to look. This function requires 1 input and has 2 other optional inputs. It should look like

invNorm(α [,μ , σ]) = Z

The α is the area to the left that you are looking to find a z-score for. If you are dealing with a normal distribution that has a mean that isn't zero and/or a standard deviation that is not one, then you will need values for μ (the mean) and σ (the standard deviation.) If you are dealing with a standard normal distribution, you can just put in α. It would look like

invNorm(α) = z

or more specifically for your case

invNorm(0.12)

Remember that the areas are percentages represented as decimals. You will need to convert from 12% to 0.12

If you are using a table, then you are looking for the area on the table that is closest to 0.1200. Most z-score tables that I have used for my students are areas to the left of the given z-score. If you take a look at this pdf, it is a pretty in depth z-score table.

https://www.stat.tamu.edu/~lzhou/stat302/standardnormaltable.pdf

On the -1.1 row, you will find 0.12100 next to 0.11900. They are in the ".07" and ".08" columns respectively. The area of 0.12100 is in the -1.1 row and the .07 column, so the z-score of -1.17 would be the z-score that cuts off the lowest 12.1% (0.12100 as a percent), while -1.18 would be the z-score that cuts off the lowest 11.9% (0.11900 as a percent). Since the percent that we are looking for is exactly halfway between the two, it would be appropriate to go halfway between the two z-scores at z = -1.175