Answer:
z = -0.97
Explanation:
For this question, you should use a standard normal table, which gives the area under a curve between the mean and the z score. (I am assuming if you have been asked this question you should have been given one, but if not they are online).
In order to solve these questions, it is important to understand how the standard normal table works. For each z score (up to 3.49 ish), the table tells you how much area is between that z score and the mean.
That means, you normally cannot just use the chart, you have to do some extra math. Something to keep in mind to make this 'extra math' a little bit easier:
- the mean is directly in the middle of a distribution, meaning 50% (0.5) of the distribution area is below it, and 50% (0.5) is above it
The only way that this can truly make sense is if you draw out the distribution, and identify what area you are trying to find. I have attached the drawing for this question to hopefully help this explanation make more sense.
Now, back to the question. If we want to find the z score that has 16.6% of the values to the left, we know this will be the same as the z score that has 33.4% of the values between itself and the mean.
How? Well we know 50% of the area is below the mean. So, if we put a z value anywhere below the mean, we know the area below and above that z score must add up to equal 50%. (again, if this isn't making sense try to draw it out).
So, what z score has 33.4% of the values between itself and the mean? We just look at our standard normal table, which tells us that when z = 0.97, 0.3340 of the area is between z and the mean.
However, it is important to remember that the standard normal table does not include negative z scores, since normal distribution is symmetrical and the areas are the same above and below the mean. You just have to know that if a z score is below the mean, it will automatically be negative. This is again why it's important to draw out the question.
Therefore, the z-score that has 16.6% of the distribution area to its left is z = - 0.97.
(Since z scores are symmetrical, z = 0.97 has 16.6% of the values to its right.)
(It is important to note, some standard normal tables do not tell you the area between z and the mean, they just tell you the total area below the z score. I don't like using those, but if you are, not that every area given will be 0.5 bigger, since it includes all the area below the mean (50%).)