Question

Find the values of z which cut off
(a) the top 10%
(b) the bottom 15%
(c) the middle 50%
of the standard Normal distribution.

Accepted Answer

In the standard normal distribution, the mean is 0 and the z scores represent how many standard deviations away from the mean a certain value is. The area underneath the curve of the normal distribution has a total value of 1, so the area underneath any part of the curve represents a proportion. This information can be used to find the z scores that correspond to certain parts of the standard normal distribution. Without knowing any other information, it will be easiest to use a z score table to find the correct z scores. One very important piece of information is that a z score table gives you the area under the curve to the left of the z score.
For the top 10%, you will be finding the z score that corresponds to an area of 0.9 under the curve. Use the z score table to find 0.9 in the body of the table, and use the left column and top row to find the z score. You will end up in between 1.28 and 1.29. In this case you can use the middle value so your answer will be a z score of 1.285. 
For the bottom 15%, use the table to find an area of 0.15. You should find a z score of -1.035.
For the middle 50%, remember that the mean of the standard normal distribution is 0 which falls right in the middle of the curve, so the z score is 0. You can use a table to verify this.
A z score table is included in the link below in the reference link.

Math

Find the values of z which cut off (a) the top 10% (b) the bottom 15% (c) the middle 50% of the standard Normal distribution.

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