Question 1134998: Let z denote a variable that has a standard normal distribution. Determine the value z* to satisfy the following conditions. (Round all answers to two decimal places.)
P(z > z* or z < −z*) = 0.1974
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! in order for the probability of z being greater than z*, the area to the right of z* must be equal to .1974.
in order for the probability of z being less than -z*, the area to the left of z* must be equal to .1974
so you're looking for a z* that has an area to the right of it of .1974 and you're looking for a -z* that has an area to the left of it of .1974.
since the normal distribution is symmetric about the mean, z* and -z* have the same absolute value.
you can use a normal distribution calculator to get the answer for you or you can manually try to derive the answer using the z-score table.
the use of a calculator is recommended.
one such calculator can be found at https://stattrek.com/online-calculator/normal.aspx
this calculator works similar to the way you would use the normal distribution tables in that it gives you the area to the left of the indicated z-score.
in my first use of this calculator, i gave it a cumulative probability of .1974 and it told me that the z-score with an area under the normal distribution curve of .1974 to the left of it was -.851.
here is a display of the result.
to find the area of .1974 to the right of the z-score, i had to generate the area to the left of the z-score.
if the area to the right of the z-score is .1974, then the area to the left of the z-score is equal to 1 - .1974 = .8026.
since this calculator only gives you the area to the left of the z-score, then in my second use of this calculator, i gave it a cumulative probability of .8026 and it told me that the z-score with an area under the normal distribution curve of .8026 to the left of it was .851.
here is a display of the result.
i now have a z-score with an area of .1974 to the left of it equal to -.851 and a z-score with an area of .1974 to the right of it equal to .851.
since the normal distribution curve is symmetric about the mean, this is the result i expected to see.
this is also the answer to your problem.
z* is equal to .851
-z* is equal to -.851
while the calculator shown above can be used and it is similar to the procedure you would use to manually find the z-score, there is another online calculator that is much more useful and visually attractive.
that calculator can be found at http://davidmlane.com/hyperstat/z_table.html
using this calculator, i was able to find the answer directly by asking the calculator to give me the z-score with the area of .1974 to the left of it and by asking it to give me the z-score with the area of .1975 to the right of it.
here's the result of asking it to find me the z-score with an area of .1974 to the left of it.
here's the result of asking it to find me the z-score with an area of .1974 to the right of it.
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