Question 1134998
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 <a href = "https://stattrek.com/online-calculator/normal.aspx" target = "_blank">https://stattrek.com/online-calculator/normal.aspx</a>


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.


<img src = "http://theo.x10hosting.com/2019/021801a.jpg" alt="$$$" >


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.


<img src = "http://theo.x10hosting.com/2019/021802a.jpg" alt="$$$" >


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 <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">http://davidmlane.com/hyperstat/z_table.html</a>


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.


<img src = "http://theo.x10hosting.com/2019/021803.jpg" alt="$$$" >


here's the result of asking it to find me the z-score with an area of .1974 to the right of it.


<img src = "http://theo.x10hosting.com/2019/021804.jpg" alt="$$$" >