Question 1167005
first look up in the z-score tables for an area to the left of the z-score of .2.


the z-score table that i used can be found at <a href = "https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf" target = "_blank">https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf</a>


you will find that you have two areas, one above .2 and one below .2


specifically, .....


z-score of -.84 has an area of .20045 to the left of it.
z-score of -.85 has an area of .19766 to the left of it.


you could pick a z-score of -.84 because the area of .20 is closest to it, or you could try to get a little closer by interpolation.


the difference between .20045 and .19766 is equal to .00279.


the difference between a z-score of -.84 and -.85 is equal to .01


.2 is different from .20045 by .00045 units.


the ratio between this and .00279 is equal to .1612903226.


apply this ratio to the difference between the z-scores of -.84 and -.85 = .01 to get .1612903226 * .01 = .0016129032.


since -.84 is more than .2 and since -.85 is less than .2, you have to get somewhere between -.84 and -.85.


add .0016129032 to .84 to get a z-score of -.8416129032.


that's your z-score to the closest you can get to it manually.


now you can use the z-score formula to find the raw score.


z = (x - m) / s
z is the z-score
x is the raw score
m is the raw mean
s is the standard deviation or the standard error.
in this case, s is the standard deviation.


when z = -.8416129032, the formula becomes:
-.8416129032 = (x - 12) / 1.1
multiply both sides of this equation by 1.1 and then add 12 to both sides of this equation to get x = 11.07422581.


that's your answer to the closest you can get with manual interpolation.


it's also the hard way to get your answer.


the easy way to get your answer is by using a normal distribution z-score calculator.


one that i found online is the omni normal distribution calculator.


it can be found at <a href = "https://www.omnicalculator.com/statistics/normal-distribution" target = "_blank">https://www.omnicalculator.com/statistics/normal-distribution</a>


you clear it by hitting the round arrow button in the middle of the display under the input and results area.


for your problem, you would enter the following:


mean = 12
standard deviation = 1.1
p(x < X) = .2


the calculator will automatically fill in the rest of the fields.


here's a display of what i got after making the above entries.


<img src = "http://theo.x10hosting.com/2020/101101.jpg" >


it told me that my raw score is equal to 11.07407 and that my z-score is -.841756.


that compares favorably to the z-score that i derive manually of -.8416129032.


it also compares favorable to the raw score that i derived manually of 11.07422581.


it also does it with a lot less effort.


unless you are instructed to do it manually, use the calculator.


i also have a normal distribution calculator in  my TI-84 Plus.
it told me that the raw score is 11.07421664.


a word about these calculators; .....


they don't all use the same algorithms, so there will be slight differences between them as to what the answer should be.


usually the differences pop up after 2 or 3 decimal places.


in most cases, your answer to 2 decimal places is close enough.