Statistics: Standard Normal Distribution

  1. 1. The problem statement, all variables and given/known data
    Find the Z value that corresponds to the given area.
    [​IMG]


    3. The attempt at a solution
    What I did was go to Table E and find the closest number to 0.0166 which was 0.0160, and the Z numbers were 0.04 and 0.0. I then added them up together to get the answer of -0.04 (negative since it's less than 0) but it's wrong. The actual answer was: -2.13. I'm suspecting it's because the area is negative infinite to Z, and that's where I messed up at. Normally if it's between the median (which is 0) and Z, i would just add up both numbers.

    Therefore, how do I solve this problem if the area is between infinite to Z?

    2. The problem statement, all variables and given/known data
    Find the z value to the left of the mean so that 98.87% of the area under the distribution curve lies to the right of it.

    I didn't understand the wording of this problem at all, but I did give my attempt at drawing the graph (not sure if it's correct).
    [​IMG]

    Actual Answer
    The answer of this problem is: -2.28


    Please help!!!
     
  2. jcsd
  3. fzero

    fzero 2,914
    Science Advisor
    Homework Helper
    Gold Member

    I have no idea what your Table E contains, but the area under the normal distribution is given by

    [tex]\Phi(z) = \frac{1}{\sqrt{2\pi}} \int_{\infty}^z e^{-x^2} dx = \frac{1}{2} \left(1+\text{erf}\left(\frac{z}{\sqrt{2}}\right)\right), [/tex]

    where [tex]\text{erf}(t)[/tex] is the error function. The error function can be computed by Wolfram Alpha or looked up in tables.
     
  4. 1) I'm not sure what table you're using but it should be one like this http://www.math.uh.edu/~bekki/CUIN 6342/zscoretable.pdf I suspect you're either using the wrong table or reading it wrong.

    What the chart shows is the z-score on the left (up to first decimal) and top (second decimal). The numbers in the body of the chart show the area under the normal distribution curve less than the indicated z-score. This is the same as saying the area to the left of the z-score.

    For example, say you were asked to find the z-score for which 0.54% of the area under the normal distribution curve lies to the left. The first thing to notice is that .54% = 0.0054. Then you look for 0.0054 in the body of the table. Once you find it, look at the corresponding value in the left-most column first to get -2.5 then look at the corresponding value in the top row to get .05. So the z-score that answers the question is -2.55

    Apply the same methodology to your question to get the answer.

    2) No, your interpretation is off. In some ways this is just the opposite of the first question where it was effectively asking you for the z value to the left of the mean so that 1.66% of the area under the distribution curve lies to the left of it. The "to the left of the mean" part indicates that the z value will be negative.

    So your diagram should be shaded red all the way to positive infinity. What you need to figure out is what area is to the left of the required z if 98.87% is to the right. Once you have that area you can do what you did in 1) to get the answer.
     
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