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Homework Help: Finding z-scores

  1. Nov 28, 2009 #1
    1. The problem statement, all variables and given/known data

    Students who are in the top 16% and top 2.5% will graduate with special honours. Where should the limits be set in terms of z-scores?

    Mean (xbar) = 2.7
    Standard deviation (s) = 0.5

    2. Relevant equations

    z = x - xbar / s


    -1 to 1 = 68% of all grades
    -2 to 2 = 95% of all grades
    -3 to 3 = 99.7% of all grades

    (don't know if that's ^ relevant)

    3. The attempt at a solution

    I tried multiplying 3 (the highest z score) by 0.84 (100 - 16) but that didn't seem to work. I know there's one thing I have to do before I use the z-score formula, but it's just not clicking.
  2. jcsd
  3. Nov 28, 2009 #2


    Staff: Mentor

    You need to look at a table of z-scores. You want the numbers z1 and z2 for which P(z < z1) = .84 and for which P(z < z2) = .975.

    After you get these numbers, you need to use the transformation formula to convert to x scores. The one you gave converts x-scores to z-scores. To get the transformation that goes from z-score to x-score, solve that formula for x.

    BTW, the formula you gave should be written as z = (x - mu)/sigma; i.e., you need parentheses, and it involves the population mean and population standard deviation, not the sample mean and standard deviation.
  4. Nov 29, 2009 #3
    I searched for a z-score table, but I don't know how to read it. On one axis they have z-scores and along the other axis are decimal values, but what are these values? How do I find 0.84 and 0.975 if they're not listed?
  5. Nov 30, 2009 #4


    Staff: Mentor

    One row will have 0.8. Look for the column with 4 in it. In the cell in the row with 0.8 and the column 4 is the probability that z < .84. The table I'm looking at has 0.7995 at that position.
  6. Nov 30, 2009 #5
    Alright, so 0.7995 corresponds to 0.84 and 0.83523 corresponds to 0.9775. But when I plug these values in and solve for x, I get virtually the same answers (3.1 and 3.11). How can that be?
  7. Nov 30, 2009 #6


    Staff: Mentor

    0.7995 is the probability that corresponds to a z-score of .84. Look in the body of the table for .9775 and find the z-score that corresponds to it?

    After you get the two z-scores, solve for x in the formula z = (x - mu)/sigma, and then use that new formula to calculate the two x-scores.
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