Find Z-Scores Needed for Top 16% & 2.5% Graduation Honours

[60051]

Homework Statement

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

Homework Equations

z = x - xbar / s

and

-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)

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.

 

[Mark44]

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.

 

[60051]

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.

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?

 

[Mark44]

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.

 

[60051]

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?

 

[Mark44]

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?

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.