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

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In summary, students in the top 16% and top 2.5% will graduate with special honours. The limits for these percentages can be determined by using z-scores, and a table of z-scores can be used to find the corresponding decimal values. By solving for x in the formula z = (x - mu)/sigma, the x-scores can be calculated for the given probabilities.
  • #1
60051
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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.
 
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  • #2
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.
 
  • #3
Mark44 said:
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?
 
  • #4
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.
 
  • #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?
 
  • #6
60051 said:
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.
 

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

1. What is a Z-score and how is it used to determine graduation honours?

A Z-score is a statistical measure that shows how many standard deviations a data point is above or below the mean. In the context of graduation honours, Z-scores are used to rank students based on their academic performance. The higher the Z-score, the better the student's performance compared to their peers.

2. How is the top 16% of students determined using Z-scores?

To determine the top 16% of students, we first need to calculate the mean and standard deviation of the entire student population's Z-scores. Then, we can use a Z-score table or a statistical calculator to find the Z-score that corresponds to the 84th percentile (16% of students below). This Z-score will be used as the cut-off point to determine the top 16% of students.

3. What Z-score is needed to be in the top 16% for graduation honours?

The Z-score needed to be in the top 16% for graduation honours will vary depending on the distribution of Z-scores for the entire student population. In general, it will be the Z-score that corresponds to the 84th percentile, which can be calculated using a Z-score table or a statistical calculator.

4. How is the Z-score for top 16% different from the Z-score for top 2.5%?

The Z-score for top 16% is the cut-off point that separates the top 16% of students from the rest of the population. On the other hand, the Z-score for top 2.5% is the cut-off point that separates the top 2.5% of students from the rest of the population. This means that the Z-score for top 2.5% will be higher than the Z-score for top 16% as it represents a smaller proportion of students.

5. How can Z-scores be used to determine graduation honours for multiple courses or subjects?

Z-scores can be calculated and used for each individual course or subject to determine graduation honours. For example, a student may have a high Z-score in one subject but a lower Z-score in another. By considering the Z-scores for each subject, a more comprehensive and accurate representation of the student's academic performance can be achieved when determining graduation honours.

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