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Z-Scores Application

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Z-Scores Application
1. The graduate selection committee wants to select the top 10% of applicants. On a standardized test with a mean of 500 and a standard deviation of 100, what would be the cutoff score for selecting the top 10% of applicants, assuming that the standardized test is normally distributed?
2. The average commute time via train from the Chicago O'Hare Airport to downtown is 60 minutes with a standard deviation of 15 minutes. Assume that the commute times are normally distributed. What proportion of commutes would be:
a. longer than 80 minutes?
b. less than 50 minutes?
c. between 45 and 75 minutes?
3. Bob takes an online IQ test and finds that his IQ according to the test is 134. Assuming that the mean IQ is 100, the standard deviation is 15, and the distribution of IQ scores is normal, what proportion of the population would score higher than Bob? Lower than Bob?

Z-scores are statistical tools for determining relative performance and comparison across different tests and results (Steinberg, 2011). Z-score helps to re-scale any raw data into standard deviation units. Therefore, different combinations of raw results transform to easy-for-comparison data.
1. Solution: A) Find the z-score, which is separating 10% of the best results. B) Find the value of X
Z= (X-Mean)/ 100
Z for 10% or 0.1 from the table Appendix A (Steinberg, 2011) is 1.28
1.28 = (X-500)/100, Therefore X = (1.28 x 100) + 500 = 628
This score (628) separates 10% of the best scores from the rest results, in other words committee wants students with score 628 or higher.
2. A)Z(80)= (80-60)/15= 4/3= 1.33 (approximately)
From the Table Appendix A (Steinberg, 2011), for Z=1.33, P (x>80) is 0.092 commutes will be longer than 80 minutes. In other words, 9.2% of commutes expected to be longer than 80 minutes
Similarly:
B) Z(50) = (50-60)/15=-2/3=-0.67

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