Using z-scores to solve normal distribution problems

Score

Z Scores

Z-Scores

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?

The cut off score is determined by identifying the z-score associated with that percentile and computing the raw for that z-score (Aron, Coups, & Aron, 2011).

Z=

Transposing for X

The cut off score is 629

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:

Longer than 80 minutes?

Less than 50 minutes?

Between 45 and 75 minutes?

The proportion of commutes would be determined by computing the z-score for the relevant times and examining the proportion of the curve that are beyond, below and between the computed z-scores (Levin, Fox, &, Forde 2010).

a) Longer than 80 minutes

Using formula Z=X- ? + ?

Z=80-60/15

=20/15

=1.33

P= Area of the curve beyond 1.33

=0.0918 or 9.18%

b) Less than 80 minutes

Using formula Z=X-? + ?

Z=50-60/15

=10/15

=-0.67

P= Area of the curve below -0.67

P=0.2514

%=25.14%

Between 45-75

Using the formula Z=X-? + ?

Z (45) =45-60/15

=-1

Z (75) = 75-60/15

=1

P between -1 and 1

= 0.3413 + 0.3413

=0.6826

% =68.26%

Question 3