1. How to Find Data Values (X) Given Specific Probabilities
1. The scores on a test have a mean of 100 and a standard deviation of 8. The variable is also normally distributed. If only the top 15% of the applicants are selected, find the cutoff score.
σ=8
Top 15%
μ=100

2. 1. The scores on a test have a mean of 100 and a standard deviation of 8. The variable is also normally distributed. If only the top 15% of the applicants are selected, find the cutoff score.
Draw or visualize the distribution and data
Convert your percentage to a z-score
σ=8
Top 15%
μ=100

3. σ=8
Top 15%
μ=100
Draw or visualize the distribution and data
Convert your percentage to a z-score
Z-score is

4. Draw or visualize the distribution and data
Convert your percentage to a z-score
Convert z-score to data point (X)
σ=8
Top 15%
μ=100
X=z σ + μ X=
X=108.29

5. 1. The scores on a test have a mean of 100 and a standard deviation of 8. The variable is also normally distributed. If only the top 15% of the applicants are selected, find the cutoff score.
Draw or visualize the distribution and data
Convert your percentage to a z-score
Convert z-score to data point (X)
Interpret your data point in light of the problem situation
X=108.29
This means the top 15% of scores will be scores of and above.

6. 2. A contractor decided to build homes that will include the middle 80% of the market. If the average size (in square feet) of homes built is 1,810, find the maximum and minimum sizes of the homes the contractor should build. Assume that the standard deviation is 92 square feet and the variable is normally distributed.
Draw or visualize the distribution and data
σ=92
Middle 80%
μ=1,810

7. σ=92
Middle 80%
μ=1,810
Draw or visualize the distribution and data
Convert your percentage to a z-score

8. X=z σ + μ X=−1.281551567 92 +1,810 X=1,692 square feet
Draw or visualize the distribution and data
Convert your percentage to a z-score
Convert z-score to data point (X)
σ=92
Middle 80%
μ=1,810
X=z σ + μ
X=− ,810
X=1,692 square feet

9. X=z σ + μ X=1.281551567 92 +1,810 X=1,928 square feet
Draw or visualize the distribution and data
Convert your percentage to a z-score
Convert z-score to data point (X)
σ=92
Middle 80%
μ=1,810
X=z σ + μ
X= ,810
X=1,928 square feet

10. X=1,692 square feet X=1,928 square feet
2. A contractor decided to build homes that will include the middle 80% of the market. If the average size (in square feet) of homes built is 1,810, find the maximum and minimum sizes of the homes the contractor should build. Assume that the standard deviation is 92 square feet and the variable is normally distributed.
X=1,692 square feet
Draw or visualize the distribution and data
Convert your percentage to a z-score
Convert z-score to data point (X)
Interpret your data point in light of the problem situation
X=1,928 square feet
This means the middle 80% of the market includes homes between 1,692 square feet and 1,928 square feet