Z - Scores

Z - Scores makes it easy to compare scores from distributions using different scales

Formula #1

Formula #2

Formula #3 for raw score computation is defined by 

Formula #4 for Standard Error

Trying out problems in link 

Problem 2. Suppose X is a normal random variable with a mean of 120 and a standard deviation of 20. Determine the probability that X is greater than 135.

Mean = 120
SD = 135
Z score = (135-120)/20 = 0.75

z score from attached link

Find P(Z < 0.75) = 0.7734
1 - 0.7734 = 0.2266

Problem 4. If the test scores of 400 students are normally distributed with a mean of 100 and a standard deviation of 10, approximately how many students scored between 90 and 110?
Mean = 100
SD = 10

For x = 90, z = (90-100)/10 = -1
For x = 110, z = (110-100)/10 = 1
For Z (< -1), 

= 0.1587
For Z (<1), 

= 0.8413
= 0.8413-0.1587
= 0.6826

Multiply this percentage by 400. After rounding, we get 273 students.

Problem 16. A traffic study shows that the average number of occupants in a car is 1.5 and the standard deviation is .35. In a sample of 45 cars, find the probability that the mean number of occupants is greater than 1.6.

Mean = 1.5
SD = .35

Applying Formula #2

P(mean > 1.6) = 1- P(mean < 1.6)
Z(1.6) = ((1.6-1.5)*sqrt(45)) / 0.35
          = 1.916

P(Z<1.6) =  0.9719
P(Z>1.6) = 1- 0.9719 = 0.0281

Happy Learning!!!