Question

Proportions (percentages) in a Z Distribution
A large population of scores from a standardized test are normally distributed with a population mean (μ) of 50 and a standard deviation (σ) of 5. Because the scores are normally distributed, the whole population can be converted into a Z distribution. Because the Z distribution has symmetrical bell shape with known properties, it’s possible to mathematically figure out the percentage of scores within any specified area in the distribution. The Z table provides the percentages corresponding to any Z score.

Helpful tips:
To answer the following questions, it helps to draw a Z distribution (bell curve) and place an individual’s Z score on the distribution as a visual aid. Use the Z table for converting between Z score and area (percentage) of the distribution.

a. John has a score of 55. What is John’s Z score?

b. What is the percentage of students that score lower than John?

c. Based on the Z table, if 1000 students take the test, how many of them would likely score above John’s score? (Round the answer to a whole number)

d. Tom has a score of 40. What is Tom’s Z score?

e. What is the percentage of students that score lower than Tom?

f. What is the percentage of students that score between John and Tom?

g. Based on the Z table, if 1000 students take the test, how many of them would likely score below Tom’s score?

h. Anna scores at the 99th percentile on this exam, what is her Z score?
Hint: A score at 99th percentile means 99% of the scores are below this score.

i. Based on the result of the previous question, what is Anna’s actual score on the exam?

j. What would be the median score on this exam?

Answer 1

µ = 50

σ = 5

Z = (X - µ) / σ

a) John's Z score = (55 - 50) / 5 = 1

b) P(Z < 1) = 0.8413 = 84.13%

c) Number of students who would likely to score above John's score = (1 - 0.8413) * 1000 = 158.7 = 159

d) Tom's Z score = (40 - 50) / 5 = -2

e) P(Z < -2) = 0.0228 = 2.28%

f) P(-2 < Z < 1) = P(Z < 1) - P(Z < -2) = 0.8413 - 0.0228 = 0.8185 = 81.85%

g) Number of students who would likely to score below Tom''s score = 0.0228 * 1000 = 22.8 = 23

h) Anna's Z score = Z_0.99 = 2.33

i) (X - 50) / 5 = 2.33

or, X = 50 + 5 * 2.33

or, X = 61.65

j) Median = 50