AP Stats Normal Distribution and Z-Scores 1 — Questions and Answers
Question 1: Which of the following is NOT a property of the normal distribution?
- It is symmetric about the mean.
- The mean, median, and mode are all equal.
- Approximately 95% of data falls within 2 standard deviations of the mean.
- It has a finite range determined by the minimum and maximum values. (Correct answer)
Correct answer: It has a finite range determined by the minimum and maximum values.
The normal distribution extends from negative infinity to positive infinity, so it does not have a finite range.
Question 2: A standardized test has scores that follow a normal distribution with mean 500 and standard deviation 100. What is the z-score for a student who scored 650?
- 1.0
- 1.5 (Correct answer)
- 2.0
- 0.65
Correct answer: 1.5
z = (650 - 500) / 100 = 150 / 100 = 1.5.
Question 3: According to the Empirical Rule, approximately what percentage of data in a normal distribution falls within one standard deviation of the mean?
- 50%
- 68% (Correct answer)
- 95%
- 99.7%
Correct answer: 68%
The Empirical Rule states that approximately 68% of data falls within one standard deviation of the mean in a normal distribution.
Question 4: A normal distribution has mean μ = 40 and standard deviation σ = 5. What value corresponds to z = -2?
- 30 (Correct answer)
- 35
- 45
- 50
Correct answer: 30
x = μ + zσ = 40 + (-2)(5) = 40 - 10 = 30.
Question 5: Which of the following describes a z-score of -1.8?
- The value is 1.8 standard deviations above the mean.
- The value is 1.8 standard deviations below the mean. (Correct answer)
- The value is 1.8 units below the mean.
- The value is in the 1.8th percentile.
Correct answer: The value is 1.8 standard deviations below the mean.
A negative z-score indicates the value is below the mean, and the magnitude (1.8) tells us how many standard deviations below.
Question 6: A class has test scores normally distributed with mean 75 and standard deviation 8. What is the probability that a randomly selected student scored above 83?
- 0.1587 (Correct answer)
- 0.3413
- 0.8413
- 0.6826
Correct answer: 0.1587
83 corresponds to z = (83-75)/8 = 1.0; P(Z > 1) = 1 - 0.8413 = 0.1587.
Question 7: If X is normally distributed with μ = 60 and σ = 12, which of the following is the area under the normal curve between x = 48 and x = 72?
- 0.50
- 0.3413
- 0.6826 (Correct answer)
- 0.9544
Correct answer: 0.6826
48 and 72 are each exactly one standard deviation from the mean (z = ±1), so the area is approximately 68.26%.
AP Stats Normal Distribution and Z-Scores 1
Which of the following is NOT a property of the normal distribution?