Normal Distribution and Z-Scores Flashcards

Which of the following is NOT a property of the normal distribution?

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.

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?

Answer: 1.5

z = (650 - 500) / 100 = 150 / 100 = 1.5.

According to the Empirical Rule, approximately what percentage of data in a normal distribution falls within one standard deviation of the mean?

Answer: 68%

The Empirical Rule states that approximately 68% of data falls within one standard deviation of the mean in a normal distribution.

A normal distribution has mean μ = 40 and standard deviation σ = 5. What value corresponds to z = -2?

Answer: 30

x = μ + zσ = 40 + (-2)(5) = 40 - 10 = 30.

Which of the following describes a z-score of -1.8?

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.

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?

Answer: 0.1587

83 corresponds to z = (83-75)/8 = 1.0; P(Z > 1) = 1 - 0.8413 = 0.1587.

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?

Answer: 0.6826

48 and 72 are each exactly one standard deviation from the mean (z = ±1), so the area is approximately 68.26%.