In a normal distribution, about 68 percent of values lie within one standard deviation of the mean.
Correct Answer
True
True. This is the first part of the empirical rule: about 68 percent of values fall between mu - sigma and mu + sigma. Within two and three standard deviations the shares are about 95 and 99.7 percent.
Category: Empirical Rule
Question 2
The total area under a normal curve equals 1.
Correct Answer
True
True. A normal curve is a probability density, so its total area is 1. That is why the area to the right of any value is 1 minus the area to its left.
Category: Normal Curve
Question 3
For the standard normal Z and any positive number a, P(Z < -a) equals P(Z > a).
Correct Answer
True
True. The standard normal curve is symmetric about 0, so the tail below -a is a mirror image of the tail above a. This is how left-tail areas are found from right-tail areas.
Category: Standard Normal
Question 4
On a test with mean 100 and standard deviation 15, a score of 85 earns a z-score of -1.
Correct Answer
True
True. The z-score is (85 - 100)/15 = -1, which means the score is one standard deviation below the mean.
Category: Z-Scores
Question 5
If X ~ N(mu, sigma), then (X - mu)/sigma has the standard normal distribution.
Correct Answer
True
True. Subtracting the mean centers the distribution at 0, and dividing by sigma rescales the spread to 1. The result is Z ~ N(0, 1), which is why one standard normal table or calculator command works for every normal distribution.
Category: Standard Normal
Question 6
For a normal random variable X, P(X < 10) equals P(X <= 10).
Correct Answer
True
True. The normal distribution is continuous, so the single point 10 has probability 0. Including or excluding an endpoint does not change an area.
Category: Normal Probabilities
Question 7
The 10th percentile of a normal distribution lies below the mean.
Correct Answer
True
True. Half of a normal distribution lies below the mean, so a value with only 10 percent of the area below it must be left of center. Its z-score is negative.
Category: Normal Percentiles
Question 8
If X ~ N(10, 2), the value 13 has a z-score of 1.5.
Correct Answer
True
True. The z-score is (13 - 10)/2 = 1.5, so 13 is 1.5 standard deviations above the mean.
Category: Z-Scores
Question 9
A normal curve gets closer and closer to the horizontal axis in both directions but does not touch it.
Correct Answer
True
True. The normal density is positive for every real number, so its tails extend without end in both directions, approaching the axis without reaching it. Values far from the mean are very unlikely but not impossible.
Category: Normal Curve
Question 10
Under a normal curve, the area to the right of any value x equals 1 minus the area to its left.
Correct Answer
True
True. The areas to the left and to the right of x together make up the whole area of 1. Calculators and tables usually give the left area, so right-tail areas are found by subtracting from 1.
Category: Normal Probabilities
Question 11
If X ~ N(40, 5), then P(X > 40) equals 0.25.
Correct Answer
False
False. A normal curve is symmetric about its mean, so exactly half of the area lies above it. P(X > 40) equals 0.5.
Category: Normal Probabilities
Question 12
The standard normal distribution has a mean of 1.
Correct Answer
False
False. The standard normal distribution has a mean of 0 and a standard deviation of 1. Its z-values are centered at 0.
Category: Standard Normal
Question 13
For a variable with mean 30 and standard deviation 5, the value 20 is 2 standard deviations above the mean.
Correct Answer
False
False. The z-score is (20 - 30)/5 = -2, so 20 is 2 standard deviations below the mean, not above it. Values below the mean have negative z-scores.
Category: Z-Scores
Question 14
If IQ scores are N(100, 15), a score of 130 is at about the 95th percentile.
Correct Answer
False
False. A score of 130 is two standard deviations above the mean. About 95 percent of scores lie between 70 and 130, leaving about 2.5 percent above 130, so 130 is near the 97.5th percentile.
Category: Normal Percentiles
Question 15
The empirical rule applies to any distribution, whether or not it is bell-shaped.
Correct Answer
False
False. The 68, 95 and 99.7 percent benchmarks describe normal, bell-shaped distributions. A strongly skewed distribution can have very different shares within one, two or three standard deviations.
Category: Empirical Rule
Question 16
Two normal distributions with the same mean must have the same standard deviation.
Correct Answer
False
False. A normal distribution is set by two separate parameters, mu and sigma. Two normal curves can share a mean and still have different standard deviations; the one with the larger sigma is wider and flatter.
Category: Normal Curve
Question 17
A z-score measures how far a value is from the mean in units of the variance.
Correct Answer
False
False. A z-score measures distance from the mean in units of the standard deviation, z = (x - mu)/sigma. The variance is in squared units and is not the yardstick.
Category: Z-Scores
Question 18
If X ~ N(20, 4), the value 28 lies three standard deviations above the mean.
Correct Answer
False
False. The z-score is (28 - 20)/4 = 2, so 28 lies two standard deviations above the mean. Three standard deviations above would be 20 + 3(4) = 32.
Category: Z-Scores
Question 19
If X ~ N(50, 10), the 84th percentile is about 40.
Correct Answer
False
False. About 50 percent of values lie below 50 and another 34 percent lie between 50 and 60, so about 84 percent lie below 60. The 84th percentile is about 60; the value 40 is near the 16th percentile.
Category: Empirical Rule
Question 20
A normal distribution is completely determined by its mean and its median.
Correct Answer
False
False. A normal distribution is determined by its mean and its standard deviation. The median of a normal distribution always equals the mean, so it adds no information about spread.
Category: Normal Curve