The z-score is (x - mu)/sigma = (60 - 50)/5 = 2, so 60 lies two standard deviations above the mean. The value 10 is the raw distance from the mean, not divided by sigma. The value -2 has the wrong sign, since 60 is above the mean. The value 0.5 divides sigma by the distance instead of the other way around.
Category: Z-Scores
Question 2
A value has a z-score of -1.5. What does this tell you about the value?
It is 1.5 units below the mean
It is 1.5 standard deviations below the mean
✓
It is 1.5 standard deviations above the mean
It is 15 percent below the mean
Correct Answer
It is 1.5 standard deviations below the mean
A z-score counts standard deviations from the mean, and a negative sign means the value is below the mean. It is 1.5 units below the mean confuses standard deviations with the original units. It is 1.5 standard deviations above the mean has the direction reversed. It is 15 percent below the mean treats the z-score as a percentage.
Category: Z-Scores
Question 3
X ~ N(70, 8). Which value of x has a z-score of 1.5?
58
71.5
82
✓
78
Correct Answer
82
Converting back uses x = mu + z sigma = 70 + 1.5(8) = 82. The value 58 subtracts 12 instead of adding it, which is the value with z = -1.5. The value 71.5 adds the z-score to the mean without multiplying by sigma. The value 78 adds one standard deviation instead of 1.5.
Category: Z-Scores
Question 4
IQ scores are normally distributed with mean 100 and standard deviation 15. By the empirical rule, about what percent of scores fall between 70 and 130?
68%
99.7%
90%
95%
✓
Correct Answer
95%
The values 70 and 130 are two standard deviations below and above the mean, and the empirical rule puts about 95% of a normal distribution within two standard deviations. 68% is the share within one standard deviation, from 85 to 115. 99.7% is the share within three standard deviations, from 55 to 145. 90% is not one of the empirical rule's benchmarks.
Category: Empirical Rule
Question 5
X ~ N(100, 15). By the empirical rule, about what percent of values lie above 115?
16%
✓
32%
34%
84%
Correct Answer
16%
About 68% of values lie between 85 and 115, leaving 32% outside, split evenly into 16% in each tail, so about 16% lie above 115. 32% counts both tails instead of one. 34% is the share between 100 and 115. 84% is the share below 115.
Category: Empirical Rule
Question 6
What are the mean and standard deviation of the standard normal distribution Z?
Mean 1 and standard deviation 0
Mean 0 and standard deviation 1
✓
Mean 0 and standard deviation 0.5
Mean 100 and standard deviation 15
Correct Answer
Mean 0 and standard deviation 1
The standard normal distribution, written Z ~ N(0, 1), has mean 0 and standard deviation 1, and it is the distribution of z-scores. Mean 1 and standard deviation 0 swaps the two numbers, and a standard deviation of 0 would mean no spread at all. Mean 0 and standard deviation 0.5 has the right center but too little spread. Mean 100 and standard deviation 15 describes a common IQ scale, not Z.
Category: Standard Normal
Question 7
Z is standard normal and P(Z < 1) = 0.8413. What is P(Z > 1)?
0.8413
0.3413
0.1587
✓
0.6826
Correct Answer
0.1587
The total area is 1, so the area to the right of 1 is 1 - 0.8413 = 0.1587. The value 0.8413 is the area to the left of 1. The value 0.3413 is the area between 0 and 1. The value 0.6826 is the area between -1 and 1.
Category: Normal Probabilities
Question 8
Given that P(Z < 1.5) = 0.9332, use symmetry to find the area to the left of z = -1.5.
0.9332
0.4332
0.8664
0.0668
✓
Correct Answer
0.0668
By symmetry, the area to the left of -1.5 equals the area to the right of 1.5, which is 1 - 0.9332 = 0.0668. The value 0.9332 is the area to the left of positive 1.5. The value 0.4332 is the area between 0 and 1.5. The value 0.8664 is the area between -1.5 and 1.5.
Category: Normal Probabilities
Question 9
For the standard normal Z, P(Z < 1) = 0.8413 and P(Z < 2) = 0.9772. Find P(-1 < Z < 2).
0.8185
✓
0.1359
0.9772
0.6826
Correct Answer
0.8185
By symmetry, P(Z < -1) = 1 - 0.8413 = 0.1587, so P(-1 < Z < 2) = 0.9772 - 0.1587 = 0.8185. The value 0.1359 is 0.9772 - 0.8413, the area between 1 and 2. The value 0.9772 is the whole area to the left of 2, including the part below -1. The value 0.6826 is the area between -1 and 1.
Category: Normal Probabilities
Question 10
Scores X ~ N(80, 5). Given that P(Z < 2) = 0.9772, what proportion of scores are above 90?
0.9772
0.0228
✓
0.0500
0.4772
Correct Answer
0.0228
The score 90 has z = (90 - 80)/5 = 2, and the area to its right is 1 - 0.9772 = 0.0228. The value 0.9772 is the proportion below 90. The value 0.0500 is the empirical rule's rough share in both tails beyond two standard deviations, not the single upper tail. The value 0.4772 is the area between the mean and 90.
Category: Normal Probabilities
Question 11
Scores X ~ N(500, 100). The 90th percentile of the standard normal is z = 1.28. What is the 90th percentile of X?
372
501.28
628
✓
600
Correct Answer
628
A percentile converts with x = mu + z sigma = 500 + 1.28(100) = 628. The value 372 subtracts 128, which gives the 10th percentile. The value 501.28 adds the z-score without multiplying by sigma. The value 600 adds one full standard deviation, which is about the 84th percentile.
Category: Normal Percentiles
Question 12
Which description fits the graph of a normal distribution?
Skewed right with its peak at zero
Flat across an interval from a to b
Skewed left with a long left tail
Symmetric and bell-shaped about the mean
✓
Correct Answer
Symmetric and bell-shaped about the mean
A normal curve is a symmetric, bell-shaped curve centered at mu, with tails extending in both directions. Skewed right with its peak at zero describes an exponential distribution. Flat across an interval from a to b describes a uniform distribution. Skewed left with a long left tail describes a lopsided shape, not a normal one.
Category: Normal Curve
Question 13
Ana scored 85 on a test with mean 75 and standard deviation 5. Ben scored 90 on a test with mean 80 and standard deviation 10. Whose score is better relative to their class?
Ana's, with a z-score of 2
✓
Ben's, with a z-score of 1
They are equally good
Ben's, because 90 is higher
Correct Answer
Ana's, with a z-score of 2
Ana's z-score is (85 - 75)/5 = 2, and Ben's is (90 - 80)/10 = 1, so Ana is farther above her class mean in standard deviation units. Ben's, with a z-score of 1, is the smaller standardized score. They are equally good is wrong because both are 10 points above the mean but the spreads differ. Ben's, because 90 is higher, compares raw scores from tests with different means and spreads.
Category: Z-Scores
Question 14
How does increasing sigma, with mu held fixed, change the graph of a normal distribution?
It becomes narrower and taller
It becomes wider and flatter
✓
It shifts to the right
It becomes skewed to the right
Correct Answer
It becomes wider and flatter
A larger standard deviation spreads the area over a wider range, and because the total area stays 1, the peak must drop. It becomes narrower and taller is what happens when sigma decreases. It shifts to the right is the effect of increasing mu. It becomes skewed to the right cannot happen, since every normal curve is symmetric.
Category: Normal Curve
Question 15
The mean of a normal distribution rises from 50 to 60 while its standard deviation stays the same. What happens to its graph?
It becomes 10 units wider
It becomes taller
It moves 10 units to the right
✓
It moves 10 units to the left
Correct Answer
It moves 10 units to the right
The mean sets the center, so raising mu by 10 slides the whole curve 10 units to the right without changing its shape. It becomes 10 units wider would require a larger standard deviation. It becomes taller would require a smaller standard deviation. It moves 10 units to the left is the effect of lowering the mean.
Category: Normal Curve
Question 16
In a normal distribution, how are the mean, median and mode related?
The mean is greater than the median
The median is greater than the mean
The mode is greater than the mean
They are all equal
✓
Correct Answer
They are all equal
A normal curve is symmetric and has a single peak at its center, so the mean, median and mode all fall at mu. The mean is greater than the median describes a right-skewed distribution. The median is greater than the mean describes a left-skewed distribution. The mode is greater than the mean also describes a left-skewed shape.
Category: Normal Curve
Question 17
X ~ N(60, 10). Using z = 1.96, which interval contains the middle 95 percent of values?
40.4 to 79.6
✓
50.0 to 70.0
34.2 to 85.8
20.8 to 99.2
Correct Answer
40.4 to 79.6
The middle 95 percent lies within 1.96 standard deviations of the mean: 60 - 1.96(10) = 40.4 and 60 + 1.96(10) = 79.6. The interval 50.0 to 70.0 is only one standard deviation on each side, about 68 percent. The interval 34.2 to 85.8 uses z = 2.576, the middle 99 percent. The interval 20.8 to 99.2 doubles the margin by mistake.
Category: Normal Percentiles
Question 18
In any normal distribution, which value has a z-score of 0?
Its standard deviation, sigma
Its center, mu
✓
Its largest observed value
The value one unit above mu
Correct Answer
Its center, mu
A z-score of 0 means a distance of 0 standard deviations from mu, so the value is the center of the distribution, mu itself. Its standard deviation, sigma, is a measure of spread, not a location with z = 0. Its largest observed value would have a large positive z-score. The value one unit above mu has z = 1/sigma, which is not 0.
Category: Z-Scores
Question 19
Which variable is most reasonably modeled by a normal distribution?
Minutes until the next random customer arrival
Number of heads in three coin flips
Heights of adult women in a large population
✓
Household incomes in a large country
Correct Answer
Heights of adult women in a large population
Heights of adult women in a large population are a classic example of a roughly symmetric, bell-shaped measurement. Minutes until the next random customer arrival are skewed right and fit an exponential model. Number of heads in three coin flips is a discrete count with only four possible values. Household incomes in a large country are strongly skewed right by a few very high earners.
Category: Normal Curve
Question 20
X ~ N(100, 15). The quartiles of Z are -0.67 and 0.67. What is the interquartile range of X? Round to one decimal place.
10.1
1.3
30.0
20.1
✓
Correct Answer
20.1
The quartiles of X are 100 - 0.67(15) = 89.95 and 100 + 0.67(15) = 110.05, so the interquartile range is 110.05 - 89.95 = 20.1. The value 10.1 is only the distance from the mean to one quartile. The value 1.3 is the distance between the z quartiles, not converted to the units of X. The value 30.0 is two standard deviations.
Category: Normal Percentiles