Describing the Distribution of Quantitative Variables - AP Statistics

1. A real estate agent is analyzing the prices of homes in a neighborhood that contains a few extremely expensive mansions among otherwise moderately priced houses. Which of the following statements is most likely to be true about the distribution of home prices?
   Answer: The mean price is greater than the median price, and the distribution is skewed to the right. The correct answer is A. The few extremely expensive mansions are high outliers. These outliers will pull the mean price up but will have little effect on the median price. When the mean is greater than the median, the distribution is typically skewed to the right.
2. A dataset of the last digit of 1,000 telephone numbers is collected. Which of the following best describes the likely shape of the distribution of these digits?
   Answer: Approximately uniform, because each digit from 0 to 9 is expected to occur with roughly equal frequency. The correct answer is C. In a large, random set of telephone numbers, there is no reason to believe any one digit (0-9) would appear more frequently than another as the last digit. Therefore, the distribution of these digits is expected to be approximately uniform, with each digit having a roughly equal count.
3. A teacher recorded quiz scores (out of 10 points) for 35 students. The dotplot below shows the distribution of scores. Which statement best describes the distribution?
   Answer: The distribution is left-skewed, with many high scores and a tail toward lower scores. This question tests identifying distribution shape from a dotplot of quiz scores. The dotplot shows many students scored high (8-10 points), with progressively fewer students earning lower scores, creating a tail extending toward the left (lower scores). This pattern indicates left-skewness, where the tail points toward smaller values. The distribution is not right-skewed (choice B) because the tail extends left not right, not symmetric (choice C) despite having a mode at 8, not uniform (choice D) because frequencies vary greatly, and not bimodal (choice E) since one low score doesn't create a second mode. Quiz scores often show left-skewness when most students perform well but a few struggle.
4. When describing the distribution of a quantitative variable, why is it essential to address unusual features like outliers or gaps?
   Answer: They can significantly influence non-resistant summary statistics and may represent important contextual information. The correct answer is B. Unusual features are critical to describe because they can have a strong effect on summary statistics like the mean and standard deviation. They can also provide important insights into the data, representing unique cases or subgroups that should be investigated rather than ignored.
5. What does this shape imply about the air quality in the city?
   Answer: Air quality is generally poor (high AQI), with a few days of exceptionally good air quality (low AQI). The correct answer is A. A distribution that is skewed to the left has its tail on the left side, meaning there are a few unusually low data values. The bulk of the data is on the right side (higher values). Since high AQI means poor air quality, the bulk of the days have poor air quality, and the few unusual days in the left tail have exceptionally good (low AQI) air quality.
6. The lifetimes of a certain brand of light bulb are claimed to be approximately normally distributed. A random sample of these bulbs is tested. Which description of the sample data's distribution would best support the manufacturer's claim?
   Answer: Approximately symmetric, unimodal, and bell-shaped with no outliers. The correct answer is C. A normal distribution is characterized by being symmetric, unimodal (having one peak), and bell-shaped. A sample from a normally distributed population should exhibit these same characteristics. The other descriptions are for distinctly non-normal distributions.
7. A statistics teacher records the height of all students in a large introductory class. The distribution of heights is found to have a single peak, with the left side being a near mirror image of the right side. Which of the following is the most accurate description of the distribution's shape?
   Answer: Unimodal and approximately symmetric. The correct answer is A. The term 'single peak' corresponds to a unimodal distribution. The description 'the left side being a near mirror image of the right side' corresponds to an approximately symmetric distribution. Therefore, the shape is best described as unimodal and approximately symmetric.
8. A consumer group tests the battery life of a sample of smartphones. The group's report states that the distribution of battery life is "roughly symmetric with a prominent peak around 12 hours." What does the term "prominent peak" signify in the description of this distribution?
   Answer: That the distribution is unimodal, with a central tendency near 12 hours. The correct answer is D. In the context of a distribution's shape, a 'peak' refers to a mode, which is a value that occurs with high frequency. A 'prominent peak' indicates a clear mode and signifies that the distribution is unimodal. It describes the location where the data are most concentrated, which is a measure of central tendency.
9. A runner tracked the time (in minutes) it took to complete a particular 5K route on 26 different days. The dotplot below shows the distribution of times. Which statement best describes the distribution?
   Answer: The distribution is roughly symmetric and unimodal, centered around about 28–30 minutes, with no clear outliers. This question asks you to describe a distribution from a dotplot of running times. The dotplot shows times clustered around 28-30 minutes, with values spreading out fairly evenly on both sides (from 22 to about 36 minutes). The data appears to have a single peak and decreases relatively symmetrically on both sides, with no values standing far apart from the main cluster. This creates a roughly symmetric, unimodal distribution with no clear outliers. When assessing symmetry, imagine folding the distribution at its center - symmetric distributions look similar on both sides.
10. A manager recorded the number of minutes late for each of 30 employees on a particular Monday. The dotplot below shows the distribution of minutes late (0 means on time). Which statement best describes the distribution?
    Answer: The distribution is right-skewed, with most employees between 0 and 6 minutes late and a possible high outlier around 18–20 minutes. This question tests your ability to describe the shape, center, and unusual features of a distribution from a dotplot. Looking at the dotplot of minutes late, we see most employees clustered between 0 and 6 minutes late, with the frequency decreasing as lateness increases, and one or two employees arriving much later (around 18-20 minutes). This creates a right-skewed distribution because the tail extends toward larger values. The single employee who was very late could be considered a high outlier. When describing distributions, always check for symmetry versus skewness by looking at how the data trails off on each side of the center.
11. The distribution of the number of hours students at a large high school spent on homework last week is unimodal and skewed to the right. Which of the following is a plausible explanation for the shape of this distribution?
    Answer: Most students spend a relatively small to moderate number of hours, but a few students spend an exceptionally large number of hours. The correct answer is C. A distribution that is skewed to the right has a main body of data on the left (lower values) and a tail extending to the right (higher values). This shape corresponds to a situation where most students spend a small to moderate amount of time on homework, and a few students spend a very large amount of time, creating the right tail.
12. A researcher studying the weights of a species of adult bird finds two distinct subgroups within the data: one for males and one for females, where males are typically heavier than females. Which of the following features would most likely be observed in the overall distribution of weights for the combined group of birds?
    Answer: Two distinct clusters of data, likely resulting in a bimodal shape. The correct answer is D. Since there are two subgroups with different typical weights, the combined distribution will show these two groups as separate clusters of data. This will likely result in a bimodal distribution, with one peak corresponding to the typical female weight and another, higher peak for the typical male weight.
13. A city planner samples 50 commute distances (in miles) for workers who drive to downtown. A histogram of the distances shows most commutes between 2 and 12 miles, with frequencies decreasing steadily as distance increases, and a tail extending out to 40 miles. Which statement best describes the distribution?
    Answer: The distribution is right-skewed because there are many short commutes and fewer long commutes, creating a tail to the right. In AP Statistics, describing commute distances via histograms requires analyzing shape, center, spread, and outliers for driving data. The distribution is right-skewed with most commutes short (2-12 miles) and a decreasing tail to longer distances up to 40 miles. Choice C is a distractor, labeling it left-skewed because small distances are near 0, but left skew has the tail on the low end, not high. Key lesson: Right skew is common in distances or times with minimums near zero and long tails; bimodality needs two peaks, not just varied lengths. Uniformity requires flat bars, symmetry balanced tails, and histograms' bin widths don't determine shape alone. Assess frequency patterns for skew.
14. A track coach recorded the 100-meter dash times (in seconds) for 28 athletes. The dotplot below shows the distribution. Which statement best describes the distribution?
    Answer: The distribution is approximately symmetric and unimodal, centered around about 13 seconds. This question asks about describing a distribution of sprint times from a dotplot. The data appears to cluster around 13 seconds with roughly equal numbers of times slightly faster and slightly slower, creating a bell-shaped pattern. This indicates an approximately symmetric, unimodal distribution. The distribution is not right-skewed (choice B) or left-skewed (choice C) because there's no pronounced tail in either direction, not uniform (choice D) because values cluster in the middle rather than spreading evenly, and not bimodal (choice E) which would require two distinct peaks. Symmetric distributions have similar shapes on both sides of the center and are common for athletic performance data.
15. A biology teacher recorded the number of seeds germinated (out of 50) for each of 32 trays under the same conditions. The dotplot below shows the distribution of germinated-seed counts. Which statement best describes the distribution?
    Answer: The distribution is approximately symmetric and unimodal, centered around about 41–43 seeds, with one unusually low tray near 28. This question asks about describing a distribution of seed germination counts from a dotplot. The data shows most trays had 41-43 seeds germinate out of 50, forming a symmetric bell shape around this center, with one unusually low value at 28 that stands apart as an outlier. This creates an approximately symmetric, unimodal distribution with an outlier. The distribution is not right-skewed (choice A) or left-skewed (choice B) because the main cluster is symmetric, not uniform (choice D) because values cluster rather than spread evenly, and not bimodal (choice E) because an outlier doesn't create a second mode. When describing distributions, identify outliers separately from the overall shape of the main data cluster.
16. A company tracks the number of customer support tickets received each day for 30 days. The dotplot shows most days between 18 and 26 tickets, but there are two days with 55 and 58 tickets, far from the rest. Which statement best describes the distribution?
    Answer: The distribution is right-skewed with two high outliers around 55–58 tickets. Describing distributions in AP Statistics for variables like daily tickets requires evaluating dotplots for shape, center, spread, and outliers. The distribution is right-skewed, with most values between 18-26 tickets but two high outliers at 55 and 58, creating a tail to the right. Choice C distracts by calling it left-skewed based on low values below 18, but the tail is actually on the high end. Mini-lesson: Outliers are extreme values that don't fit the pattern; right skew means the mean is pulled rightward by high extremes. Symmetry has balanced tails, while bimodality shows two modes—here, the two high points aren't a separate peak but outliers. Uniformity implies even spread without clustering.
17. What component of a complete description is missing from the student's statement?
    Answer: A description of unusual features, such as potential outliers. The correct answer is C. The student's description includes shape ('skewed to the right'), center ('median salary of $55,000'), and spread ('range from$30,000 to $1,200,000'). However, it does not explicitly mention any unusual features. Given the skew and the very high maximum salary, a comment on potential outliers would be a necessary part of a complete description.
18. A distribution of data is approximately symmetric and unimodal with no significant outliers. Which statement accurately describes the expected relationship between the mean and median?
    Answer: The mean and the median will be approximately equal. The correct answer is C. In a symmetric distribution, the data are balanced around the center. As a result, the mean (the balance point) and the median (the middle value) are located at the same central position and will be approximately equal.
19. A gym records the number of minutes 60 members spent on a treadmill during one visit. The dotplot below shows the distribution of times (in minutes). Which statement best describes the distribution? Dotplot counts by minute: - 10: 1 - 12: 1 - 14: 2 - 16: 3 - 18: 4 - 20: 6 - 22: 7 - 24: 8 - 26: 7 - 28: 6 - 30: 5 - 32: 4 - 34: 3 - 36: 2 - 38: 1 (Each count represents how many members had exactly that time.)
    Answer: The distribution is approximately symmetric and unimodal, centered around about 24–26 minutes, with no clear outliers. This question assesses the skill of describing the distribution of a quantitative variable, specifically treadmill times, by examining shape, center, spread, and outliers in a dotplot. The distribution is approximately symmetric and unimodal, with the peak around 24-26 minutes and no clear outliers, as the data mirror evenly on both sides without isolated points. A common distractor is choice A, which incorrectly identifies the distribution as strongly right-skewed solely based on the largest value being 38 minutes, ignoring the overall symmetry. In describing distributions, remember to look for symmetry where the left and right sides are mirror images, skewness where there's a tail on one side, and modality based on the number of peaks. Outliers are values that stand apart from the main body of data, which aren't present here. Always consider the entire pattern rather than isolated points.
20. A local gym records the ages of its members. The data reveal a large number of members between 20 and 35 years old and another large number of members between 55 and 70 years old, with very few members in between these two ranges. Which unusual feature is present in this distribution?
    Answer: A gap, representing the range of ages where there are very few members. The correct answer is B. A gap is a region in a distribution where there are no, or very few, data values. The description of 'very few members in between' two concentrated groups of ages indicates a gap. The distribution is also likely bimodal with two clusters, but the gap is the feature that describes the space between them.

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