Appendix A — Answer Key, Chapter 17: The Data Cycle and Histograms

SOL 7.PS.2 · Covers textbook Chapter 17 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both.

Interval convention. Every interval is $a \le x < b$: the lower endpoint is included and the upper endpoint belongs to the next interval. Every frequency below was tallied value by value and the frequencies were verified to sum to the size of the data set.

Open-ended items. Items that ask a student to formulate a question, plan a collection, design a sample, or write a conclusion have many acceptable responses. For those items the key gives one sample acceptable response and says so; grade on whether the response meets the stated requirements, not on whether it matches the wording here.

Lesson 17.1 — The Data Cycle

Guided practice

1. Formulate questions; collect or acquire data; organize and represent data; analyze data and communicate results.
2. Organize and represent data.
3. Formulate questions. The data will be numerical, because each response is a height in centimeters.
4. The cycle does not end there. The histogram must still be analyzed, and the conclusions communicated; and the results usually raise a new question, which sends you back to the first stage.
5. Analyze data and communicate results.

Independent practice

6. a) Collect or acquire data (by measurement) b) Formulate questions c) Organize and represent data d) Analyze data and communicate results
7. Formulate questions; collect or acquire data; organize and represent data; analyze data and communicate results.
8. Collect or acquire data — they are deciding whom to gather data from. (Accept a student who argues the plan belongs to the first stage, if the reasoning is that the question's population has not yet been fixed.)
9. Answers will vary. One acceptable response: "How many minutes did each student in our class spend on a screen after dinner last night?" The response must ask for a number, not a category, and must apply to a group rather than one person.
10. a) Categorical — no histogram b) Numerical — yes, a histogram could display it c) Numerical — yes, a histogram could display it d) Categorical — no histogram
11. Answers will vary. One acceptable response: Formulate — "How many hours of sleep, to the nearest half hour, did each seventh grader get last night?" Collect — survey a random sample of 60 seventh graders drawn from the roster. Organize — group the hours into intervals of width 1 hour and draw a histogram. Analyze — observe where the values pile up, compare that to the recommended 9 to 11 hours, and report the finding to families and teachers.
12. Answers will vary. One acceptable response: Every answer produces a new question, so the process never truly stops. For example, a histogram of reading minutes shows that most students read fewer than 30 minutes. That immediately raises the question "Does this change on weekends?", which the existing data cannot answer, so the class returns to the first stage and formulates it. A straight line would suggest the work ends at the last stage, and it does not.

Exit ticket 17.1

13. Formulate questions; collect or acquire data; organize and represent data; analyze data and communicate results.
14. Organize and represent data.
15. Numerical. Yes — the values are counts of pages, so they can be grouped into equal-width intervals on a number line and displayed in a histogram.
16. Answers will vary. One acceptable response: Representing the data turns a long list into a picture with a shape. Patterns such as where the values pile up, how spread out they are, and whether there are gaps are nearly impossible to see in the list and obvious in the picture, so the representation is what makes the analysis possible.

Lesson 17.2 — Asking a Question and Collecting Data

Guided practice

17. No. It asks about one person on one day, so it has a single answer and the data do not vary.
18. Yes. Different seventh graders practice for different amounts of time, so the data vary and must be collected.
19. Because a histogram is built on a number line. Each response has to be a number that can be placed on that line and grouped into an interval. A choice from a list is a category, and categories have no position on a number line.
20. Answers will vary. One acceptable response: "How many hours of sleep, to the nearest half hour, did each seventh grader at our school get last night?"
21. The population; the quantity together with its units; the time frame or condition.

Independent practice

22. a) Not statistical b) Statistical c) Statistical d) Statistical
23. a) Categorical — no histogram b) Numerical — yes c) Numerical — yes d) Categorical — no histogram
24. Answers will vary. One acceptable response: "How many miles, to the nearest tenth of a mile, does each seventh grader at our school live from the school building?" It must name the population (seventh graders at our school), the units (miles), and the precision (to the nearest tenth).
25. a) Measurement b) Observation c) Survey d) Experiment
26. The population is far too large for one class to measure — there are tens of thousands of seventh graders in Virginia — so the data must be acquired from an agency that already collects it. Two questions to ask: Who collected the data, and from which students? When was it collected, and by what method or instrument? (Accept any two reasonable questions about source, date, population, or method.)
27. Answers will vary. One acceptable response: Sharpened question — "How many sit-ups can each seventh grader at our school complete in one minute, using the same starting position and the same one-minute timer?" Data needed — one count per student, taken under identical conditions. Method — measurement (a timed count) or an experiment-style controlled trial; observation with a stopwatch and a counter is acceptable. Why — the quantity is something you can watch and count directly, so no survey is needed and no one has to remember or estimate.
28. The three responses are categories, not numbers, so there is no number line on which to place them and no way to group them into equal-width intervals. Rewritten: "How many minutes did you read last night, to the nearest minute?" The job that moved is the grouping: instead of asking the respondent to decide what counts as "some," you collect the exact number and decide the intervals yourself in the organize-and-represent stage, where the decision can be stated and changed.

Exit ticket 17.2

29. Yes. Different seventh graders spend different amounts of time on homework, so the data vary, and each response is a number of minutes, so the data are numerical.
30. Observation, measurement, survey, experiment.
31. Answers will vary. One acceptable response: "What is the height, in centimeters to the nearest centimeter, of each seventh grader at our school?"
32. Because the answers are names of subjects — categorical data. A histogram groups numerical values into equal-width intervals on a number line, and there is no number line on which "science" sits between "math" and "art." A bar graph is the correct display for that question.

Lesson 17.3 — Samples, Sample Size, and Randomness

Guided practice

33. Population: all 360 seventh graders. Sample: the 45 students measured.
34. Answers will vary. One acceptable response: A sample is representative when its values reflect the larger population well enough that a conclusion drawn from the sample is likely to be true of the whole population. That happens when every member of the population had the same chance of being chosen and the sample is large enough that a few unusual values do not control the result.
35. Because whoever is standing near you was not chosen by chance — it was decided by where people happen to be, and by who chose to be near you. Students far away, in other classes, or absent from that spot had no chance of being selected, so the chances are not equal.
36. 60 chosen at random. Both are random, so neither is tilted, but with only 8 students a single unusual value moves the result a great deal, while with 60 the unusual values are balanced by the rest. Larger random samples give steadier results.
37. No. The sample size is large, but the selection is not random: students at a basketball game are more likely than average to be athletes or fans of sports, so their exercise minutes will run high. Size cannot correct a tilt in who was chosen.

Independent practice

38. a) Population: all 500 students in the school. Sample: the 30 students measured. b) Population: all 48 members of the track team. Sample: the 12 members timed.
39. Any two: (i) With only 5 values, one or two unusual answers change the result enormously, so the sample is unstable. (ii) Five students cannot cover the variety in a grade of 300 — different homerooms, activities, and schedules — so whole parts of the population are unrepresented. (iii) A small sample makes it likely that by chance the five chosen are unlike the grade as a whole.
40. In a random sample every member of the population has the same chance of being chosen; in a convenience sample members are chosen because they were easy to reach. Example of random: number all 400 students on the roster and use a random number generator to select 40. Example of convenience: survey the students in your own homeroom.
41. Any three: (i) Write all 400 names or numbers on identical slips, mix them thoroughly, and draw 40. (ii) Use a random number generator to produce 40 different whole numbers from 1 to 400 and select the students with those roster numbers. (iii) Use a table of random digits, reading off three-digit groups and keeping the first 40 distinct values from 001 to 400. (iv) Have a spreadsheet assign each of the 400 students a random number, sort by that column, and take the top 40.
42. Answers will vary. One acceptable response: Number every student in the grade on the official roster, use a random number generator to choose the required number of roster numbers, and then find and survey exactly those students wherever they are, following up with anyone who is absent. This gives every student in the grade — not only students who go to the library — an equal chance of being selected.
43. Answers will vary. One acceptable response: Survey 80 of the 320 students. Number the full seventh-grade roster from 1 to 320 and use a random number generator to select 80 different numbers. This is likely representative because every seventh grader had the same chance of selection, all eight homerooms can be reached by the draw, and 80 is large enough that a few students with an unusual homework night will not control the result.
44. The second sample is biased because of who could be selected, not how many were. Only students leaving the band room could be chosen, and band students practice music far more than students who are not in a music program; every non-music student in the grade had zero chance of selection. Size does not repair that, because adding more band students only makes the tilted group larger. The reported practice minutes would be pushed too high.

Exit ticket 17.3

45. Population: all 400 students in the school. Sample: the 25 students measured.
46. Answers will vary. One acceptable response: A larger sample is steadier — a single unusual value has much less influence on the result, so repeated samples of the same size would give answers closer to one another and closer to the population's.
47. Answers will vary. One acceptable response: Randomness decides whether the sample points at the right answer at all. If the selection favors a certain kind of student, every additional person selected the same way repeats the same tilt, so a large biased sample is simply a confidently wrong sample.
48. Chess club members were chosen because they are in the chess club, and students who join the chess club are likely to spend their time differently from athletes, so the sample is tilted and probably reports too few sports hours. Every student not in the chess club had no chance of being selected. Better plan: number the full seventh-grade roster, use a random number generator to select a sample of roughly 60 students, and survey exactly those, following up with any who are absent.

Lesson 17.4 — Building a Histogram

Guided practice

49. Data: $3, 7, 11, 14, 16, 19, 21, 24, 25, 29, 33, 38$.

Check: $2 + 4 + 4 + 2 = 12$.

50. $30 \le x < 40$. The first interval means at least 20 and less than 30, and 30 is not less than 30. The second means at least 30 and less than 40, and 30 is at least 30. Under this chapter's convention the lower endpoint is included and the upper endpoint is not.
51. $5 + 8 + 6 + 2 = 21$ values.
52. Because the horizontal axis is a number line and a number line is continuous. There is no space on the number line between $20 \le x < 30$ and $30 \le x < 40$, so leaving a gap between their bars would falsely suggest that some values could not occur there.
53. $6 + 7 + 4 = 17$, and $25 - 17 = 8$. The missing frequency is 8. Check: $6 + 7 + 8 + 4 = 25$.

Independent practice

54. Bus-ride data: $4, 6, 9, 10, 12, 13, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 31, 33, 37, 44$.

Check: $3 + 7 + 6 + 3 + 1 = 20$.

55. A drawing. The histogram has ticks on the horizontal axis at 0, 10, 20, 30, 40, and 50, labeled "Minutes riding the bus this morning," and a vertical axis labeled "Frequency" scaled from 0 to at least 8 in steps of 1. Bar heights, left to right: 3, 7, 6, 3, 1. Every bar spans its full interval from edge to edge, so consecutive bars share a side and touch. A suitable title is "Bus-Ride Times for 20 Students."
56. Fewer than 30 minutes: $2 + 4 + 7 = 13$ students. Thirty minutes or more: $5 + 4 + 2 = 11$ students. Check: $13 + 11 = 24$, the whole data set.
57. a) Bar graph — pizza toppings are categories, and nothing lies between one topping and the next, so the bars have gaps. b) Histogram — times in seconds are numerical and can be grouped into equal-width intervals on a number line, so the bars touch. c) Histogram — numbers of books are numerical values that can be grouped into equal-width intervals. (A dot plot is also defensible for a small set, but among the two choices offered, a histogram is correct.) d) Bar graph — eye colors are categories.
58. The graph is drawn as a histogram but formatted as a bar graph. Numerical data grouped into intervals belongs on a continuous number line, and the intervals have no gaps between them, so the bars must not either. The gaps wrongly suggest to a reader that there are values between the intervals that could not occur, or that the horizontal axis is a list of separate categories rather than a number line.
59. Science-test scores, intervals of width 10 starting at 50.

Check: $3 + 5 + 7 + 5 + 4 = 24$. Description: five bars with ticks at 50, 60, 70, 80, 90, and 100. The tallest bar is $70 \le x < 80$ at a height of 7, the bars on either side are both 5, and the graph falls off to 3 on the left and 4 on the right — a single peak near the middle, roughly balanced, with no gaps and no bar of height 0.

60. Their tables differ only at the boundary values — values exactly equal to an interval edge, such as 20 or 30. The first student counts 20 in $20 \le x < 30$; the second counts it in $10 \le x < 20$. Each boundary value moved shifts one count from one interval to its neighbor, so two frequency tables can disagree by one or more in several intervals even though both totals still equal 24 and neither student miscounted. A stated convention prevents the problem because it fixes, in advance and for everyone, which interval owns each boundary value, so the same data always produces the same table.

Exit ticket 17.4

61. Data: $2, 5, 9, 10, 14, 18, 21, 25$.

Check: $3 + 3 + 2 = 8$.

62. $4 + 7 + 2 = 13$, and $18 - 13 = 5$. The missing frequency is 5. Check: $4 + 7 + 5 + 2 = 18$.
63. First, a histogram displays numerical data grouped into equal-width intervals, while a bar graph displays categorical data. Second, a histogram's bars touch, because the horizontal axis is a continuous number line, while a bar graph's bars are separated by gaps.
64. $40 \le x < 50$. The interval $30 \le x < 40$ means at least 30 and less than 40, and 40 is not less than 40; $40 \le x < 50$ means at least 40 and less than 50, and 40 is at least 40.

Lesson 17.5 — Choosing Intervals and Comparing Representations

Guided practice

65. Sit-up data regrouped into intervals of width 20 starting at 0.

Check: $10 + 12 + 2 = 24$. These are the width-10 frequencies added in pairs: $2 + 8 = 10$, $4 + 8 = 12$, and the last 2 alone.

66. $10 \le x < 20$ and $30 \le x < 40$ tie, each with a frequency of 8.
67. It hides the dip at $20 \le x < 30$ and the two separate clusters on either side of it, because the width-20 grouping absorbs each cluster into an interval with its neighbor and reports one middle-heavy group instead.
68. A stem-and-leaf plot. It records each value as a stem plus a leaf, so every original value can be read back out, while a histogram reports only how many values fell in each interval and discards the values themselves.
69. Because reading times are numerical data, and a circle graph shows parts of a whole without preserving the order of the intervals. Nothing about a circle says that $30 \le x < 40$ comes after $20 \le x < 30$, so the reader loses the sense of low, middle, and high that is most of what the data mean, along with any sense of spread.

Independent practice

70. Game scores, intervals of width 10 starting at 0.

Check: $2 + 5 + 8 + 4 + 1 = 20$.

71. Percents and central angles for the game-score intervals.

Sample work: $\tfrac{2}{20} = 10\%$ and $0.10 \times 360° = 36°$; $\tfrac{8}{20} = 40\%$ and $0.40 \times 360° = 144°$. Checks: $10 + 25 + 40 + 20 + 5 = 100$ and $36 + 90 + 144 + 72 + 18 = 360$.

72. a) 42 points, read from the row $4 \mid 2$. b) 2 games, read from the two 1s in the row $2 \mid 1\ 1\ 2\ 3\ 5\ 6\ 8\ 9$. c) 21, which occurs twice. (It is the only repeated value in this data set.)
73. a) The circle graph, because the question asks for a share of the whole and each sector shows one interval as part of one whole — here $40\%$ of the circle. b) The stem-and-leaf plot, because it preserves every individual value, so the highest score can be read exactly as 42; the histogram reports only that one game fell in $40 \le x < 50$. (A dot plot also answers this and is acceptable.) c) The histogram, because grouping keeps a large data set readable — 200 dots would stack into unreadable towers and 200 leaves would overflow the rows — while the bar heights still show where the values pile up. d) The dot plot, because it plots every value separately, so the two games at exactly 21 points appear as a stack of 2 dots.
74. Reading data regrouped into intervals of width 20 starting at 0.

Work: $2 + 4 = 6$, $7 + 5 = 12$, $4 + 2 = 6$. Check: $6 + 12 + 6 = 24$. What it hides: the single tallest interval, $20 \le x < 30$ with 7 students, disappears into a wide middle bar, so you can no longer see that the peak sits in the twenties rather than the thirties, and the graph now looks perfectly symmetric with three bars in a 6-12-6 pattern.

75. a) $5 + 4 + 2 = 11$ students read 30 minutes or more. b) Fewer than 20 minutes is $2 + 4 = 6$ students, and $\tfrac{6}{24} = \tfrac{1}{4} = 25\%$. c) Answers will vary. One acceptable response: Conclusion — a 30-minute independent reading block would meet or exceed what most of the class currently does on its own, so it is a realistic target rather than an unreachable one. Observation it rests on — the tallest bar is $20 \le x < 30$ with 7 students, and 13 of the 24 students read fewer than 30 minutes.
76. The width-20 histogram shows 12 of 24 students in $20 \le x < 40$, which does make the class look like one consistent middle group. But the width-10 histogram of the identical data shows that this interval is really two separate clusters — 4 students in $20 \le x < 30$ and 8 in $30 \le x < 40$ — and that the largest counts sit at $10 \le x < 20$ and $30 \le x < 40$ with 8 each, with a dip between them. So the class is not consistent; it splits into a lower group and a higher group. A supported rewrite: "At intervals of width 10, the sit-up data show two clusters of 8 students each, one at $10 \le x < 20$ and one at $30 \le x < 40$, with only 4 students between them, so the class splits into two performance groups rather than forming one consistent middle."

Exit ticket 17.5

77. Answers will vary. One acceptable response: A dot plot shows every individual value, including exactly which values repeat and how many times, while a histogram reports only how many values fell in each interval and never tells you any single value.
78. Answers will vary. One acceptable response: A histogram shows the overall shape of a large data set at a glance — where the values pile up, how spread out they are, and where the gaps are — while a dot plot of a few hundred values becomes an unreadable forest of stacked dots.
79. 10, 12, 2. The width-20 frequencies are the width-10 frequencies added in pairs: $2 + 8 = 10$ and $4 + 8 = 12$, with the final 2 alone in $40 \le x < 60$. Check: $10 + 12 + 2 = 24$.
80. Answers will vary. One acceptable response: The bars of a histogram show grouped counts, not the data values themselves, so changing the width changes which values get added together. Wider intervals combine neighboring counts and smooth away peaks, dips, and gaps; narrower intervals split counts apart and expose them. The data never moved — only the grouping did.

Chapter 17 Review

Part A — Formulating questions (7.PS.2a)

81. b and d are statistical questions whose data are numerical. a is numerical but not statistical, because it asks about one person on one day. c is statistical but categorical, because the answers are names of strokes.
82. Answers will vary. One acceptable response: "What is the mass, in kilograms to the nearest tenth, of each seventh grader's backpack on a Monday morning?" It must name the population, the quantity with its units, and the time frame, and must ask for a number.
83. Answers will vary. One acceptable response: Question — "How many minutes did each seventh grader at our school spend on homework last night?" Population — all seventh graders at our school. Quantity and units — time spent on homework, in minutes to the nearest minute. Time frame — last night.

Part B — Determining and collecting the data (7.PS.2b)

84. a) Measurement b) Observation c) Survey d) Experiment
85. The population is every seventh grader in the division, which is thousands of students spread across many schools, so one class cannot measure them all in any reasonable time — but the division's health records already contain the heights. Two questions to ask about the acquired data: Who collected it, and from exactly which students? When was it collected, and with what instrument or method? (Accept any two reasonable questions about the source, date, population covered, or method.)

Part C — Sample size and randomness (7.PS.2c)

86. Population: all 480 seventh graders. Sample: the 60 students whose arm spans were measured. The plan is likely to be representative because the random number generator gives every one of the 480 students the same chance of selection, so no group is systematically favored, and 60 is large enough that a few unusually long or short arm spans will not control the result. Randomness removes the tilt; the size keeps the result steady.
87. Any two: (i) Track team members train specifically for sprinting, so their times are systematically faster than the grade's, and the sample is tilted toward fast. (ii) Every student who is not on the track team had zero chance of being selected, so most of the population is unreachable by this plan. (iii) Ten values is too few to be stable for a whole grade, so a single unusual time would move the result a great deal.
88. The sample is large but not random, and the tilt is in who could be selected: only students who attended the winter concert could be chosen, and concertgoers are mostly band and chorus members who practice music far more than the average student. Every student who did not attend had zero chance of selection, and surveying more concertgoers only repeats the same tilt. The reported practice minutes would be pushed too high.

Part D — Building histograms (7.PS.2d)

89. Homework data: $8, 12, 15, 17, 19, 20, 22, 25, 27, 28, 30, 31, 34, 36, 38, 41, 44, 47, 52, 58$.

Check: $1 + 4 + 5 + 5 + 3 + 2 = 20$.

90. The horizontal axis is a number line labeled "Minutes spent on math homework," with ticks at every interval edge: 0, 10, 20, 30, 40, 50, and 60. The vertical axis is labeled "Frequency" and is scaled from 0 to at least 6 in steps of 1. Each bar spans its whole interval from edge to edge, at heights 1, 4, 5, 5, 3, and 2 from left to right. The bars touch because the horizontal axis is a continuous number line and the intervals have no gaps between them, so leaving space between the bars would falsely suggest that some values could not occur.
91. $30 \le x < 40$. Convention: intervals are written $a \le x < b$, so the lower endpoint is included in the interval and the upper endpoint belongs to the next one; 30 is not less than 30, so it cannot go in $20 \le x < 30$.
92. $3 + 6 + 7 + 2 = 18$, and $26 - 18 = 8$. The missing frequency is 8. Check: $3 + 6 + 8 + 7 + 2 = 26$.

Part E — Intervals and comparing representations (7.PS.2e, 7.PS.2f)

93. Homework data regrouped into intervals of width 20 starting at 0.

Work: $1 + 4 = 5$, $5 + 5 = 10$, $3 + 2 = 5$. Check: $5 + 10 + 5 = 20$.

94. Answers will vary. One acceptable response: The width-20 grouping hides that only 1 student spent fewer than 10 minutes while 4 spent between 10 and 20 — a steep rise that is invisible once those two counts are merged into a single bar of 5. It also produces a perfectly symmetric 5-10-5 picture, which suggests a balance the width-10 view does not show, since that view has a longer, thinner right side.
95. Because a histogram reports only how many values fell in each interval, never which values they were. The bar for $30 \le x < 40$ has height 5, which tells you that 5 students spent at least 30 and fewer than 40 minutes, but any of those five could have spent 31, 34, or 39 minutes. To answer the question you would need the original list, a dot plot, or a stem-and-leaf plot.
96. a) Dot plot — it stacks one dot for every value, so repeats are visible and countable. b) Histogram — grouping keeps 250 values readable, and the bar heights still show where the values pile up and how spread out they are. c) Circle graph — each sector shows one interval as a part of the whole class, which is exactly what a share question asks for. d) Stem-and-leaf plot — it is the only one of the four that shows the shape and preserves every exact value at the same time, and with 20 two-digit numbers the rows stay short enough to read.
97. Stem-and-leaf plot for the homework data, stems are tens digits.

Key: $2 \mid 5$ means 25 minutes. Check: $1 + 4 + 5 + 5 + 3 + 2 = 20$ leaves, matching the 20 values.

98. Advantage: it answers share-of-the-whole questions directly — the $20 \le x < 30$ sector is $\tfrac{5}{20} = 25\%$ of the circle, which can be read at a glance without any computation. Limitation: it destroys the order of the intervals, since nothing about a circle says $40 \le x < 50$ comes after $30 \le x < 40$, so the reader cannot see that the data run from low to high, cannot judge the spread, and cannot see the shape. (Accept also: it cannot show individual values, and it cannot show a gap as an empty position on a number line.)

Part F — Analyzing histograms, mixed reasoning (7.PS.2g)

99. Fewer than 30 minutes: $1 + 4 + 5 = 10$ students. Forty minutes or more: $3 + 2 = 5$ students. (Check: the remaining 5 students are in $30 \le x < 40$, and $10 + 5 + 5 = 20$.)
100. Answers will vary. One acceptable response: The histogram shows immediately that the values pile up in the middle — the two tallest intervals, $20 \le x < 30$ and $30 \le x < 40$, hold 10 of the 20 students between them — and that the shape then falls off more gently to the right than to the left, with a thin tail out to $50 \le x < 60$. Exactly half the class finishes in under 30 minutes and a quarter takes 40 minutes or more, a split that takes real work to see in a list of 20 numbers and is obvious in the picture. (Accept any correct pattern statement: the low first bar of 1, the middle-heavy shape, the tail to the right, or the absence of gaps.)
101. $20 \le x < 30$ and $30 \le x < 40$ tie, each with a frequency of 5.
102. Answers will vary. One acceptable response: Formulate — write the question "How many hours of sleep, to the nearest half hour, did each seventh grader at our school get last night?" Collect — number the seventh-grade roster and use a random number generator to select 60 students, then survey exactly those students. Organize — group the hours into equal-width intervals of half an hour, build a frequency table, verify that the frequencies sum to 60, and draw the histogram with touching bars. Analyze — observe where the values pile up and how many students fall below the recommended amount, then report the histogram and the conclusion to families and school staff.
103. A histogram never shows individual values. Its bars report only how many values fell in each interval, so a bar of height 5 means that 5 students slept at least as much as the interval's lower edge and less than its upper edge — not that all 5 slept the same amount, and not that any of them slept exactly 8 hours. A correct claim would be phrased as "5 students slept at least 8 and fewer than 9 hours," using the interval the graph actually shows. To claim an exact value you would need the original data, a dot plot, or a stem-and-leaf plot.
104. Both descriptions come from the same 24 values because a histogram displays grouped counts, and the grouping is a choice the grapher makes. At width 10 the frequencies are 2, 8, 4, 8, 2, and the two 8s with a 4 between them are the two clusters and the dip. At width 25 the two intervals hold $2 + 8 + 2 = 12$ and $2 + 8 + 2 = 12$ — the values 22 and 24 land in the first interval and 26 and 28 land in the second — so the counts really are equal and the second class reported honestly. Nothing was miscounted; the width-25 grouping simply combined the peaks with the dips and averaged them away. I would trust the width-10 grouping for describing the class, because 5 intervals sits in the workable five-to-ten range and shows a shape you can describe, while 2 intervals is too coarse to show any shape at all and reports an evenness that the finer view proves is not there.