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

SOL 8.PS.3 · Covers textbook Chapter 17 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both. Item numbers run continuously from 1 to 112 across the chapter. Reasoning answers show an acceptable response, not the only wording.

Two limits from the standard are enforced throughout: no data set exceeds 20 items, and relationships are named qualitatively as positive linear, negative linear, or no relationship. No correlation coefficient is computed anywhere in this chapter.

A note on sketched lines. Bullet (f) asks students to sketch the line of best fit, so two careful students will draw slightly different lines and get slightly different estimates from the same plot. Accept any line that follows the direction of the trend, runs through the middle of the cloud with roughly equal numbers of points above and below, stays close to the points, and is straight. The equations quoted in this key — $y = 6x + 60$ for the study data, $y = -2x + 150$ for the hot chocolate data, $y = -13x + 200$ for Plot D, and $y = x + 1$ for Plot E — are the lines drawn in the figures, and estimates within a few units of these are correct.

The data sets used.

• Study time and quiz score (12 pairs): $(0, 63)$, $(0, 57)$, $(1, 68)$, $(1, 62)$, $(2, 77)$, $(2, 70)$, $(3, 81)$, $(3, 75)$, $(4, 88)$, $(4, 80)$, $(5, 92)$, $(5, 87)$
• Temperature and cups of hot chocolate (10 pairs): $(20, 116)$, $(25, 92)$, $(30, 94)$, $(35, 75)$, $(40, 77)$, $(45, 54)$, $(50, 53)$, $(55, 36)$, $(60, 35)$, $(65, 18)$
• Letters in a last name and minutes to school (12 pairs): $(3, 22)$, $(4, 8)$, $(4, 35)$, $(5, 15)$, $(5, 28)$, $(6, 12)$, $(6, 31)$, $(7, 19)$, $(7, 5)$, $(8, 26)$, $(9, 10)$, $(10, 24)$

Lesson 17.1 — The Data Cycle with Two Variables

Guided practice

1. Formulate questions; collect or acquire data; organize and represent data; analyze data and communicate results.
2. Each item in the set contributes two numbers, recorded as an ordered pair $(x, y)$ and kept matched to the same item.
3. Univariate. Only one number — the height — is recorded per student, so there is no second coordinate and nothing to pair.
4. Organize and represent data.
5. Hours practiced goes on the horizontal axis; it is the independent variable.
6. The cycle does not end there. Drawing the scatterplot completes only the third stage. The fourth stage — analyzing the plot and communicating what it shows — still remains, and the conclusion usually raises a new question, which sends you back to the first stage.

Independent practice

7. a) Collect or acquire data (by measurement) b) Formulate questions c) Organize and represent data d) Analyze data and communicate results
8. Formulate questions; collect or acquire data; organize and represent data; analyze data and communicate results.
9. a) Univariate b) Bivariate c) Univariate d) Bivariate
10. a) Independent: hours of practice; dependent: free throws made. b) Independent: age of the car; dependent: price. c) Independent: days since planting; dependent: plant height.
11. Any question naming two numerical quantities measured on the same student. For example: "For students in our class, is there a relationship between hours of sleep last night and minutes spent on homework last night?" Quantities: hours of sleep, to the nearest half hour, and homework minutes.
12. Formulate: write the question "For eighth graders at our school, is there a relationship between hours of sleep on a school night and resting heart rate?" Collect: for each of at most 20 students, record hours slept and resting heart rate in beats per minute, keeping each student's two numbers on one row. Organize: plot each student as a point, hours on the horizontal axis and heart rate on the vertical axis. Analyze: describe the relationship as positive linear, negative linear, or no relationship, justify it from the points, and report it to the staff.
13. A histogram is built on one number line, so it can display only one quantity at a time. It would show how the study times are distributed, or how the scores are distributed, but in grouping the values into intervals it throws away which score belonged to which student. The relationship between two quantities lives entirely in that pairing, so once the pairing is lost the question cannot be answered.

Exit ticket 17.1

14. Formulate questions; collect or acquire data; organize and represent data; analyze data and communicate results.
15. Bivariate: the height and arm span of each of 15 students. Univariate: the height of each of 15 students. (Any correct pair of examples.)
16. Plant height goes on the vertical axis; it is the dependent variable.
17. A point needs both coordinates from the same item, so if the pairing is lost there is no way to know which $y$ goes with which $x$, and no point can be plotted.

Lesson 17.2 — Asking a Bivariate Question and Collecting the Data

Guided practice

18. "Is there a relationship between one numerical quantity and a second numerical quantity?" For example, hours studied and quiz score.
19. A scatterplot places each item at a position along two number lines, so both measurements must be numbers. A category such as "soccer" has no position on a number line.
20. The population; the first quantity with its units; the second quantity with its units; the time frame or condition.
21. "For the eighth graders in our class, is there a relationship between the number of hours a student studied in the week before Friday's quiz and that student's score out of 100 points on that quiz?"
22. The standard limits a data set in this chapter to no more than 20 items, so 22 pairs is too many. Collect from at most 20 students, or select 20 of the 22 pairs in a way that is not chosen to favor a result.
23. No. A point needs both coordinates. A student with a study time but no score contributes only half a pair, and inventing the missing score would be making up data.

Independent practice

24. a) No — univariate; one number per student, so a histogram or boxplot fits. b) Yes. c) No — favorite subject is categorical, and both quantities must be numerical. d) Yes.
25. a) "For members of our track team, is there a relationship between height in centimeters and 100-meter sprint time in seconds, recorded at this month's time trial?" b) "For the school store this winter, is there a relationship between the noon temperature in degrees Fahrenheit and the number of cups of hot chocolate sold that day?"
26. a) Measurement; height in centimeters and arm span in centimeters. b) Observation; the noon temperature in degrees Fahrenheit and the number of students wearing coats. c) Survey; hours of sleep last night and minutes of homework last night. d) Experiment; the ramp height in centimeters and the distance the car rolls in centimeters.
27. Rainfall and cloud-cover records for 15 cities cannot be measured by a class in Virginia, and weather services have already recorded them, so the data should be acquired. Two questions worth asking: who collected the data and over what period, and are the two columns really matched to the same city and the same year? A third good question is what instrument or definition was used — "a cloudy day" has to mean the same thing in every city.
28. The two tables describe different items. One has a row per month, the other a row per year, so there is no item that contributes both numbers, and no ordered pair can be formed. Pairing requires both measurements to come from the same item.
29. Sharpened question: "For the students on the basketball team, is there a relationship between the number of weeks a player has practiced free throws and the number of free throws made out of 20 attempts this Friday?" Data needed: two numbers per player — weeks of practice, and free throws made out of 20. Method: measurement combined with survey — the made shots are counted directly, and the weeks of practice are asked. It fits because both numbers can be recorded for each player under the same conditions, 20 attempts each.
30. The two lists were recorded in "whatever order people answered," so there is no way to tell which homework time belongs to which sleep time. The pairing is gone, and pairing is what makes data bivariate. Both numbers should have been written on the same row of a single two-column table, one row per student.

Exit ticket 17.2

31. Any question of the form "Is there a relationship between (numerical quantity) and (numerical quantity)?" naming a population and a time frame. For example: "For students in our class, is there a relationship between shoe length and height?"
32. The two quantities of the student's question, each with units. For the example: shoe length in centimeters and height in centimeters.
33. Favorite subject is categorical — the responses are names, not numbers — so there is no position for it on a number line and no coordinate to plot.

Lesson 17.3 — Building a Scatterplot

Guided practice

35. One item in the data set — one student, one day, one object — shown at the ordered pair formed by that item's two measurements.
36. $(45, 18)$
37. Horizontal axis from $0$ to $10$ in steps of $1$; vertical axis from $60$ to $200$ in steps of $20$. (Any scale that covers the data with friendly steps.)
38. As two separate points in a vertical line above $x = 3$, at their two different heights. Neither point is moved sideways, because changing an $x$-value would misreport how long that student practiced.
39. The quantity and its units, for each axis — for example "Hours studied in a week" and "Quiz score (points)."

Independent practice

41. $(20, 116)$, $(25, 92)$, $(30, 94)$, $(35, 75)$, $(40, 77)$, $(45, 54)$, $(50, 53)$, $(55, 36)$, $(60, 35)$, $(65, 18)$
42. Horizontal axis from $15$ to $70$ in steps of $5$, labeled "Outside temperature (°F)"; vertical axis from $0$ to $130$ in steps of $10$, labeled "Cups of hot chocolate sold." The two axes use different scales because they measure different quantities.
43. A scatterplot of the 10 points listed in item 41, temperature on the horizontal axis and cups on the vertical axis, both axes labeled. The points fall from upper left to lower right. (See the figure "Cups of hot chocolate sold against outside temperature.")
44. 10 items. The standard limits a data set to no more than 20 items, and $10 \le 20$.
45. It is not mathematically wrong — the same 12 students are shown, and each point still carries both numbers. What changes is the reading: the axes now say "score" across and "hours" up, so the trend must be described as scores increasing with hours read in the other order, and the independent variable is no longer where a reader expects it. Convention puts the independent variable on the horizontal axis, so the graph should be redrawn.
46. No student scored below $57$, so a vertical axis from $0$ to $100$ would compress all 12 points into the upper half and hide the trend. Starting at $50$ uses the space well. Because a cut axis makes the climb look steeper than it is, the reader must be shown clearly labeled tick marks so the actual scale is visible; the numbers may never be left off.
47. Build a two-column table — one row per student, with shoe length in centimeters in one column and height in centimeters in the next — so each student's two numbers stay on the same row. Then plot shoe length on the horizontal axis, because it is the input we are using to explain height, and height on the vertical axis. With 16 students there are 16 points, within the 20-item limit.
48. A scatterplot is a picture of data, not the graph of a function, so nothing requires one $y$ per $x$. Two students really can study the same number of hours and score differently. The vertical stack means exactly that: two items share an $x$-value and differ in $y$.

Exit ticket 17.3

49. $(8, 95)$
50. The two axes measure different quantities in different units and over different ranges — hours from $0$ to $5$ and points from $57$ to $92$, for instance — so one scale cannot fit both.
51. The columns were selected in the wrong order, so the tool read the dependent variable as $x$. Put the independent variable in the first column, redraw the chart, and label both axes.
52. The question is about how the two quantities move together, and that information lives in the individual pairs. Grouping would merge pairs and destroy the very pattern the graph is meant to show.

Lesson 17.4 — Positive, Negative, or No Relationship

Guided practice

53. Positive linear relationship; negative linear relationship; no relationship.
54. A negative linear relationship.
55. No relationship.
56. Name the relationship in the standard's language; point at the evidence in the plot, citing specific points and the direction of the cloud; say what it means in context with the actual quantities and units.
57. A relationship describes the overall pattern of the whole cloud, not any two individual points. "Tends to increase" allows individual reversals, and a single pair running the other way leaves the overall direction unchanged.
58. Finding that two quantities rise or fall together does not show that one makes the other happen; something else may be driving both, or the causation may run the other way.

Independent practice

59. Negative linear. The points fall from $(0, 95)$ at the left to $(7, 60)$ at the right, and each step to the right brings a lower score.
60. No relationship. The points show no consistent direction — at $0$ pets the heights are $150$ cm and $168$ cm, and at $2$ pets they are $171$ cm and $155$ cm, so tall and short values appear at both small and large $x$.
61. Positive linear. The points climb from $(1, 4)$ at the left to $(8, 14)$ at the right, and each step to the right brings generally more free throws made.
62. Students who had practiced more weeks tended to make more free throws out of 20 — about $4$ made after one week and about $14$ after eight weeks.
63. a) Positive linear — more practice generally goes with more made shots. b) Negative linear — older cars are generally worth less. c) No relationship — a house number is assigned by where you live and has nothing to do with height. d) Negative linear — warmer days generally bring fewer cups of hot chocolate sold.
64. Acceptable rewrite: "The relationship is negative linear. The points fall from $(20, 116)$ at the left to $(65, 18)$ at the right, so each step toward warmer temperatures brings generally fewer sales. In context, on warmer days the school store sold fewer cups of hot chocolate."
65. Positive linear: as cones sold rises from $20$ to $140$, sunburns rise from $1$ to $11$. The shop should not conclude causation, because a scatterplot shows only that the two quantities rose together, and a third quantity can raise both at once. The lurking variable is hot, sunny weather, which sends people out for ice cream and exposes them to sun.
66. The student mistook a single reversal for the absence of a trend. One pair running against the pattern does not decide the relationship; the whole cloud does. Correct description: a positive linear relationship. The points climb from $(0, 57)$ at the left to $(5, 92)$ at the right, so students who studied more hours tended to score higher on the quiz.
67. Sleep causes it: more sleep leaves a student more alert, so reactions are faster. The other direction: a student with fast reactions may be a calmer, less anxious person who therefore falls asleep more easily and sleeps longer. Lurking variable: heavy screen use late at night could both shorten sleep and slow reactions, producing the pattern without either quantity acting on the other.

Exit ticket 17.4

68. A positive linear relationship.
69. Report that there is no relationship between the two quantities. That is a real result, not a failed investigation — it answers the question that was asked, and it tells you that one quantity gives no information about the other.
70. Negative linear: the store sold $116$ cups on a $20$°F day and only $18$ cups on a $65$°F day, so warmer days went with fewer cups sold.
71. A scatterplot records what was observed, not what was controlled. The same picture is consistent with $x$ causing $y$, with $y$ causing $x$, and with a lurking variable driving both, and nothing in the plot distinguishes those. Establishing causation requires an experiment in which the value of $x$ is assigned rather than merely recorded.

Lesson 17.5 — Sketching the Line of Best Fit

Guided practice

72. The overall trend of the points — the straight-line pattern the cloud follows. It does not pass through every point and it does not connect the points in order.
73. It follows the direction of the trend, and it runs through the middle of the cloud with roughly as many points above it as below it.
74. The slope $m$. A positive slope goes with a positive linear relationship, a negative slope with a negative one.
75. For each additional degree of temperature, sales drop by about $2$ cups of hot chocolate.
76. $y = 6(4) + 60 = 24 + 60 = 84$, so about $84$ points.
77. No. There is no trend to summarize, and a drawn line would suggest a direction the data do not show. Report "no relationship" instead.

Independent practice

78. The line is far too low: with $11$ of $12$ points above it, it is running along the bottom of the cloud rather than through the middle. It should be moved up (and possibly tilted) until roughly half the points lie above it and half below, while still following the upward direction.
79. a) $y = 6(1) + 60 = 66$, about $66$ points. b) $y = 6(3) + 60 = 78$, about $78$ points. c) The line describes the overall trend, not any individual student. The two actual students at $3$ hours scored $81$ and $75$ — one above the line and one below — which is exactly what "the points scatter around the line" means. An estimate from a sketched line is an estimate, and the word "about" belongs in the answer.
80. a) $y = -2(30) + 150 = 90$, about $90$ cups. b) $y = -2(60) + 150 = 30$, about $30$ cups. c) The actual values, $94$ and $35$ cups, are each about $4$ or $5$ cups above the line. The line is close to both without matching either, which is what a good sketched fit does.
81. $y = 6(10) + 60 = 120$. The estimate should not be trusted for two reasons: $10$ hours lies far outside the collected range of $0$ to $5$ hours, so nothing in the data says the pattern continues, and $120$ points is impossible on a quiz scored out of $100$. A line has no idea where the data stopped, so the reader has to keep track.
82. Interpolation is estimating at an $x$-value inside the range of the collected data; extrapolation is estimating at an $x$-value outside it. Interpolation is more trustworthy, because the data actually show what happens in that region. Extrapolation assumes the trend continues past the last measurement, which nothing in the data supports and which can produce impossible answers such as $-30$ cups or a $120$-point quiz score.
83. Neither is wrong. The line is sketched, so it is a judgment about where the trend runs, and two reasonable judgments can differ slightly. A difference of $2$ points in the estimate is well within that. Both should be reported with the word "about," and either line is acceptable if it follows the trend, runs through the middle of the cloud, and stays close to the points.
84. On day $15$ the line gives $y = 15 + 1 = 16$ cm, so the plant was about $16$ cm tall. The slope $1$ means the plant grew about $1$ centimeter per day over this period. Day $200$ is far outside the collected range of day $2$ to day $20$, and the line would give $201$ cm — a two-meter bean plant. Plants do not grow at a constant rate forever, and the data say nothing about what happens after day $20$.
85. In $y = 6x + 60$, the y-intercept $60$ is the score the line estimates for a student who studied $0$ hours. It is meaningful because $x = 0$ is actually in the data — two students studied $0$ hours and scored $63$ and $57$, close to $60$. In $y = -2x + 150$, the intercept $150$ would be the cups sold at $0$°F, but the coldest day recorded was $20$°F, so $0$°F is outside the data entirely and the number is an extrapolation rather than a fact about the store.
86. Two problems. First, "proves" claims causation from an observational scatterplot; the plot shows only that students who studied more tended to score higher. Second, the line gives an estimate, not a guarantee for any individual — the two actual students who studied $5$ hours scored $92$ and $87$, and the line's value at $x = 5$ is $90$. Rewrite: "In our class, students who studied more hours tended to score higher, and the sketched line estimates a score of about $90$ for a student who studied $5$ hours."

Exit ticket 17.5

87. The line should run through the middle of the cloud, with roughly as many points above it as below it.
88. $y = -2(25) + 150 = 100$, so about $100$ cups. (The actual sales that day were $92$ cups, which is the usual size of the gap between a sketched line and a real point.)
89. The line is a sketch through scattered points, so the value it gives is an estimate of the trend rather than a measured fact, and "about" reports it at the precision the method actually has.
90. The value $x = 12$ is far outside the range of the collected data, which stops at $x = 5$. Nothing in the data shows that the pattern continues that far, so reading the line there is extrapolation and the answer may be badly wrong — or, as with a quiz score above $100$, impossible.

Chapter 17 Review

Part A — Formulating questions (8.PS.3a)

91. a) No — univariate, one number per student; a histogram or boxplot fits. b) Yes. c) No — favorite sport is categorical. d) Yes.
92. "For students in our class, is there a relationship between the volume of a backpack in cubic centimeters — or its capacity in liters — and its mass in kilograms to the nearest tenth, measured on a Monday morning?" Any sharpened version naming a population, two numerical quantities with units, and a time frame is acceptable.
93. Any bivariate question naming two numerical quantities, with the input on the horizontal axis and the response on the vertical axis, and a reason for the choice. For example: "Is there a relationship between minutes of exercise and resting heart rate?" — minutes of exercise on the horizontal axis, because it is the input being used to explain the heart rate.

Part B — Determining and collecting the data (8.PS.3b)

94. a) Measurement; height in centimeters and arm span in centimeters. b) Observation, or acquiring the store's sales log together with recorded temperatures; noon temperature in degrees Fahrenheit and cups sold. c) Survey; hours of sleep last night and minutes of homework last night. d) Experiment; ramp height in centimeters and rolling distance in centimeters.
95. The set exceeds the 20-item limit for this chapter. Acceptable fix: collect from at most 20 students, or select 20 of the 24 pairs by a method that does not favor any result — for example, by drawing names — and report how the 20 were chosen.
96. 13 points. The 2 unmatched study times have no partner, and pairing one of them with a different student's score would attach two numbers that never came from the same person, which invents data rather than collecting it.

Part C — Organizing and representing data in a scatterplot (8.PS.3c)

97. $(1, 4)$, $(2, 6)$, $(3, 5)$, $(4, 9)$, $(5, 8)$, $(6, 11)$, $(7, 12)$, $(8, 14)$
98. Horizontal axis from $0$ to $9$ in steps of $1$, labeled "Weeks of practice"; vertical axis from $0$ to $16$ in steps of $2$, labeled "Free throws made (out of 20)."
99. A scatterplot of the 8 points in item 97, with both axes labeled. The points climb from lower left to upper right, with the week-$3$ point dipping slightly below the week-$2$ point. (See Plot C in the figure "Three scatterplots labeled Plot A, Plot B, and Plot C.")
100. The student at week $3$ made $5$ free throws while the student at week $2$ made $6$, so that one pair runs against the trend. The table is not wrong — real data scatter, and a relationship describes the overall pattern rather than every consecutive pair. Across all 8 points the count still climbs from $4$ to $14$.

Part D — Describing the relationship (8.PS.3d)

101. A negative linear relationship.
102. A positive linear relationship.
103. No relationship. A house number is assigned by where a family happens to live, and nothing about that number is connected to how tall a student is, so the points would scatter with no direction.
104. Report that there is no relationship between the two quantities. No line of best fit should be drawn, because a line asserts a direction and there is no direction in the data.

Part E — Analyzing and justifying (8.PS.3e)

105. Negative linear. The points fall from $(1, 190)$ at the left to $(10, 70)$ at the right, and each additional year of age brings a generally lower value. In context, older cars in this data set were worth less — about $19{,}000$ dollars at one year old and about $7{,}000$ dollars at ten years old.
106. Positive linear. The points climb from $(2, 4)$ at the left to $(20, 20)$ at the right, and each step of two days brings a generally taller plant. In context, the longer the bean plant had been growing, the taller it was — about $4$ cm on day $2$ and about $20$ cm on day $20$.
107. Supported version: "In this data set, older cars tended to have lower values." Claiming that age causes the drop would require ruling out the other explanations a scatterplot cannot rule out — mileage, condition, and model year all travel with age — which means controlling those factors rather than merely recording what was observed.
108. Positive linear: as cones sold rises, sunburns rise. The lurking variable is hot, sunny weather. Neither quantity causes the other, because eating ice cream does not damage skin and a sunburn does not create an appetite for cones; the weather independently pushes both numbers up on the same days, which is enough to produce the pattern.

Part F — Sketching the line of best fit (8.PS.3f)

109. It follows the direction of the trend; it runs through the middle of the cloud with roughly as many points above as below; it stays close to the points overall; it is a single straight line drawn across the data, not a curve and not a set of segments.
110. a) $y = -13(5) + 200 = -65 + 200 = 135$, so about $135$ hundred dollars — about $13{,}500$ dollars. (The actual $5$-year-old car in the data was worth $130$ hundred dollars, or $13{,}000$ dollars.) b) For each additional year of age, the car's value drops by about $13$ hundred dollars, or about $1{,}300$ dollars. c) The data run only from $1$ to $10$ years, so $30$ years is far outside the range. The line would give $-13(30) + 200 = -190$ hundred dollars, a negative value, which is impossible — a car cannot be worth less than nothing.
111. a) $y = 15 + 1 = 16$, so about $16$ cm. b) The plant grew about $1$ centimeter per day. c) Interpolation, because day $15$ lies inside the collected range of day $2$ to day $20$.

Part G — Mixed application and reasoning

112. Formulate: write the question "For eighth graders in our class, is there a relationship between hours of sleep last night and minutes spent on homework last night?" Collect: survey at most 20 classmates, recording both numbers for each student on the same row of one table, and discard any row missing a value. Organize: plot each student as a point with hours of sleep on the horizontal axis and homework minutes on the vertical axis, labeling both axes. Analyze: describe the relationship as positive linear, negative linear, or no relationship; justify it by citing points and the direction of the cloud; if a linear trend appears, sketch a line of best fit and report any estimate as an estimate; and state the result in context without claiming that either quantity causes the other. The graph is a scatterplot.

Workbook-only items

Page 2, fill in the blanks. The four stages are formulate questions · collect or acquire data · organize and represent data · analyze data and communicate results. A data set is bivariate when two numbers are recorded for each item and kept paired (as an ordered pair). The independent variable goes on the horizontal axis; the dependent variable goes on the vertical axis.

Page 6, question shape. Is there a relationship between one numerical quantity and a second numerical quantity? Both blanks must hold a numerical quantity. The four things a sharp question names: the population; the first quantity and its units; the second quantity and its units; the time frame.

Page 10, plotting a pair. To plot $(3, 81)$, go across to $x = \mathbf{3}$, then up to $y = \mathbf{81}$. One point represents one item in the data set — here, one student, shown at that student's two measurements.

Page 11, item 43 grid. Use the right-hand grid, which runs from $15$ to $70$ across and $0$ to $130$ up. Label the horizontal axis "Outside temperature (°F)" and the vertical axis "Cups of hot chocolate sold," then plot the 10 pairs from item 41. The points fall from upper left to lower right.

Page 14, fill in the blanks. As $x$ increases, $y$ tends to increase → positive linear relationship. As $x$ increases, $y$ tends to decrease → negative linear relationship. As $x$ increases, $y$ shows no consistent direction → no relationship.

Page 18, four tests. 1) It follows the direction of the trend. 2) It runs through the middle of the cloud, with roughly as many points above it as below it. 3) It stays close to the points overall. 4) It is a single straight line. In $y = mx + b$, the slope $m$ tells you how much $y$ changes for each increase of $1$ in $x$, and the y-intercept $b$ is the value of the line at $x = \mathbf{0}$.

Page 18, item 76 frame. $y = 6(\mathbf{4}) + 60 = \mathbf{84}$.

Page 19, sketch the line. After plotting the 12 study-time pairs and sketching a line along the trend, a good sketch has about 6 points above and 6 below. The line drawn in the textbook figure is $y = 6x + 60$; any line with roughly equal counts above and below, following the upward trend, is acceptable.

Page 19, item 79 frames. a) $y = 66$. b) $y = 78$.

Page 20, item 80 frames. a) $y = 90$. b) $y = 30$.

Page 20, item 81 frame. $y = 6(10) + 60 = \mathbf{120}$, which is impossible on a quiz scored out of $100$ and lies far outside the collected range of $0$ to $5$ hours.

Page 20, item 82 blanks. Interpolation is estimating at an $x$-value inside the range of the data. Extrapolation is estimating at an $x$-value outside that range. More trustworthy: interpolation, because the collected data actually show what happens there.

Page 22, item 88 frame. $y = -2(25) + 150 = \mathbf{100}$.

Page 26, item 110 frame. a) $y = \mathbf{135}$ hundred dollars.

Page 26, item 111 frame. a) $y = \mathbf{16}$ cm.