Appendix A — Answer Key, Chapter 19: The Data Cycle: Scatterplots and Curve of Best Fit

SOL A.ST.1 (a–i) · Covers textbook Chapter 19 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 124 across the chapter. Reasoning answers show an acceptable response, not the only wording.

Conventions used in every answer below: regression equations are from technology, with coefficients rounded to the nearest hundredth. Predictions from a model may be written $\hat{y}$. Association is not causation. Sets have no more than 30 points. A different calculator may differ in the last digit of a coefficient; treat that as a rounding difference when the fit is the same pattern.

Technology results used throughout this chapter:

• Study (hours, exam score), $n=12$: $y = 6.99x + 55.27$ (linear)
• Car (age in years, value in hundreds of dollars), $n=10$: $y = -12.74x + 196.87$ (linear)
• HVAC (temperature °F, energy kWh), $n=12$: linear $y = -0.21x + 60.97$ (weak); quadratic $y = 0.04x^2 - 4.76x + 176.60$ (preferred)
• Advertising (weeks, sales in hundreds of dollars), $n=14$: $y = -0.78x^2 + 11.59x + 10.92$ (quadratic)
• Phone (hours charged, battery percent), $n=10$: $y = 21.88x + 8.67$ (linear)
• Plant (days, height cm), $n=10$: $y = 1.74x + 1.33$ (linear)

Figures: fig1 data cycle; fig2 SRS diagram; fig3 study positive linear; fig4 car negative linear; fig5 HVAC curved; fig6 no association; fig7 HVAC weak linear fit; fig8 HVAC quadratic fit; fig9 study regression; fig10 advertising quadratic; fig11 phone prediction; fig12 blank grids.

Lesson 19.1 — The Data Cycle and Bivariate Questions

Guided practice

1. Formulate questions; collect or acquire data; organize and represent data; analyze data and communicate results.
2. Two quantitative measurements are recorded for each item and kept paired as an ordered pair.
3. Formulate questions.
4. No. It asks for only one measurement per player (height), so it is univariate.
5. Accept any clear bivariate form that names a population and two quantitative variables, e.g. "For customers of our school store over two weeks, is there a relationship between dollars spent on advertising that week and weekly sales in dollars?"
6. The cycle continues: analyzing and communicating usually raises a new question, returning to formulate questions. A regression equation is part of analysis, not the end of the cycle.

Independent practice

7. a) Collect or acquire data
   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. Rewrite e.g. "For students in our class, is there a relationship between minutes exercised yesterday and resting heart rate?"
   b) Bivariate.
   c) Univariate. Rewrite e.g. "For backpacks in the classroom, is there a relationship between mass and number of books inside?"
   d) Bivariate.
10. Accept any feasible school question naming population and two measurable quantitative variables (e.g. practice minutes and free throws made for the basketball team).
11. "Happiness" and "success" are not clearly defined quantitative variables with a reproducible measurement procedure, and the question names no population.
12. The pairing was destroyed by shuffling. Without matched pairs, the ordered pairs are meaningless and any regression is nonsense.

Exit ticket 19.1

13. Formulate questions; collect or acquire data; organize and represent data; analyze data and communicate results.
14. Accept e.g. "For players on our school basketball team, is there a relationship between minutes of practice this week and free throws made out of 20?"
15. Collect or acquire data.
16. Only matched pairs form ordered pairs that can be plotted or fitted; separate lists lose which $y$ belongs with which $x$.

Lesson 19.2 — Choosing Variables and Sampling

Guided practice

17. Age of the car; independent (explanatory) variable.
18. Accept two distinct measurable pairs, e.g. (hours of sleep, quiz score) and (hours of sleep, reaction time), or (bedtime hour, GPA).
19. A sample in which every member of the population has the same chance of being chosen (and typically every sample of the planned size is equally likely).
20. No. It is a convenience sample of early arrivals, who may differ systematically from students who arrive later.
21. Number members $1$–$60$. Use a calculator random-integer feature (or equivalent) to select $12$ distinct integers; include those members.
22. A nonrepresentative sample can produce a scatterplot and regression that describe the wrong population, so conclusions will not answer the investigative question honestly.

Independent practice

23. a) $x$ = fertilizer amount (with units); $y$ = tomato yield (with units)
    b) $x$ = outside temperature; $y$ = cups of hot chocolate sold
    c) $x$ = weeks of advertising; $y$ = weekly sales
24. The sample is a convenience sample of one team and cannot support a claim about all teens in the state; selection bias is likely.
25. Accept: number the $50$ seedlings and draw an SRS of $20$ with technology; or another clear representative plan (e.g. stratified by tray if trays differ). Must measure days and height for the same seedlings.
26. Strength: saves time / may be larger or professionally measured. Weakness: may not match your population, may omit variables you need, or may be outdated.
27. Sample size alone does not fix bias. A large convenience sample can still miss whole parts of the population.
28. Which quantity is $x$ is a modeling choice that controls how slope and intercept are interpreted. Letting the tool's column order decide silently can reverse the story.

Exit ticket 19.2

29. $x$ = hours charged; $y$ = battery percent.
30. Every member of the population has the same chance to be selected.
31. Students in the cafeteria at one time may differ by grade, schedule, or free period from the full Algebra 1 roster.
32. Accept stratified sample, systematic sample, or acquire from a trustworthy existing source.

Lesson 19.3 — Analyzing Relationships in a Scatterplot

Guided practice

33. Positive linear association.
34. Negative linear association.
35. Curved (U-shaped) / quadratic candidate. A single linear direction cannot describe points that fall then rise.
36. No association.
37. Accept any two points that show higher score with higher study time, e.g. $(0.5, 58)$ and $(6, 96)$, or $(2, 70)$ and $(5, 90)$.
38. Accept: "In this sample, older cars tended to have lower resale values." (No "causes.")

Independent practice

39. $10$ pairs; yes, $10 \le 30$.
40. Plot should show ten points climbing from about $(2, 5)$ to $(20, 36)$; axes labeled days and height (cm).
41. Positive linear association.
42. Name positive linear; cite direction or endpoints; context: seedlings planted longer tended to be taller in this sample.
43. A U-shape is a real curved relationship, not "no association." No association means no consistent pattern; a clear curve is a pattern that suggests a quadratic model.
44. A scatterplot is not required to be a function. Two items can share an $x$-value and have different $y$-values; both points are kept and stack vertically.
45. The student confused graphs of functions with scatterplots of observed pairs. Scatterplots may have repeated $x$-values; that is ordinary bivariate data, not an error.

Exit ticket 19.3

46. Positive linear; negative linear; curved / quadratic candidate; no association.
47. Negative linear. Older cars in this sample tended to have lower resale values.
48. The cloud turns (U-shape or peak-then-fall), which a straight line cannot follow.
49. Accept: seeing two quantities move together does not prove one causes the other — e.g. ice cream sales and sunburns both rise on hot days.

Lesson 19.4 — Technology: Linear or Quadratic Fit

Guided practice

50. Available technology (graphing calculator, Desmos Virginia, spreadsheet, or equivalent).
51. $30$.
52. $y = -0.21x + 60.97$. It is nearly flat and misses the U-shape of the points.
53. $y = 0.04x^2 - 4.76x + 176.60$.
54. Neither a linear nor a quadratic function appropriately represents the relationship.
55. Accept graphing calculator, Desmos (Virginia), spreadsheet regression, or equivalent classroom tool.

Independent practice

56. $12$; yes.
57. Yes. The scatterplot shows a clear positive linear trend that the line tracks.
58. Yes. The points fall roughly along a straight declining trend.
59. Technology gives the quadratic $y = 0.04x^2 - 4.76x + 176.60$, which follows the U-shape. Reject $y = -0.21x + 60.97$ because the linear fit misses the curve.
60. A.ST.1d names technology; least-squares coefficients are not a hand skill in this standard, and the course treats the calculator as the instrument that produces the fit.
61. The student did not use technology to determine the curve of best fit (A.ST.1d). An eye sketch is Grade 8 work, not Algebra 1 regression.

Exit ticket 19.4

62. Accept: a table or scatterplot used for this bullet has at most thirty ordered pairs.
63. $y = 0.04x^2 - 4.76x + 176.60$.
64. When the scatterplot shows no linear or quadratic pattern (no association, or a pattern outside those two families).
65. Internal precision and rounding can shift the hundredths place even when both tools fit the same model type to the same data.

Lesson 19.5 — Regression Models and Predictions

Guided practice

66. $y = 6.99x + 55.27$.
67. Strength: points closely follow a rising line. Weakness: sample-only; extrapolation past the study-hour range (or past a max score) is unreliable.
68. $y = -0.78x^2 + 11.59x + 10.92$.
69. $\hat{y} = 21.88(3) + 8.67 = 74.31$ (about $74.31\%$).
70. Interpolation — $x = 3$ lies between $0$ and $4.5$, the data's horizontal span.
71. It exceeds $100\%$ and sits at/beyond the edge of the data; a battery percent cannot be $118\%$, so the prediction is not valid in context.

Independent practice

72. $\hat{y} = 1.74(10) + 1.33 = 18.73$ cm.
73. $\hat{y} = 1.74(30) + 1.33 = 53.53$ cm; extrapolation (data only go to day $20$); weaker validity — growth may not stay linear that far.
74. Strength: clear positive linear trend in-sample. Weakness: extrapolation risk; other growing conditions ignored.
75. $\hat{y} = -12.74(4) + 196.87 = 145.91$ (about $145.91$ hundreds of dollars).
76. Week $40$ is far outside the $0$–$13$ week window (wild extrapolation). The quadratic already turns downward inside the data; it does not support "huge forever," and association is not causation.
77. The new $x$ may be outside the sample range, the context may cap the response, or the new case may differ from the sample even when the in-sample fit looked strong.
78. Correct: $\hat{y} = 6.99(3) + 55.27 = 76.24$. The student added the slope instead of multiplying by $x$.

Exit ticket 19.5

79. $y = 21.88x + 8.67$.
80. $\hat{y} = 52.43$; yes, interpolation.
81. It describes only the sample (and the measured variables); it may not generalize, and predictions can fail outside the data span.
82. The standard's prediction bullet (A.ST.1f) specifically names linear models; quadratic fits are still taught under A.ST.1d–e for representing curved data.

Lesson 19.6 — Interpreting Parameters and Communicating Conclusions

Guided practice

83. Each additional hour studied is associated with about $6.99$ more exam points, according to the model.
84. At $0$ hours studied, the model predicts a score of about $55.27$.
85. Each additional year of age is associated with a predicted drop of about $12.74$ hundred dollars in resale value.
86. Age $0$ is outside the observed ages (data start at $1$), so the intercept is a model value, not a measured price in this sample.
87. Accept any three from: restate the question/findings; give the model or association; interpret parameters; state limits (sample, no causation, interpolation range); write a clear context sentence.
88. Accept: "In our sample, students who studied more tended to score higher. That association does not by itself prove that studying causes better grades."

Independent practice

89. Slope: about $1.74$ cm of height per additional day, in the model. Intercept: at $0$ days the model predicts about $1.33$ cm (a mathematical starting value; data begin at day $2$).
90. Slope: about $21.88$ percentage points of battery per additional hour charged. Intercept: at $0$ hours charged, about $8.67\%$ battery.
91. Accept a paragraph that includes the negative linear association, $y = -12.74x + 196.87$, slope/intercept meanings with units, sample limits, and no-causation.
92. Accept a paragraph that states the U-shape, prefers $y = 0.04x^2 - 4.76x + 176.60$ over the weak linear $y = -0.21x + 60.97$, and notes sample/context limits without overclaiming causation.
93. A.ST.1g requires meaning in context with the quantities and units, not the bare number.
94. The response unit is hundreds of dollars, so the slope is about $12.74$ hundred dollars per year (about $\$1274$ per year), not $\$12.74$ per year.

Exit ticket 19.6

95. Each additional study hour is associated with about $6.99$ more exam points in the model.
96. At $0$ hours charged, the model predicts about $8.67\%$ battery.
97. Accept: association does not prove causation / this describes the sample only / predictions are strongest inside the data range.
98. i

Chapter 19 Review

99. Accept e.g. "For school-stand sales days this spring, is there a relationship between noon temperature (°F) and cups of lemonade sold?"
100. $x$ = noon temperature (°F); $y$ = cups of lemonade sold (matching the question in 99).
101. Number the $40$ days $1$–$40$; use technology to choose $15$ distinct random integers; include those days.
102. Rainy-only days bias weather and likely sales; the sample is not representative of the whole season.
103. Positive linear; points climb from lower left to upper right (cite endpoints or trend).
104. Curved U-shape; suggests a quadratic model.
105. No association.
106. $y = 6.99x + 55.27$.
107. $y = 0.04x^2 - 4.76x + 176.60$; it follows the U-shape while the linear fit does not.
108. $y = -0.78x^2 + 11.59x + 10.92$.
109. Strength: strong positive linear tracking. Weakness: sample-only / poor for far extrapolation or scores above a test maximum.
110. $\hat{y} = 21.88(1) + 8.67 = 30.55$; yes, interpolation.
111. $\hat{y} = 6.99(4) + 55.27 = 83.23$.
112. Extrapolation beyond $6$ hours; may exceed a realistic max score; weak validity — report with strong caution or decline to treat it as trustworthy.
113. Slope: about $12.74$ hundred dollars lower value per additional year. Intercept: about $196.87$ hundred dollars at age $0$ (model value; outside observed ages).
114. Slope: about $1.74$ cm per additional day. Intercept: about $1.33$ cm at day $0$ (model starting value).
115. Accept four sentences covering positive linear association, technology equation $y = 21.88x + 8.67$, slope/intercept or a sample prediction, and a limit (sample / no causation / interpolation).
116. Accept four sentences stating no association, that neither linear nor quadratic is appropriate, that name length does not predict commute in this sample, and that this is still a valid communicated result.
117. Enter ≤30 pairs → scatterplot → linear regression on technology → record $y = mx + b$ rounded to hundredths → confirm the line tracks the cloud.
118. Enter pairs → scatterplot → run linear and quadratic regressions → compare which curve follows the U-shape → report the quadratic equation from technology (and reject the linear if it fails to track).
119. A.ST.1d–e require regression methods available through technology; calculator-free invented slopes are not the standard's curve of best fit.
120. The student replaced the technology curve of best fit with a hand-rounded convenience equation, violating A.ST.1d's technology requirement.
121. $\hat{y} = 1.74(12) + 1.33 = 22.21$ cm. Slope: about $1.74$ cm of growth per day in the model. Intercept: about $1.33$ cm at day $0$ (model value).
122. $\hat{y} = -12.74(6) + 196.87 = 120.43$ (hundreds of dollars). Caution: older cars tended to have lower values in this sample; the association does not prove that aging alone causes the drop.
123. Linear $y = -0.21x + 60.97$; quadratic $y = 0.04x^2 - 4.76x + 176.60$. Report the quadratic as the curve of best fit.
124. Accept a new bivariate question raised by the peak-then-fall, e.g. relating weeks of advertising to customer complaints, or comparing two advertising channels' weeks vs sales.

Coverage reminder

Every Knowledge and Skill letter a–i appears in the Chapter 19 textbook coverage table. This key answers items 1–124 continuously for both the textbook and the workbook.