Scatter Plots - SAT Math

1. A delivery company compared the number of packages delivered (packages) and the total time spent delivering (minutes) for 10 routes. The scatterplot includes a line of best fit. Based on the line of best fit, what is the predicted delivery time for a route with $50$ packages?
   Answer: About 100 minutes This question asks for the predicted delivery time for 50 packages using the line of best fit. Looking at the scatterplot, we find x = 50 on the horizontal axis and trace up to the line of best fit. At x = 50 packages, the line appears to intersect at approximately y = 100 minutes on the vertical axis. The positive slope indicates that more packages require more delivery time, which makes logical sense. A common mistake would be extrapolating beyond the data range or misreading the scale - here 50 packages is within the observed range. When reading predictions from a graph, carefully check the scale on both axes and ensure you're reading from the line, not data points.
2. A music teacher recorded, for 10 students, the number of practice sessions completed and the number of mistakes in a performance. The scatterplot is shown. Which point is most likely an outlier that weakens the negative association?
   Answer: $(9, 16)$ This question asks which point is most likely an outlier that weakens the negative association between practice sessions and mistakes. The scatterplot shows a general negative trend - more practice leads to fewer mistakes. An outlier that weakens this association would be a point that goes against this trend. Looking at the options: (9, 16) represents many practice sessions (9) but also many mistakes (16), which contradicts the expected pattern of fewer mistakes with more practice. This point appears far from where the negative trend line would be, making it the outlier. Points (2, 18), (4, 14), and (6, 10) all follow the expected downward pattern. When identifying outliers in associations, look for points that deviate most from the overall pattern.
3. A nutritionist recorded the number of sugary drinks consumed per week (drinks) and a health score (points) for 8 clients. The scatterplot shows drinks on the x-axis and health score on the y-axis. Which statement best describes the correlation?
   Answer: Strong negative correlation The question asks which description fits the relationship between weekly sugary drinks and health score for 8 clients. Direction comes from what happens to the y-values as x increases: here health scores decline as sugary drink consumption rises, which makes the relationship negative, and because the points track that downward pattern closely instead of scattering widely, the correlation counts as strong rather than weak. The description of a strong positive correlation reverses the direction, which would mean more drinks accompany higher health scores. The description of no correlation would require health scores to show no systematic change as drinks increase, contradicting the trend in the data. And calling the relationship perfect and concluding that sugary drinks cause the score change fails twice: real data rarely fall exactly on a line, and no correlation, however tight, establishes causation from recorded observations.
4. A student investigated whether the number of hours of sleep (hours) is related to reaction time (milliseconds) for 9 trials. The scatterplot shows sleep on the x-axis and reaction time on the y-axis. Which statement best describes the pattern?
   Answer: Reaction time decreases as sleep increases (negative association). The question asks which statement describes the pattern in nine paired measurements of sleep and reaction time. Reading the plot from left to right, the points drift downward: trials with more sleep go with lower reaction times, so the description of reaction time decreasing as sleep increases, a negative association, fits the data. The statement that reaction time increases as sleep increases reverses the direction of the trend. The statement that reaction time is constant regardless of sleep would require a roughly flat, patternless cloud of points, which is not what a clear downward drift looks like. And the statement that the plot proves more sleep causes faster reactions for everyone overreaches twice: nine trials in one student's investigation show an association rather than causation, and no scatterplot can establish that a pattern holds for every person.
5. A student measured the length of a candle burned (centimeters) and the time it burned (minutes) for 8 trials. The scatterplot shows time on the x-axis and length burned on the y-axis. Which statement best describes the association and its strength?
   Answer: Strong positive association This question asks about the association between time a candle burned and length burned. The scatterplot shows that as time increases (moving right), length burned increases (moving up), indicating a positive association. The points follow a very clear linear pattern with little scatter, making this a strong positive association. This relationship makes physical sense - the longer a candle burns, the more of its length is consumed. When describing association strength, consider how closely the points follow a linear pattern; tightly clustered points indicate strong association.
6. A researcher recorded the outside temperature (x, in °C) and the number of hot chocolates sold (y, in cups) at a café on 12 different afternoons. The scatterplot shows temperature on the x-axis and cups sold on the y-axis, with a dashed line of best fit. Which statement is best supported by the scatterplot?
   Answer: As temperature increases, hot chocolate sales tend to decrease. The question asks which statement is best supported by the scatterplot of temperature versus hot chocolate sales. The scatterplot exhibits a downward trend from left to right, with points showing moderate spread around the dashed line of best fit, indicating a negative association without apparent curvature. This pattern supports that as temperature increases, sales tend to decrease, as in choice A, based on the negative slope of the line. Reference the data by noting that at lower temperatures, points cluster higher on the y-axis, and vice versa. Choice D errs by claiming direct causation and universality, ignoring variability and that association does not imply cause; B suggests a positive trend against the evidence, and C denies any relationship. Promote describing observed patterns factually while avoiding causal inferences unless experimentally supported. A helpful strategy is to cover the choices and first describe the trend in your own words to build objective interpretation.
7. An environmental scientist measured the percentage of tree canopy cover and the average daytime temperature in 10 neighborhoods. The scatterplot shows canopy cover (%) on the x-axis and temperature (°F) on the y-axis with a line of best fit. Which statement is best supported by the scatterplot?
   Answer: Neighborhoods with higher canopy cover tend to have lower daytime temperatures. This question asks which statement is best supported by the scatterplot of tree canopy cover and daytime temperature. The scatterplot indicates a negative association, with temperatures tending to decrease as canopy cover percentage increases, and points loosely following the downward line of best fit. This supports that neighborhoods with higher canopy cover tend to have lower temperatures, as seen in the overall pattern. Arrive at this by noting the direction of the trend without assuming causation or exact predictions. Errors include reversing the association (choice A), overgeneralizing to 'always' or causation (choices B and C), often from ignoring variability. Focus on descriptive language like 'tend to' to differentiate correlation from causation, enhancing careful interpretation of visual data.
8. A driver recorded the speed of a car and the distance needed to come to a complete stop for 10 trials. The scatterplot shows Speed (mph) versus Stopping Distance (feet), and a line of best fit is drawn. Using the line of best fit, what is the predicted stopping distance when the speed is $50$ mph?
   Answer: 130 ft This question asks us to predict stopping distance at 50 mph using the line of best fit. We need to find x = 50 on the horizontal axis (speed) and trace up to the line of best fit to read the corresponding y-value (stopping distance). At 50 mph, the line of best fit passes through approximately 130 feet on the y-axis. Common errors include reading from actual data points instead of the line, or misreading the scale on either axis. When making predictions, always use the line of best fit rather than trying to estimate between data points.
9. A delivery company recorded the number of packages delivered (packages) and total driving time (minutes) for 12 shifts. The scatterplot shows packages on the x-axis and time on the y-axis, with a line of best fit. Based on the line of best fit, about how many additional minutes of driving time are predicted for each additional package delivered?
   Answer: About 2 minutes per package This question asks about the slope of the line of best fit in the context of packages delivered versus driving time. The slope represents the additional minutes of driving time per additional package. Looking at the line, we can estimate the slope by finding two points on the line and calculating rise over run. The line appears to increase by about 20 minutes for every 10 packages, giving a slope of 20/10 = 2 minutes per package. This means each additional package is associated with about 2 more minutes of driving time. When interpreting slope in context, always include the appropriate units.
10. A student compared the number of practice problems completed and the time to finish a quiz for 12 classmates. The scatterplot shows Practice Problems (problems) versus Quiz Time (minutes). Which statement best describes the relationship?
    Answer: Negative association: more practice corresponds to shorter quiz times. The question asks which description fits the pattern between practice problems completed and quiz time. Reading the scatterplot from left to right, the points trend downward: classmates who completed more practice problems generally finished the quiz in fewer minutes. A pattern where one variable increases while the other decreases is a negative association, so the statement that more practice corresponds to shorter quiz times is the right description. The claim of no association fails because quiz time clearly changes with practice instead of staying the same for every practice amount. The claim of a positive association reverses the direction, predicting longer quiz times as practice increases. The claim of a perfect association overstates the evidence: scattered points that merely trend downward do not all sit exactly on one line, and perfect requires exactly that.
11. A botanist recorded the amount of fertilizer used (grams) and plant height after 6 weeks (centimeters) for 9 plants of the same species. The scatterplot is shown. Which statement is best supported by the scatterplot?
    Answer: There is a positive association; taller plants tend to correspond to more fertilizer. This question asks which statement is best supported by the scatterplot of fertilizer amount versus plant height. Looking at the data, as fertilizer amount increases (moving right), plant height generally increases (points trend upward), showing a positive association. Choice B correctly describes this pattern without making causal claims. Choice A incorrectly claims causation and suggests an exact relationship, while the scatterplot shows variation. Choice C incorrectly states a negative association, and choice D is wrong as there is clearly an upward trend. When interpreting scatterplots, describe the association (positive/negative) without assuming causation - correlation does not imply causation.
12. A fitness coach recorded each client's average daily steps (in thousands) and resting heart rate (in beats per minute). The scatterplot shows 12 clients. Which statement is supported by the scatterplot?
    Answer: As steps increase, resting heart rate tends to decrease. This question asks about the relationship between daily steps and resting heart rate shown in a scatterplot. The data would show a negative association if higher step counts correspond with lower heart rates, which makes physiological sense as more active people tend to have lower resting heart rates. Choice A correctly describes this negative trend without making causal claims or stating exact numerical relationships. Choice D incorrectly suggests a precise causal relationship (exactly 5 bpm drop) which cannot be determined from a scatterplot alone. Scatterplots show trends and associations, not exact cause-and-effect relationships or precise numerical changes. When analyzing health data, remember that correlation suggests a relationship but doesn't prove one variable causes changes in the other.
13. A company compared advertising spending (in thousands of dollars) and weekly sales (in thousands of dollars) for 12 weeks. The scatterplot shows a generally increasing trend with one unusually low sales week. If that outlier week were removed, what would most likely happen to the correlation?
    Answer: The positive correlation would likely become stronger. This question asks about the effect of removing an outlier on correlation. The scatterplot shows a generally increasing trend (positive correlation) with one unusually low sales week that deviates from the pattern. When an outlier pulls away from an otherwise strong linear pattern, removing it typically strengthens the correlation because the remaining points follow the trend more closely. Choice A correctly predicts that removing this low outlier would likely make the positive correlation stronger. Choice D is incorrect because outliers can significantly affect correlation, especially in small datasets. When analyzing the impact of outliers, consider whether they weaken or disrupt the overall pattern - removing disruptive points usually strengthens the association.
14. A school tracked, for 10 students, the number of absences (days) and their final course grade (percent). The scatterplot shows the data and a dashed line of best fit. For the student with $x=8$ absences, approximately how much greater is the actual grade than the grade predicted by the line of best fit?
    Answer: About 8 percentage points This question asks how much greater the actual grade is compared to the predicted grade for a student with 8 absences. First, we locate x = 8 on the horizontal axis and find both the actual data point and the predicted value on the line. The actual data point at x = 8 appears to be at approximately 68%, while the line of best fit predicts about 60% for 8 absences. The difference is 68% - 60% = 8 percentage points, making the actual grade about 8 percentage points greater than predicted. This represents a positive residual, where the student performed better than the model predicted. When calculating residuals, always subtract predicted from actual: residual = actual - predicted.
15. A runner tracked training weeks and 5K race time for 10 races. The scatterplot shows Week (number) versus Time (minutes). The line of best fit decreases as weeks increase. Which conclusion is most reasonable?
    Answer: As training weeks increase, 5K time tends to decrease. This question asks for a reasonable conclusion when training weeks increase and 5K time decreases (shown by a decreasing line of best fit). A decreasing line means that as x increases, y decreases - so more training weeks correspond to faster (lower) 5K times. Option A correctly describes this relationship without claiming causation. Option D overstates the conclusion by claiming proof of causation for every runner, which goes beyond what observational data can show. Scatterplots reveal associations in the observed data but cannot prove universal causal relationships.
16. A mechanic recorded the mileage of a car (thousands of miles) and its stopping distance from 60 mph (feet) for 10 brake tests. The scatterplot shows the data and a line of best fit. For the test at $x=90$ (thousand miles), the actual stopping distance is about how many feet less than the value predicted by the line of best fit?
    Answer: About 4 feet This question asks how much less the actual stopping distance is compared to the predicted value at x = 90 thousand miles. First, we locate x = 90 on the horizontal axis and find both the actual data point and the predicted value on the line. The actual data point at x = 90 appears to be at approximately 66 feet, while the line of best fit predicts about 70 feet. The difference is 70 - 66 = 4 feet, meaning the actual stopping distance is about 4 feet less than predicted. This represents a negative residual. When calculating how much less an actual value is than predicted, compute: predicted - actual.
17. A botanist measured the amount of fertilizer applied and the height of 10 plants after 6 weeks. The scatterplot shows fertilizer (grams) versus plant height (cm), with a line of best fit. What does the slope of the line of best fit represent in this context?
    Answer: The predicted increase in plant height (cm) for each additional gram of fertilizer. The question asks what the slope of the line of best fit means when fertilizer in grams is on the x-axis and plant height in centimeters is on the y-axis. Slope is change in y per one-unit change in x, so it predicts how many additional centimeters of height accompany each extra gram of fertilizer, which is the reading given as the predicted increase in plant height per additional gram. The description of a predicted increase in fertilizer for each additional centimeter of height reverses the two variables and reports grams per centimeter, the reciprocal quantity. The description of plant height when no fertilizer is applied refers to the y-intercept rather than the slope, and it also attaches grams to a height. And the number of plants measured is just the sample size, which no feature of the line describes.
18. A school counselor recorded the number of hours 12 students studied for a math test and their resulting scores. The scatterplot shows hours studied (hours) on the x-axis and test score (points) on the y-axis, along with a dashed line of best fit. Based on the line of best fit, what is the predicted test score when a student studies for 6 hours, and is this prediction an interpolation or extrapolation?
    Answer: About 78 points; interpolation This question asks for the predicted test score based on the line of best fit for a student who studies 6 hours and whether this prediction is an interpolation or extrapolation. The scatterplot displays a positive linear trend, with data points generally increasing from lower hours and scores to higher ones, clustered moderately around the dashed line of best fit. To find the prediction, locate x=6 on the hours axis and trace up to the line of best fit, which intersects at approximately y=78 points on the score axis; since 6 hours falls within the range of observed data points (assuming from about 0 to 8 hours), this is an interpolation. A common error is misreading the graph scale or confusing interpolation (within data range) with extrapolation (beyond data range), leading to incorrect choices like 70 or 84 points with extrapolation. Another mistake might involve eyeballing the line incorrectly, resulting in overestimations like 90 points. When dealing with scatterplots, always verify if the prediction point is inside or outside the observed x-range to distinguish between interpolation and extrapolation, promoting careful visual interpretation over hasty assumptions.
19. A farmer recorded the amount of rainfall (inches) and the number of apples harvested (bushels) for 9 seasons. The scatterplot shows the data and a line of best fit. The farmer uses the line to predict harvest when rainfall is 14 inches, which is beyond the observed range. This prediction is an example of
    Answer: extrapolation This question asks about the type of prediction being made when using rainfall of 14 inches, which is beyond the observed data range. When we use a line of best fit to predict values outside the range of observed x-values, this is called extrapolation. Interpolation would be predicting within the data range, causation refers to cause-and-effect relationships, and "no correlation" would mean no relationship exists. Extrapolation carries more uncertainty than interpolation because we're assuming the linear pattern continues beyond where we have data. When making predictions, always note whether you're interpolating (within data range) or extrapolating (beyond data range), as extrapolation is less reliable.
20. A city analyst compared the outside temperature and the number of hot chocolates sold at a café on 11 different days. The scatterplot shows Temperature (°F) on the x-axis and Hot Chocolates Sold (cups) on the y-axis. Which statement is best supported by the scatterplot?
    Answer: As temperature increases, hot chocolate sales tend to decrease. This question asks which statement is supported by the relationship between temperature and hot chocolate sales. Looking at the scatterplot, as temperature increases (moving right on x-axis), hot chocolate sales decrease (moving down on y-axis), showing a negative association. This makes intuitive sense - people buy less hot chocolate when it's warmer. Option C incorrectly states a specific causal relationship that cannot be determined from a scatterplot alone. Remember that scatterplots show associations, not causation, and avoid making overly specific claims about exact changes.

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