PSAT Math Quiz: Scatter Plots

What this quiz covers

This quiz focuses on Scatter Plots, giving you a quick way to practice the rules, question types, and explanations that matter most for PSAT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. P
2. Q
3. R (correct answer)
4. S

Explanation: Residual magnitude is the vertical distance from a point to the line of best fit. Point R is visually farthest from the line in the vertical direction (about 8 units below the line). P is on the line, Q is about 2 units above, and S is about 4 units below. Thus R has the greatest |residual|.

Question 2

1. $-40$ (correct answer)
2. $40$
3. $520$
4. $1{,}000$

Explanation: Predicted sales: $14(80) - 600 = 1120 - 600 = 520$. Residual = actual − predicted = $480 - 520 = -40$. Choice B reverses the sign. Choice C is the predicted value. Choice D comes from $14 \times 80 - (-600)$, a sign error in the intercept.

Question 3

1. The predicted increase in plant height (cm) for each additional gram of fertilizer. (correct answer)
2. The predicted increase in fertilizer (g) for each additional centimeter of height.
3. The plant height when fertilizer is $1$ gram.
4. The total height increase for all plants combined.

Explanation: This question asks about the meaning of the slope in the context of fertilizer and plant height. In a scatterplot with fertilizer (grams) on the x-axis and height (cm) on the y-axis, the slope represents the change in y per unit change in x. Therefore, the slope tells us the predicted increase in plant height (cm) for each additional gram of fertilizer applied. A common error is reversing the interpretation (choice B) - remember that slope is always rise over run, or change in y over change in x. The slope does not represent a specific height value (choice C) or a total (choice D), but rather a rate of change.

Question 4

1. There is a strong positive linear association between the two variables. (correct answer)
2. There is a strong negative linear association between the two variables.
3. There is no apparent association between the two variables.
4. The association is positive but clearly nonlinear (curved).

Explanation: The points rise steadily from left to right, forming an upward-sloping cloud that is roughly straight, indicating a strong positive linear relationship. B is wrong because the pattern is increasing, not decreasing. C is wrong because a clear trend is visible. D is wrong because the pattern does not show noticeable curvature; it is approximately linear.

Question 5

1. 2 (correct answer)
2. 4
3. -2
4. 10

Explanation: The drawn best-fit line crosses the y-axis at approximately (0,2). Therefore, the y-intercept is about 2. The other numbers do not match the visible intercept point.

Question 6

1. $14$
2. $28$ (correct answer)
3. $42$
4. $104$

Explanation: The predicted difference is the slope times the difference in x: $3.5(12-4) = 3.5(8) = 28$. Choice A uses 4 sessions instead of 8. Choice C uses 12. Choice D is the predicted score at x=12 ($3.5(12)+62=104$), not the difference.

Question 7

1. Raising the price will always cause sales to drop by the same amount.
2. Weeks with higher prices tend to be associated with fewer units sold. (correct answer)
3. Weeks with higher prices tend to be associated with more units sold.
4. Price and units sold show no relationship in the data.

Explanation: This question asks which claim is supported by the scatterplot of price versus units sold, without assuming causation. The plot shows a negative association, with points trending downward, meaning higher prices link to fewer units sold across the weeks. Referencing the data pattern, the supported claim is that weeks with higher prices tend to be associated with fewer units sold, emphasizing correlation over cause. This avoids overreaching into causation like choice A. Common errors include assuming causation or misinterpreting the direction as positive (choice C) or none (choice D). Another mistake is implying constancy in the relationship, ignoring variability. When evaluating scatterplots, prioritize descriptive statements about associations to distinguish them from inferential claims in data literacy.

Question 8

1. The predicted plant height when $0$ grams of fertilizer are used.
2. The increase in plant height (cm) for each additional gram of fertilizer, on average. (correct answer)
3. The number of grams of fertilizer needed for each additional centimeter of height.
4. The difference between the tallest and shortest plant heights in the data set.

Explanation: The question asks what the slope of the line of best fit represents in the context of fertilizer amount versus plant height. The scatterplot shows a positive linear trend, with points spreading around the line, indicating that more fertilizer is generally linked to taller plants after 6 weeks. The slope, calculated as the change in height per unit change in fertilizer, represents the average increase in plant height (in cm) for each additional gram of fertilizer. This interpretation references the line's steepness and its real-world meaning in the data pattern. Common errors include confusing slope with y-intercept (choice A) or inverting the relationship (choice C). Another mistake is interpreting it as a range rather than a rate (choice D). When examining lines of best fit, focus on slope as a rate of change to build skills in interpreting visual trends descriptively.

Question 9

1. $y \approx -0.03x + 40$
2. $y \approx -0.3x + 4$
3. $y \approx -3x + 40$ (correct answer)
4. $y \approx 3x - 40$

Explanation: The points fall along a line that decreases about 3 units in $y$ for every 1 unit increase in $x$ and crosses the y-axis near 40, matching choice C. Choice A has too small a slope, B has correct intercept but wrong scale, and D has the wrong sign for the slope.

Question 10

1. $r \approx -0.85$ (correct answer)
2. $r \approx -0.20$
3. $r \approx 0.30$
4. $r \approx 0.90$

Explanation: The points cluster tightly around a downward-sloping line, indicating a strong negative association, so $r$ should be close to −1, with −0.85 the best of the given values. B shows only a weak negative correlation, C a weak positive, and D a strong positive—none of which match the graph.

Question 11

1. (5, 52)
2. (9, 31)
3. (12, 80)
4. (14, 15) (correct answer)

Explanation: The existing data show a moderate positive trend: as $x$ increases, $y$ roughly increases. Point (14, 15) lies far below the trend line at a large $x$-value, making it a leveraged outlier that would pull the correlation downward. Points A and B fall near the present trend, while C also extends the trend upward, so they would either maintain or strengthen the correlation.

Question 12

1. The value of $y$ when $x = 0$
2. The value of $y$ when $x = 3$
3. The value of $y$ when $x = 4$
4. The value of $y$ when $x = 7$ (correct answer)

Explanation: The plotted data cover $x$-values from 1 to 6. Predicting at $x = 7$ goes furthest beyond this range, making it the least reliable extrapolation. The prediction at $x = 0$ is also outside the data but is closer than 7. Choices B and C are within the data range and therefore interpolations.

Question 13

1. 1 (correct answer)
2. -1
3. 7
4. -7

Explanation: For $x = 6,$ the line predicts $y = 2(6) + 1 = 13.$ The actual plotted value is 14, so the residual (actual − predicted) is 14 − 13 = 1. B gives the negative of this value, and C and D confuse the difference in location with the residual.

Question 14

1. $r$ would increase because the point continues the upward trend. (correct answer)
2. $r$ would decrease because the point is an outlier weakening the trend.
3. $r$ would remain the same because adding points never changes $r$.
4. $r$ would become 0 because the data would be uncorrelated.

Explanation: The existing points form an upward trend that appears to approach (9, 85). The new point (10, 90) is consistent with that pattern, so it would strengthen the linear relationship and make $r$ larger in magnitude. B is incorrect because the point is not an outlier relative to the trend. C is false; adding any point can change $r$. D is obviously false because the relationship would not become uncorrelated.

Question 15

1. $9,000
2. $10,000 (correct answer)
3. $11,000
4. $13,000

Explanation: The best-fit line descends roughly $2,000 for every additional year and crosses about $24,000 when the car is new. Substituting 7 years gives $24,000 − $2,000(7) = $10,000. The other values are either too low or too high relative to the plotted trend.

Question 16

1. The plant's height increases about 4.5 centimeters for each additional day after planting. (correct answer)
2. The plant's height increases about 32 centimeters each day after planting.
3. The plant's height increases about 4.5 days for every additional centimeter of height.
4. On day 0, the plant's height is about 4.5 centimeters.

Explanation: Slope is rise over run, so 4.5 means the height (centimeters) rises 4.5 for every 1 day. B confuses slope with intercept. C inverts the units, and D confused slope with y-intercept.

Question 17

1. The predicted mean SAT math score at a school where no students receive free or reduced-price lunch is $640$. (correct answer)
2. The mean SAT math score decreases by $640$ points for each 1% increase in students receiving free or reduced-price lunch.
3. The mean SAT math score remains constant at $640$ points, regardless of the percentage of students receiving lunch assistance.
4. At a school where the mean SAT math score is $640$, the model predicts that no students receive free lunch.

Explanation: The y-intercept is the predicted $y$-value when $x=0$, which here means the predicted mean SAT math score when $0\%$ of students receive free/reduced-price lunch. Choice B confuses the intercept with the slope. Choice C ignores that the line has nonzero slope. Choice D reverses the roles of $x$ and $y$ and misstates a prediction as a certainty.

Question 18

1. Curve A: $y = 2x + 3$
2. Curve B: $y = 0.5x^2 + 1$
3. Curve C: $y = 8\sqrt{x}$ (correct answer)
4. Curve D: $y = 3(1.5)^x$

Explanation: The data points (1, 8), (4, 16), (9, 24), (16, 32) fit $y=8\sqrt{x}$ exactly: $8\sqrt{1}=8, 8\sqrt{4}=16, 8\sqrt{9}=24, 8\sqrt{16}=32$. Curve A (linear) underpredicts at small x and misses curvature. Curve B (parabolic upward) grows much too fast (at x=16 it predicts 129). Curve D (exponential) grows explosively (at x=16 it predicts about 1,500). Only curve C matches the concave-down, slowly increasing shape of the data.

Question 19

1. Positive and linear
2. Negative and linear
3. Nonlinear (U-shaped) (correct answer)
4. No association

Explanation: Electricity usage is high at low temperatures (heating) and high at high temperatures (air conditioning), with a minimum in the middle range. This produces a U-shaped (nonlinear) pattern. A purely linear description—positive or negative—misses the minimum at moderate temperatures. There is clearly an association, so choice D is wrong.

Question 20

1. As length increases, period increases linearly at a constant rate.
2. As length increases, period increases, but at a decreasing rate. (correct answer)
3. As length increases, period decreases.
4. There is no association between length and period.

Explanation: The scatter shows period increasing with length but with a clearly concave-down (square-root-like) shape: going from L = 0.25 to 1.0 produces a larger change than from L = 1.5 to 2.25. This indicates an increasing but decelerating relationship, consistent with $T = 2\pi\sqrt{L/g}$. Choice A (linear) ignores the curvature. Choice C contradicts the positive trend. Choice D ignores the strong association.