College Statistics Quiz: Scatterplots And Association

20 questions · exam conditions

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Scatterplots And AssociationQuestion 1 of 20

A scatterplot of two variables shows that for low values of the explanatory variable, the response variable tends to be high. As the explanatory variable increases, the response variable tends to decrease. The points on the plot are clustered in a broad, cloud-like pattern around a discernible downward trend. Which of the following is the most appropriate description?

A strong, negative, linear association.

A weak, negative, linear association.

A weak, positive, non-linear association.

No association between the variables.

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College Statistics Quiz

College Statistics Quiz: Scatterplots And Association

Practice Scatterplots And Association in College Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Scatterplots And Association, giving you a quick way to practice the rules, question types, and explanations that matter most for College Statistics.

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.

All questions

Question 1

A strong, negative, linear association.
A weak, negative, linear association. (correct answer)
A weak, positive, non-linear association.
No association between the variables.

Explanation: The description that the response variable decreases as the explanatory variable increases indicates a negative direction. The fact that the points are in a 'broad, cloud-like pattern' suggests that the relationship is not precise, meaning it is weak. A 'discernible downward trend' confirms that there is an association, and in the absence of information about curvature, a linear form is the default description for a general trend.

Question 2

A regression analysis of student scores on a final exam versus the number of classes they missed produced an $R^2$ value of 0.64. A scatterplot of the data shows a clear downward linear trend. Which statement is the best description of the association?

A weak, negative, linear association.
A strong, positive, linear association.
A non-linear association, because $R^2$ is not equal to 1.
A moderately strong, negative, linear association. (correct answer)

Explanation: When analyzing correlation and regression results, you need to interpret both the coefficient of determination (

R^2

) and the direction of the relationship shown in the scatterplot. The

R^2

value of 0.64 tells you that 64% of the variation in final exam scores is explained by the number of classes missed. To interpret the strength of this association, remember that

R^2

values around 0.25 indicate weak associations, values around 0.64 suggest moderately strong associations, and values above 0.81 indicate strong associations. The scatterplot shows a "clear downward linear trend," meaning as classes missed increases, exam scores decrease—this is a negative linear relationship. Answer D correctly identifies this as a moderately strong, negative, linear association, combining the strength indicated by

R^2 = 0.64

with the negative direction from the scatterplot. Answer A is wrong because

R^2 = 0.64

indicates moderate strength, not weakness. Answer B incorrectly identifies the direction as positive when the scatterplot clearly shows a downward trend. Answer C makes a fundamental error—

R^2

doesn't need to equal 1 for a linear association to exist.

R^2 = 1

would indicate a perfect linear relationship, but real-world data rarely achieves this. Remember this rule:

R^2

tells you strength (how tightly points cluster around the line), while the scatterplot's visual trend tells you direction (positive or negative slope). You need both pieces of information to fully describe the association.

Question 3

Two dietitians study the relationship between daily calorie intake and weight loss. Dietitian A studies subjects on a 1200-1500 calorie diet, while Dietitian B studies subjects on a 2000-2300 calorie diet. Both find a strong, negative, linear association (higher intake corresponds to less weight loss). The slope of the regression line for Dietitian A's group is -0.1, and for Dietitian B's group it is -0.3. Which conclusion is justified?

The association between calorie intake and weight loss is stronger for Dietitian B's group.
The data for Dietitian B's group must be more linear (less scattered) than the data for Dietitian A's group.
The range of calorie intake was greater for Dietitian A's group, leading to a less steep slope.
A steeper slope means that in Dietitian B's group, an increase in calorie intake is associated with a larger decrease in weight loss. (correct answer)

Explanation: The slope of a regression line represents the average change in the response variable (weight loss) for a one-unit change in the explanatory variable (calorie intake). A steeper slope (a value further from zero, in this case -0.3 vs -0.1) indicates a greater change. Slope does not measure the strength or linearity of an association; those are measured by the correlation coefficient

r

. The problem states both found a 'strong' association, so we cannot conclude one is stronger than the other based on the slope.

Question 4

A university analyzes the relationship between the number of extracurricular activities a student participates in and their GPA. When all students are analyzed together, there is a weak negative association. However, when students are separated into two groups—those with part-time jobs and those without—analysts find a moderate positive association between activities and GPA within each group. Which statement is the most likely explanation for this observation?

The overall negative association is the most reliable result, and the positive association within the groups is a statistical anomaly.
The form of the relationship must be non-linear, which causes the perceived direction of association to change.
The overall negative association is an example of Simpson's Paradox, likely caused by the lurking variable of having a part-time job. (correct answer)
A data entry error must have occurred; it is mathematically impossible for the direction of association to reverse when groups are combined.

Explanation: This scenario describes Simpson's Paradox, where a trend that appears in different groups of data reverses when the groups are combined. Here, having a job is a lurking variable. Students with jobs might have less time for both extracurriculars and studying, leading to lower GPAs and fewer activities on average than students without jobs. This creates an overall negative trend when the groups are combined, even if within each group, students who manage more activities also tend to have higher GPAs.

Question 5

A researcher creates a scatterplot of student study hours per week versus their grade point average (GPA). The plot shows a moderate, positive, linear association. A new data point is added for a student whose data was mistyped: 25 study hours and a GPA of 0.5. This point is a significant outlier compared to the general trend of the original data. How will this new point most likely affect the description of the association?

The strength of the positive association will increase, and the form will appear more linear.
The strength of the positive association will decrease, and the form may appear less linear. (correct answer)
The direction of the association will become negative, but the form will remain linear.
The association will be largely unaffected, as a single outlier rarely changes the overall trend significantly.

Explanation: The new point (high study hours, very low GPA) is an influential outlier that contradicts the existing positive trend. It will 'pull' the regression line towards it, decreasing the slope and thus weakening the positive correlation (decreasing the value of

r

). The presence of this single point far from the general pattern could also make the overall form appear less linear.

Question 6

A scatterplot shows a moderately strong, positive, linear association between the weight of a package in pounds and its shipping cost in dollars. If the package weights are converted from pounds to kilograms (1 kg ≈ 2.2 lbs) and the costs are converted from US dollars to Euros (€1 ≈ $1.10), how will the association be affected?

The association will become weaker because the numerical values of the variables have changed.
The direction of the association will become negative due to the conversion formulas.
The strength and direction of the association will remain the same, but the slope of the regression line will change. (correct answer)
The scatterplot will look identical, and all quantitative measures of the association will be unchanged.

Explanation: The correlation coefficient (

r

), which measures the strength and direction of a linear association, is unaffected by linear transformations (like converting units) of the variables. Therefore, the strength and direction remain the same. However, the slope of the regression line (

\text{slope} = r \frac{s_y}{s_x}

) depends on the standard deviations (

s_y

and

s_x

) of the variables, which change when units are converted. Thus, the slope will change.

Question 7

A city finds a strong positive association between the number of firefighters responding to a fire and the amount of damage caused by the fire. A city official, reasoning that the firefighters are causing the damage, proposes sending fewer firefighters to fires to reduce damages. What is the primary flaw in this reasoning?

The strength of the association is likely exaggerated and not sufficient to support a policy change.
The official has reversed the direction of the association; less damage causes fewer firefighters to be sent.
The observed association is due to a lurking variable; the size of the fire influences both the number of firefighters and the amount of damage. (correct answer)
The form of the association is likely non-linear, so the conclusion that more firefighters leads to more damage is invalid.

Explanation: This is a classic example of confusing correlation with causation. A third variable, the size or severity of the fire (a lurking variable), is responsible for the association. Larger fires require more firefighters to be sent and they cause more damage. The firefighters are not causing the damage; they are both a response to the fire's size.

Question 8

A biologist studies the relationship between the ambient temperature of a habitat and the activity level of a certain species of lizard. Which of the following descriptions of the association is most plausible from a biological standpoint?

A strong, positive, linear association, as lizards are more active in warmer temperatures.
A strong, negative, linear association, as heat causes lizards to become lethargic.
A non-linear (curved) association, where activity increases with temperature up to an optimal point, then decreases as it gets too hot. (correct answer)
No association, as lizard activity is determined by food availability, not temperature.

Explanation: Many biological processes exhibit an optimal range. Lizards are cold-blooded, so their activity increases as temperature rises from a low point. However, if the temperature becomes excessively high, they will seek shelter to avoid overheating, causing their activity to decrease. This would create a non-linear, inverted U-shaped relationship on a scatterplot.

Question 9

An education board reports that for high school students, there is a moderate, positive association between the number of advanced placement (AP) courses taken and their final high school GPA. Which of the scatterplots described below would best represent this finding?

Points are tightly clustered around a line that slopes downward from top-left to bottom-right.
Points are somewhat scattered in a band that goes from the bottom-left to the top-right. (correct answer)
Points are widely scattered in a random cloud with no clear direction or form.
Points form a distinct V-shape, starting high on the left, dipping low in the middle, and ending high on the right.

Explanation: A positive association means the trend goes from bottom-left to top-right. A moderate association means the points are 'somewhat scattered' but still form a discernible linear pattern. 'Tightly clustered' (Choice A) would imply a strong association. 'Widely scattered' (Choice C) would imply a weak or no association. A V-shape (Choice D) is non-linear.

Question 10

An analyst wants to investigate the relationship between the number of cylinders in a car's engine (a quantitative variable) and the car's country of origin (a categorical variable). They create a scatterplot with number of cylinders on the y-axis and country of origin on the x-axis. The resulting plot shows three distinct vertical lines of points. What is the most appropriate conclusion?

There is a strong positive linear association between the two variables.
There is no association between the variables because the points do not form a single line or curve.
The association is non-linear, as indicated by the multiple vertical lines.
A scatterplot is an inappropriate graph for this data, and conclusions about association should not be drawn from it. (correct answer)

Explanation: When analyzing relationships between variables, the type of variables involved determines which visualization methods are appropriate. This question tests your understanding of when scatterplots should and shouldn't be used. A scatterplot requires both variables to be quantitative (numerical) because it plots points along continuous numerical axes. Here, you have number of cylinders (quantitative) and country of origin (categorical). When you force categorical data onto a numerical axis, you get the described pattern: vertical lines of points. This happens because all cars from each country cluster at discrete x-positions, with cylinders varying only vertically. The correct answer is D because scatterplots simply aren't designed for mixed variable types. The three vertical lines don't represent any meaningful pattern about association—they're an artifact of using the wrong graph type. Drawing conclusions about relationships from this inappropriate visualization would be misleading. Answer A is wrong because vertical lines don't indicate linear association—that would require points trending diagonally. Answer B incorrectly assumes the lack of a single line/curve means no association exists, but the issue is the graph type, not the relationship itself. Answer C misinterprets the vertical lines as indicating non-linear association, when they actually just show the categorical nature being forced onto a continuous axis. Study tip: Always check variable types before choosing visualizations. Use scatterplots only when both variables are quantitative. For mixed types like this, consider side-by-side boxplots or comparative histograms instead.

Question 11

A researcher studies the relationship between age and blood pressure across a wide range of adults (ages 20-80) and finds a moderate, positive, linear association. If the researcher restricts the analysis to only include adults in a narrow age range, such as 40-45 years old, how will the strength of the association likely change for this subgroup?

The association will likely become much stronger because the relationship is more focused.
The association will likely become much weaker, possibly close to zero. (correct answer)
The association will have the exact same strength as in the full population.
The direction of the association will likely reverse from positive to negative.

Explanation: Restricting the range of the explanatory variable (age) tends to weaken the correlation. The original moderate correlation was visible because of the large variation in age from 20 to 80. Within a very narrow 5-year range, the corresponding variation in blood pressure will be much smaller and will likely look more like a random cloud of points. This will cause the correlation coefficient to be much closer to zero, indicating a weak or nonexistent linear association in that specific subgroup.

Question 12

Which of the following statements most accurately describes the distinction between the form and the strength of an association on a scatterplot?

Form refers to the general pattern of the points (e.g., linear, curved), while strength refers to how closely the points follow that pattern. (correct answer)
Form describes whether the association is positive or negative, while strength describes whether it is linear or curved.
Strength is measured by the slope of the regression line, while form is determined by the y-intercept.
Strength describes whether the association is causal, while form describes whether it is merely correlational.

Explanation: When analyzing scatterplots, you need to distinguish between two key characteristics of associations: form and strength. These concepts help you describe the relationship between variables systematically. Form describes the overall shape or pattern that the data points create on the scatterplot. This could be linear (points roughly follow a straight line), curved (quadratic, exponential, etc.), or have no discernible pattern. Strength measures how tightly the points cluster around whatever pattern exists - whether the points closely follow the form or are widely scattered around it. Answer A correctly captures this distinction: form identifies the general pattern (linear, curved, etc.), while strength indicates how closely points adhere to that pattern. A strong linear relationship shows points tightly clustered around a straight line, while a weak linear relationship has the same straight-line pattern but with points more scattered. Answer B confuses form with direction. Whether an association is positive or negative describes the direction, not the form. Answer C incorrectly links strength to slope and form to y-intercept - these are components of the regression equation, not fundamental characteristics of scatterplot associations. Answer D misunderstands both concepts entirely by bringing in causation versus correlation, which relates to interpretation of relationships, not their visual characteristics on scatterplots. Study tip: When examining any scatterplot, always ask yourself two questions in order: "What pattern do I see?" (form), then "How closely do the points follow that pattern?" (strength). This systematic approach will help you accurately describe any association.

Question 13

The strength of the positive association will increase, and the form will appear more linear.
The strength of the positive association will decrease, and the form may appear less linear. (correct answer)
The direction of the association will become negative, but the form will remain linear.
The association will be largely unaffected, as a single outlier rarely changes the overall trend significantly.

r

). The presence of this single point far from the general pattern could also make the overall form appear less linear.

Question 14

The association will become weaker because the numerical values of the variables have changed.
The direction of the association will become negative due to the conversion formulas.
The strength and direction of the association will remain the same, but the slope of the regression line will change. (correct answer)
The scatterplot will look identical, and all quantitative measures of the association will be unchanged.

Explanation: The correlation coefficient (

r

\text{slope} = r \frac{s_y}{s_x}

) depends on the standard deviations (

s_y

and

s_x

) of the variables, which change when units are converted. Thus, the slope will change.

Question 15

The strength of the association is likely exaggerated and not sufficient to support a policy change.
The official has reversed the direction of the association; less damage causes fewer firefighters to be sent.
The observed association is due to a lurking variable; the size of the fire influences both the number of firefighters and the amount of damage. (correct answer)
The form of the association is likely non-linear, so the conclusion that more firefighters leads to more damage is invalid.

Question 16

A strong, positive, linear association, as lizards are more active in warmer temperatures.
A strong, negative, linear association, as heat causes lizards to become lethargic.
A non-linear (curved) association, where activity increases with temperature up to an optimal point, then decreases as it gets too hot. (correct answer)
No association, as lizard activity is determined by food availability, not temperature.

Question 17

A scatterplot shows three data points: A(2, 2), B(4, 5), and C(6, 3). Which of the following fourth points, when added to the plot, would most substantially weaken any existing linear association?

(5, 9) (correct answer)
(8, 4)
(4, 3.5)
(1, 1)

Explanation: The original three points form a weak, slightly positive association. To weaken an association, one should add a point that deviates significantly from the established trend. Point (8, 4) is roughly consistent with the trend. Point (4, 3.5) is in the center of the data and would have little effect. Point (1, 1) is also reasonably consistent with the trend. Point (5, 9) is a significant outlier in the y-direction, lying far above the trend of the other points. Adding it would pull the regression line strongly upwards and increase the scatter, thus substantially weakening the linear association and decreasing the correlation coefficient.

Question 18

An analyst creates a scatterplot with a person's height as the explanatory variable (X-axis) and their foot length as the response variable (Y-axis). The plot shows a strong, positive, linear association. If the analyst swaps the axes, what will be the effect on the Pearson correlation coefficient $r$ and the slope of the least-squares regression line?

The correlation coefficient $r$ will remain the same, but the slope will change. (correct answer)
Both the correlation coefficient $r$ and the slope will remain the same.
The correlation coefficient $r$ will change, but the slope will remain the same.
Both the correlation coefficient $r$ and the slope will change.

Explanation: When you encounter questions about swapping variables in correlation and regression analysis, focus on the fundamental mathematical properties of these statistical measures. The Pearson correlation coefficient

r

measures the strength and direction of the linear relationship between two variables. Crucially, correlation is symmetric — it doesn't matter which variable you call X and which you call Y. The formula

r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2 \sum(y_i - \bar{y})^2}}

produces the same result regardless of variable order. So swapping height and foot length won't change

r

. The slope of the regression line, however, is not symmetric. When height predicts foot length, the slope tells you how much foot length increases per unit increase in height. When foot length predicts height, you get a completely different slope representing how much height increases per unit increase in foot length. Mathematically, if the original slope is

b

, the new slope becomes

\frac{r^2}{b}

(not simply

\frac{1}{b}

). Looking at the options: B is wrong because while

r

stays the same, the slope definitely changes. C is wrong because

r

is symmetric and doesn't change when axes swap. D is wrong because it incorrectly assumes both measures change. Study tip: Remember "correlation is symmetric, regression is not." The correlation coefficient treats both variables equally, but regression slopes depend entirely on which variable is predicting which. This distinction appears frequently on statistics exams.

Question 19

The correlation coefficient $r$ will remain the same, but the slope will change. (correct answer)
Both the correlation coefficient $r$ and the slope will remain the same.
The correlation coefficient $r$ will change, but the slope will remain the same.
Both the correlation coefficient $r$ and the slope will change.

r

r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2 \sum(y_i - \bar{y})^2}}