Aimathic
Login | English | Deutsch

Statistics math worksheets

Build your own math worksheets from 35,000+ problems for grades 3 to 12, from fractions to AP Calculus. Every problem comes with step-by-step solutions.

Scatterplots and association

Learning goals

  • Draws a scatterplot for two quantitative variables.
  • Reads paired values from a scatterplot.
  • Describes direction, form, and strength of association.
  • Identifies unusual points or clusters.
  • Distinguishes association from causation.

Click problems to add them to your worksheet.

Scatterplots and association 1
A study asks whether distance from a stage in feet helps explain measured sound level in decibels. Which variable belongs on each axis of the scatterplot, and what does one point represent?

Hints

- Ask which measurement could help predict or explain the other measurement. - Put the possible explanatory variable on the x-axis and the response variable on the y-axis. - A scatterplot point must pair the two measurements from the same location.

Solution

1. Distance from the stage is the possible explanatory variable, so it belongs on the x-axis. 2. Measured sound level is the response variable, so it belongs on the y-axis. 3. Each point pairs the distance and sound-level measurements taken at the same observation location.

Answer

x-axis: distance from the stage in feet; y-axis: measured sound level in decibels. Each point represents the distance and sound level recorded at one observation location.
Scatterplots and association 2
Suppose the number of practice serves for players in one sample is plotted against the number of successful serves out of \(20\) for players in a different, unrelated sample. Explain why this would not form a meaningful scatterplot for studying association.

Hints

- Identify what one point in the proposed scatterplot would claim about a single player. - Check whether each practice-serve count and success count came from the same person. - Consider whether a pattern made from arbitrary pairings describes any real observational units.

Solution

1. Each bivariate observation must contain both quantitative measurements from the same player. 2. Pairing a practice-serve value from one sample with a success value from an unrelated sample creates arbitrary ordered pairs. 3. Because the pairs do not describe actual players, the plotted pattern cannot represent an association between the two variables.

Answer

The two measurements must be paired for the same players. Arbitrary pairings from unrelated samples do not represent a meaningful association.
Scatterplots and association 3
While describing a scatterplot of hours of daylight and electricity used for indoor lighting in kilowatt-hours, consider the statement: “The y-axis variable should always be the variable measured second.” Identify the error and give a corrected statistical statement.

Hints

- Ask which variable could help explain changes in the other. - The explanatory variable normally goes on the x-axis. - The response variable normally goes on the y-axis, regardless of when it was measured.

Solution

1. Measurement order does not determine scatterplot axes. 2. Hours of daylight is the plausible explanatory variable, so it belongs on the x-axis. 3. Electricity used for indoor lighting is the response variable, so it belongs on the y-axis.

Answer

The error is using measurement order to choose axes. Hours of daylight belongs on the x-axis as the explanatory variable, and indoor-lighting electricity use belongs on the y-axis as the response variable.
Scatterplots and association 4
Consider a proposed scatterplot with weekly study time, in hours, on the x-axis and final course letter grade \(A\), \(B\), \(C\), \(D\), or \(F\) on the y-axis. Explain why this is not an ordinary scatterplot of two quantitative variables, and identify what would need to change for a standard quantitative scatterplot to be appropriate.

Hints

- Classify each variable before choosing a graph. - A scatterplot places numerical measurements on both axes. - Think of a response that preserves numerical distance rather than category labels.

Solution

1. Weekly study time is quantitative, but letter grade is categorical rather than quantitative. 2. A standard scatterplot represents paired values of two quantitative variables. 3. A quantitative response such as numerical exam score or course percentage could be paired with study time in a standard scatterplot.

Answer

The proposed display is not a standard quantitative scatterplot because letter grade is categorical. A quantitative response variable would be needed for an ordinary two-quantitative-variable scatterplot.
Scatterplots and association 5
A study records the number of customer-support tickets submitted by each business in a week and the business's average response time, in minutes. Consider the statement: “A scatterplot is inappropriate because the number of tickets is a whole-number count rather than a continuous measurement.” Evaluate the statement.

Hints

- Classify the variables as quantitative or categorical. - Do not confuse “discrete” with “categorical.” - Ask whether numerical differences in the ticket count have meaningful size.

Solution

1. The number of tickets is a quantitative variable even though it takes discrete whole-number values. 2. Average response time is also quantitative. 3. A scatterplot is appropriate for paired observations of two quantitative variables; the explanatory variable does not have to be continuous. 4. Repeated ticket counts may create vertical stacks of points, but that does not make the scatterplot invalid.

Answer

The statement is incorrect. A whole-number count is still quantitative, so a scatterplot is appropriate for paired ticket counts and response times.
Scatterplots and association 6
A data file contains \(50\) students. Every student has a recorded study-time value, but \(7\) students are missing an exam-score value. A scatterplot of study time versus exam score contains \(43\) points. Consider the claim: “The graphing software lost seven observations.” Evaluate the claim.

Hints

- Identify what information is needed to locate one point in a scatterplot. - Count how many observations have both coordinates available. - Distinguish missing measurements from points that a graphing program failed to display.

Solution

1. A point in the scatterplot requires both an explanatory value and a response value for the same student. 2. The \(7\) students with missing exam scores do not have complete ordered pairs, so they cannot be placed on this scatterplot. 3. The \(43\) plotted points are therefore consistent with the available complete pairs; the missing responses should be noted as missing data rather than treated as plotting failures.

Answer

The graph is consistent with the data. Only the \(43\) students with both study time and exam score can appear as points; the other \(7\) lack complete pairs.
Scatterplots and association 7
The scatterplot shows a data set in which the response values vary. Can this plot be used to describe a positive or negative association between \(x\) and \(y\)? Explain.
Figure for problem 548159

Hints

- Count the distinct horizontal coordinates represented in the plot. - Association requires comparing responses across different explanatory values. - Variation in \(y\) alone does not determine a direction of association with \(x\).

Solution

1. Every point has horizontal coordinate \(5\), so the plot is a vertical stack at \(x=5\). 2. There is no variation in the explanatory variable, so the plot provides no information about how \(y\) changes as \(x\) changes. 3. Therefore, a positive or negative association cannot be assessed from these data.

Answer

No. Because every observation has \(x=5\), the explanatory variable does not vary, so the data cannot show a positive or negative association between \(x\) and \(y\).
Scatterplots and association 8
By what factor is the original input \(x\) larger at B than at A?
Figure for problem 563062

Hints

- The graph plots the logarithm of the input. - Which original numbers have base-10 logarithms of 1 and 2?

Solution

1. A is at \(\log_{10}(x)=1\), so its original input is \(x=10\). 2. B is at \(\log_{10}(x)=2\), so its original input is \(x=100\). 3. The factor is \(100\div10=10\).

Answer

A factor of \(10\).
Scatterplots and association 9
Each dot shows how many practice sessions a singer completed and how many of six songs they performed correctly. What ordered pair represents point B? State which number is the input.
Figure for problem 563063

Hints

- Read the horizontal axis first. - A point's second coordinate comes from its height.

Solution

1. Read B's horizontal coordinate as \(4\) practice sessions. 2. Its vertical coordinate is \(3\) songs performed correctly. 3. The point is \((4,3)\), with practice sessions as the input.

Answer

\((4,3)\). The input is \(4\) practice sessions.
Scatterplots and association 10
A plant study recorded the pairs \((1,2)\), \((2,4)\), \((3,3)\), and \((4,5)\), where each pair gives weeks and height in inches. One observation is missing from the scatterplot. Which one?
Figure for problem 563064

Hints

- Compare each listed input with the dots on the graph. - Make sure both coordinates match before deciding a pair is present.

Solution

1. The graph contains \((1,2)\), \((2,4)\), and \((3,3)\). 2. The unplotted observation is \((4,5)\).

Answer

\((4,5)\).
Scatterplots and association 11
Use the scatterplot of kiln firing time, in hours, and surface hardness score. Describe the association by form, direction, strength, and unusual features.
Figure for problem 539409

Hints

- Read the plotted points from left to right and note how hardness changes as firing time increases. - Decide whether the points lie close to a straight line or follow a curved pattern. - Check whether any point is far from the overall trend.

Solution

1. As kiln firing time increases from \(1\) to \(7\) hours, the hardness scores generally increase. 2. The points cluster closely around an upward-sloping straight line, so the association is strong, positive, and linear. 3. No point lies noticeably away from the overall pattern.

Answer

A strong positive linear association with no clear unusual points.
Scatterplots and association 12
Use the scatterplot of distance along a hiking trail, in miles, and elevation, in feet. Describe the association by form, direction, strength, and unusual features.
Figure for problem 539410

Hints

- Follow the plotted elevations as trail distance increases. - Judge whether the points lie near one downward-sloping straight line. - Look for a point that breaks the overall pattern by a large amount.

Solution

1. As trail distance increases from \(1\) to \(7\) miles, elevation generally decreases from about \(2650\,\text{ft}\) to \(1150\,\text{ft}\). 2. The points cluster closely around a downward-sloping straight line, so the association is strong, negative, and linear. 3. No point lies noticeably away from the overall pattern.

Answer

A strong negative linear association with no clear unusual points.
Scatterplots and association 13
Use the scatterplot of the number of volunteers and the number of boxes packed in \(30\) minutes. Consider these statements: A. “The variables have a strong positive linear association with one unusual low point.” B. “Every increase in the number of volunteers causes the number of boxes packed to increase.” Which statement is better supported, and why?
Figure for problem 539418

Hints

- Compare each statement with all four parts of a scatterplot description: form, direction, strength, and unusual features. - Decide whether a scatterplot alone can establish that changing the number of volunteers causes a change in output. - A strong trend can still contain an observation that does not increase with the explanatory variable.

Solution

1. Statement A accurately describes the form, direction, strength, and unusual feature shown in the scatterplot. 2. Statement B is not supported because an association does not by itself prove causation, and a strong association does not mean the response increases for every individual observation.

Answer

Statement A. It describes the strong positive linear association and the unusual low point without making an unsupported causal or exception-free claim.
Scatterplots and association 14
Use the scatterplot of launch angle, in degrees, and horizontal distance traveled, in yards. Consider these statements: A. “The variables have a strong nonlinear association.” B. “The variables have no association because the overall direction is neither positive nor negative.” Which statement is better supported, and why?
Figure for problem 539419

Hints

- Trace the pattern from smaller to larger launch angles and note that the distance first rises and then falls. - Strength describes how closely points follow a pattern; the pattern does not have to be a straight line. - Decide whether “no overall direction” is the same as “no association.”

Solution

1. Statement A is supported because the points follow a clear curved pattern closely. 2. Statement B is not supported because association can be strong even when a curved pattern has no single positive or negative direction.

Answer

Statement A. The tight arch shows a strong nonlinear association even though the overall direction is neither entirely positive nor entirely negative.
Scatterplots and association 15
Use the scatterplot of the number of rotations on a pottery wheel and wall thickness, in millimeters. a) Describe the association in terms of direction, form, and strength. b) Explain why the width of the displayed range of wheel rotations is not, by itself, evidence that the association is strong.
Figure for problem 539420

Hints

- Use left-to-right movement to determine whether wall thickness tends to rise or fall. - Use the amount of scatter around the trend—not the width of the x-value range—to judge strength. - Decide whether the overall shape is roughly straight or clearly curved.

Solution

1. The points generally move downward as wheel rotations increase, so the direction is negative. 2. The overall pattern is roughly straight rather than clearly curved. 3. The points stay fairly close to that decreasing straight-line pattern, so the linear association is fairly strong. 4. The numerical range of the explanatory variable does not determine association strength. Strength depends on how tightly the observations follow the overall pattern.

Answer

a) The association is fairly strong, negative, and roughly linear. b) A wide range of wheel-rotation values does not make an association strong; strength is determined by how closely the points follow the overall pattern.
Scatterplots and association 16
Use the scatterplot of distance from a lamp, in inches, and light intensity, in lux. Consider these statements: A. “The plot has two clusters, so one overall trend may hide group differences.” B. “The two variables must be unrelated because each cluster has little internal trend.” Which statement is better supported, and why?
Figure for problem 539421

Hints

- Look first for separated groups of distance-and-intensity points. - Compare variation within each group with the difference between the two groups. - Consider whether combining distinct groups can hide a meaningful feature of the data.

Solution

1. Statement A is supported because the most prominent feature is two compact, separated clusters. 2. Statement B is not supported because little trend within each cluster does not rule out an overall pattern or meaningful differences between the groups. 3. A useful interpretation should investigate what distinguishes the two clusters rather than summarize all points with one trend.

Answer

Statement A. The two clusters suggest that one overall trend could hide important group differences, even if each cluster has little internal trend.
Scatterplots and association 17
The scatterplot compares practice time with the average number of performance errors per show. Describe the direction, strength, and form of the association. Would one straight line describe it well? Explain from the points.
Figure for problem 539422

Hints

- Look at whether the points rise or fall as practice time increases. - How closely do they follow a pattern? - Does that pattern look like a straight line or a curve?

Solution

1. As practice time grows, the average number of errors falls, giving a negative association. 2. The dots stay close to a curved pattern, so the association is strong but nonlinear. 3. One straight line would miss the curve.

Answer

Strong, negative, and nonlinear. One straight line would not describe the curved pattern well.
Scatterplots and association 18
The scatterplot compares the number of rowing strokes with distance traveled in meters. The cross is a new observation. How does it change the visual association?
Figure for problem 539423

Hints

- Imagine extending the trend of the original points to the right. - Compare the cross with that extension. - Describe a change you can actually see, rather than assuming a tiny numerical difference.

Solution

1. The first six observations follow a very strong positive linear trend. 2. The new point lies close to the continuation of that trend. 3. The association still looks very strong and positive. Any small change in its numerical correlation is not reliably visible here.

Answer

The new point extends the existing strong positive linear pattern. The visual strength changes little, if at all.
Scatterplots and association 19
The scatterplot compares rainfall in inches with river height in feet. The cross is a new observation. How does it change the visual association?
Figure for problem 539425

Hints

- Trace the original six points to estimate the river height usually associated with that rainfall value. - Is the new point unusual mainly horizontally or vertically? - Describe its effect on the strength as well as the direction of the pattern.

Solution

1. The original six points follow a strong positive linear pattern. 2. The new point has an ordinary rainfall value but an unusually large river height relative to that pattern. 3. It creates a high vertical outlier and weakens the positive linear association, without reversing the overall direction.

Answer

The cross creates a high vertical outlier that weakens the positive linear association, although the overall trend remains positive.
Scatterplots and association 20
The scatterplot shows the original relationship between the number of singers and a sound-intensity index. The red cross is a new observation. Describe how the new point changes the visual association.
Figure for problem 539426

Hints

- Extend the trend from \(10\) through \(60\) singers out to \(90\) singers. - Compare the intensity index \(20\) with the value suggested by that extension. - Consider both the increased x-range and whether the new point lies close to the existing line.

Solution

1. The original points form a very strong positive, roughly linear pattern. 2. Extending the original fitted trend to \(90\) singers gives a sound-intensity index of about \(18.1\). 3. The new point \((90, 20)\) is close to that extension, so it expands the x-range and reinforces the overall positive linear pattern, although the fit becomes only slightly less tight.

Answer

The point \((90, 20)\) extends the positive linear pattern to a larger x-value. The association remains very strong, though slightly less tight.
Scatterplots and association 21
While describing a scatterplot of number of artifacts in a display case and time visitors spend at the case in seconds, consider the statement: “The scatterplot has a slope of about \(0.8\), so its association is strong.” Identify the error and give a corrected statistical statement.

Hints

- Separate what slope tells you about direction and rate from what strength tells you about scatter. - Think about how changing the units of time would change the numerical slope. - Ask whether two scatterplots could have the same slope but very different amounts of scatter.

Solution

1. A slope of about \(0.8\) describes the rate and positive direction of a fitted line in the variables' units; it does not measure how tightly the points follow that line. 2. Association strength must be judged from the scatter around the pattern or from a scale-free measure such as correlation. 3. Without information about the scatter, the strength cannot be determined from the slope alone.

Answer

The error is using slope to judge strength. A slope of about \(0.8\) indicates a positive rate of change, but the association's strength cannot be determined without knowing how closely the points follow the pattern.
Scatterplots and association 22
While describing a scatterplot of ramp height in centimeters and toy-car speed in centimeters per second, consider the statement: “Because the points curve upward, there is no association.” Identify the error and give a corrected statistical statement.

Hints

- Decide whether the points follow any recognizable pattern, not only whether they follow a straight line. - “Linear” and “associated” are not synonyms. - Use the curve's overall movement and the amount of scatter to complete the description.

Solution

1. A curved pattern can show a clear association even though it is not linear. 2. “Curves upward” describes the form as nonlinear; it does not imply that the variables are unrelated. 3. A complete statement should describe the curve's direction and strength from the full scatterplot rather than declare no association.

Answer

The error is treating nonlinearity as no association. An upward-curving pattern is a nonlinear association; its direction and strength should be described from the full plot.
Scatterplots and association 23
A student looks at a scatterplot of the number of ingredients in a recipe and the preparation time. The student says, “There is one unusual point, so we cannot describe any association.” Explain why that conclusion may be wrong.

Hints

- What does one unusual point tell you about the rest of the data? - Can the remaining points follow a recognizable trend? - Separate a description of the main pattern from a description of the unusual point.

Solution

1. An unusual point is an observation far from the main pattern. 2. The remaining points may still show a clear direction, form, and strength. 3. Describe the main pattern and mention the unusual point separately. Its presence does not automatically erase the association.

Answer

One unusual point does not necessarily remove the overall association. Describe the main pattern and report the unusual observation separately.
Scatterplots and association 24
A quality-control study plots print-head feed rate \(x\), in millimeters per second, against surface roughness \(y\), in micrometers. A student says, “The two clusters mean there is no association between feed rate and roughness.” Describe the association by form, direction, strength, and unusual features. Is the student right?
Figure for problem 547942

Hints

- Follow all the points from smaller to larger feed rates. - Describe the overall form before focusing on gaps. - Decide whether clusters prevent the data from showing a direction.

Solution

1. As feed rate increases, roughness tends to increase in a roughly linear pattern. 2. The points lie fairly close to the overall pattern, so the association is strong and positive. 3. There are two separated clusters, but the gap between them does not make the association disappear. The student's claim is wrong.

Answer

The association is strong, positive, and roughly linear, with two separated clusters. The clusters do not mean there is no association.
Scatterplots and association 25
A study records machine setting \(x\) and vibration level \(y\) for five production runs. <table> <thead><tr><th>Machine setting \(x\)</th><th>Vibration level \(y\)</th></tr></thead> <tbody> <tr><td>\(2.0\)</td><td>\(5.1\)</td></tr> <tr><td>\(3.0\)</td><td>\(6.4\)</td></tr> <tr><td>\(3.0\)</td><td>\(7.0\)</td></tr> <tr><td>\(4.0\)</td><td>\(8.2\)</td></tr> <tr><td>\(5.0\)</td><td>\(9.1\)</td></tr> </tbody> </table> Consider the statement: “These data cannot be shown in a scatterplot because two observations have the same \(x\)-value.” Is the statement correct? Explain how those two observations appear and what each plotted point represents.

Hints

- A scatterplot represents paired observations, not a function that must assign only one response to each explanatory value. - Think about where two points with the same horizontal coordinate but different vertical coordinates would appear. - Keep the observational unit in mind when interpreting a point.

Solution

1. Repeated explanatory-variable values are allowed in bivariate quantitative data. 2. The observations \((3.0, 6.4)\) and \((3.0, 7.0)\) appear as two distinct points vertically aligned at \(x=3.0\). 3. Each point represents the paired measurements from one production run, so both observations belong in the scatterplot.

Answer

No. Both observations can be plotted. They appear as two vertically aligned points at \(x=3.0\), and each point represents one run's paired \(x\)- and \(y\)-measurements.
Scatterplots and association 26
The scatterplot shows tree height in feet on the x-axis and canopy width in feet on the y-axis. The canopy widths are converted from feet to inches, and a new scatterplot is made using the same observations. Describe what changes and what stays the same in the plot’s form, direction, strength, and unusual features.
Figure for problem 547959

Hints

- First identify the form, direction, strength, and any unusual feature in the displayed data. - Converting feet to inches multiplies every vertical coordinate by the same positive constant. - Decide which visual properties depend on scale and which depend on the relative arrangement of the observations.

Solution

1. The original scatterplot shows a moderate positive, roughly linear association with one observation whose canopy width is unusually large for its tree height. 2. Converting canopy width from feet to inches multiplies every y-coordinate by the same positive constant, \(12\). 3. The vertical scale changes, but the relative arrangement of the points does not. 4. The association remains moderate, positive, and roughly linear, and the same observation remains unusual relative to the pattern.

Answer

Only the vertical scale and y-axis units change. The scatterplot still shows a moderate positive, roughly linear association, and the same point remains unusually high relative to the pattern.
Scatterplots and association 27
A survey records the number of days students exercised last week and the number of days they slept at least \(8\) hours. Both variables are whole-number counts from \(0\) through \(7\). The data set contains \(120\) students, but the scatterplot shows only \(24\) distinct dot locations. Consider the claim: “The graph must be wrong because \(120\) observations should produce \(120\) visible dots.” Evaluate the claim.

Hints

- Think about whether two different observations are allowed to have identical coordinates. - Consider what happens when points are plotted at exactly the same location. - Distinguish the number of observations from the number of distinct ordered pairs.

Solution

1. Different students can have the same ordered pair of exercise days and sleep days. 2. When identical ordered pairs are plotted with ordinary point markers, their dots lie directly on top of one another. 3. Therefore, \(120\) observations can produce far fewer than \(120\) distinct visible locations. The scatterplot can be correct even though overplotting hides repeated pairs.

Answer

The claim is incorrect. Multiple students can have the same ordered pair, so their points overlap. A scatterplot of \(120\) observations can therefore show only \(24\) distinct dot locations.
Scatterplots and association 28
Panels a) and b) display exactly the same paired observations and use the same numerical axis ranges, but the panels have different physical shapes. Consider the claim: “Panel b) shows a stronger association because the points rise at a steeper angle.” Evaluate the claim.
Figure for problem 548003

Hints

- Compare the numerical coordinates and axis ranges in the two panels. - Ask whether any ordered pair changed between the displays. - Strength describes how closely points follow a pattern, not the screen angle of that pattern.

Solution

1. Both panels contain exactly the same ordered pairs, so the underlying association is unchanged. 2. Panel b) is horizontally compressed, which makes the same point pattern look steeper without changing how closely the points follow a pattern. 3. Strength concerns how tightly the points follow the association, not the visual angle of the point cloud on the screen.

Answer

The claim is incorrect. Panel b) only makes the same data look steeper because of the display shape. The direction, form, strength, and unusual features of the association are unchanged.
Scatterplots and association 29
A scatterplot shows backpack weight and walking speed for \(30\) hikers. Suppose the plotted points are connected in the order the hikers appear in the data file, with the claim that the connected zigzag line shows the relationship more clearly. Evaluate the graphing choice.

Hints

- Ask whether consecutive rows represent consecutive values of a meaningful sequence. - Think about what would happen to the connecting line if the rows were randomly reordered. - Focus on what features of the point cloud describe association.

Solution

1. Each point represents a different hiker, and the row order in the data file has no mathematical meaning for the relationship between backpack weight and walking speed. 2. Connecting consecutive rows creates line segments determined by an arbitrary file order rather than by the association between the two quantitative variables. 3. The scatterplot should be interpreted from the point cloud itself, including its form, direction, strength, and unusual features.

Answer

Connecting the points in file order is misleading because that order is arbitrary. The association should be read from the scatter of paired observations, not from a zigzag path created by unrelated row order.
Scatterplots and association 30
In the scatterplot, package weight \(x\) was recorded only to the nearest \(10\) pounds, and shipping time \(y\) was recorded in days. Consider the statement: “The vertical bands prove there are five natural clusters of packages.” Evaluate the statement.
Figure for problem 548038

Hints

- Compare the visible horizontal coordinates with the measurement precision stated in the problem. - Ask whether nearby true weights could be recorded as the same value. - Distinguish a plotting pattern caused by measurement from a genuine grouping in the population.

Solution

1. The vertical bands occur at \(20\), \(30\), \(40\), \(50\), and \(60\) pounds, exactly the values allowed by the rounding rule. 2. Many packages with different true weights can be recorded at the same rounded \(x\)-value, causing points to stack vertically. 3. The bands can therefore be a measurement-resolution artifact rather than evidence of five distinct underlying groups. 4. Any description of the association should note the heaping caused by rounded explanatory-variable measurements.

Answer

The statement is not justified. The vertical bands can result from rounding package weights to the nearest \(10\) pounds, so they do not by themselves establish five natural clusters.
Scatterplots and association 31
The scatterplot compares production speed with the average number of defects per batch, based on several batches tested at each speed. A student says, “The association is stronger at high speeds because the points are more spread out there.” Explain the mistake and describe the change in spread.
Figure for problem 548045

Hints

- Describe the overall pattern first. - Compare the vertical distances between points at the same speed. - Decide whether a wider group of responses means more consistency or less.

Solution

1. The overall association is positive and roughly linear. 2. At slower settings, average defect counts are close together. At faster settings, they are spread farther apart at similar speeds. 3. That widening shows more variability in the response. It does not mean the association is stronger at higher speeds.

Answer

The association is positive and roughly linear, but the vertical spread grows at higher speeds. More spread means less consistent defect averages, not stronger association.
Scatterplots and association 32
The scatterplot shows paired values of two quantitative variables. Consider the statement: “There is no association because the points do not trend upward or downward.” Describe the form and strength of the association, and evaluate the statement.
Figure for problem 548058

Hints

- Look for a recognizable shape before deciding whether an association exists. - Direction is only one feature of a bivariate pattern. - Ask whether the points are randomly scattered or organized around a consistent form.

Solution

1. The points follow a clear closed, roughly circular pattern rather than a straight line. 2. The association is strong in form because the points are tightly organized around that nonlinear pattern. 3. There is no single overall positive or negative direction around the circle, but lack of one direction does not mean lack of association. 4. The statement therefore confuses absence of a monotone direction with absence of a relationship.

Answer

The plot shows a strong nonlinear, roughly circular association with no single positive or negative direction. The statement is incorrect: the variables have a clear relationship even though it is not an increasing or decreasing one.
Scatterplots and association 33
A small motor warms while it runs. Describe the association between operating time and motor temperature. Would “strong positive linear association” be a complete description?
Figure for problem 548091

Hints

- Describe direction and strength before deciding on form. - Compare the change in temperature from hour to hour early and late in the graph. - Think about whether one line could represent both parts equally well.

Solution

1. Motor temperature generally increases with operating time, so the direction is positive. 2. The points stay close to a clear pattern, so the association is strong. 3. Temperature rises slowly at first, then more quickly after about \(6\) hours. The change in slope means the form is not described well by one straight line.

Answer

The association is strong and positive, with a noticeable change in slope near \(6\) hours. Calling it simply linear misses that change.
Scatterplots and association 34
A scatterplot uses age in whole years on the horizontal axis, so many observations have exactly the same \(x\)-coordinate and some points overlap. For display only, software adds a tiny random horizontal “jitter” to separate overlapping points. Should the correlation be computed from the jittered coordinates or from the original data? Explain.

Hints

- Decide whether the shifted horizontal positions are measured values or only a plotting choice. - Correlation should describe the actual paired observations. - Separate a visual technique for revealing overlap from a change to the data themselves.

Solution

1. Jitter is a display device used to reveal overlapping observations; it is not a new measurement of age. 2. Computing correlation from the jittered coordinates would introduce artificial random changes into the explanatory variable. 3. The numerical correlation should therefore be computed from the original paired measurements, while jitter may be used only for visualization.

Answer

Use the original data to compute correlation. Jitter may help display overlapping points, but its random coordinate shifts are not part of the measured data and should not enter the calculation.
Scatterplots and association 35
A data set records each student's locker number and mathematics test score. Both columns contain numbers, and a scatterplot of locker number versus test score is proposed for describing the direction of association. Is that an appropriate use of a scatterplot? Explain.

Hints

- Ask whether the numbers in each column represent measured quantities or merely labels. - A meaningful quantitative axis requires differences between values to have an interpretable size. - Do not assume a variable is quantitative just because it is written with digits.

Solution

1. A scatterplot is designed for two quantitative variables whose numerical values represent meaningful magnitudes. 2. Locker numbers are numerical labels; differences such as \(420-210\) do not represent a meaningful amount of “locker number.” 3. Treating locker number as a quantitative explanatory variable would make distances and direction on the horizontal axis meaningless. 4. Therefore, the proposed scatterplot should not be used to interpret quantitative association between these variables.

Answer

No. Locker number is an identifier, not a quantitative measurement, so a scatterplot treating it as a numerical explanatory variable would be misleading.
Scatterplots and association 36
An experiment uses only two explanatory-variable settings, \(x=10\) and \(x=30\), with many response observations at each setting. The average response at \(x=30\) is higher than at \(x=10\). Can the resulting scatterplot establish that the mean response changes linearly for all values between \(10\) and \(30\)? Explain.

Hints

- Count how many distinct explanatory-variable settings were actually observed. - Ask what information the graph provides about values between those settings. - Two endpoints alone cannot distinguish a straight relationship from many possible nonlinear ones.

Solution

1. The data show how the response behaves at the two observed explanatory settings. 2. With no observations at intermediate \(x\)-values, many different curves and lines could connect the two group centers. 3. Therefore, the plot can show a difference between the two settings but cannot establish a linear form throughout the interval between them.

Answer

No. The data support a comparison between \(x=10\) and \(x=30\), but with no intermediate settings they do not establish that the relationship is linear across the full interval.
Scatterplots and association 37
A scatterplot uses a logarithmic horizontal axis with equally spaced tick labels \(1\), \(10\), \(100\), and \(1000\). What do equal horizontal distances represent on the original \(x\)-scale?

Hints

- Compare the numerical differences between consecutive tick labels. - Then compare the ratios of consecutive labels. - A logarithmic axis converts multiplicative changes into equal visual spacing.

Solution

1. On a logarithmic axis, equal horizontal distances represent equal changes in \(\log(x)\), not equal additive changes in \(x\). 2. The displayed ticks increase by a factor of \(10\): \(1\) to \(10\), \(10\) to \(100\), and \(100\) to \(1000\). 3. Therefore, equal horizontal spacing represents equal multiplicative changes in the original variable.

Answer

Equal horizontal distances represent equal multiplicative factors in \(x\); here each tick is \(10\) times the preceding tick.
Scatterplots and association 38
A city recorded the number of electric vehicles registered and the average movie-ticket price once each year for \(20\) years. The scatterplot shows the paired yearly values. Why would it be risky to interpret this association as evidence that electric-vehicle registrations cause movie-ticket prices to rise?
Figure for problem 548162

Hints

- Identify the variable that orders all \(20\) observations even though it is not on either axis. - Two quantities can rise together because each changes over time. - Separate a descriptive association from evidence of a causal mechanism.

Solution

1. The scatterplot shows a strong positive association between the two recorded quantities. 2. Both variables are measured over the same sequence of years and can increase because of separate long-term trends. 3. The association can therefore arise because both variables are related to time, even without a causal connection between them. 4. The observational time trend and other possible confounding factors prevent the stated causal conclusion.

Answer

The positive association may be driven by both variables changing over time rather than by one causing the other. The scatterplot alone does not justify a causal interpretation.
Scatterplots and association 39
The panels use the same three positive input values but different horizontal scales. Which panel places the input values at equal distances apart, and why?
Figure for problem 563065

Hints

- Compare the horizontal distances between neighboring dots in each panel. - Equal multiplicative changes become what kind of changes after taking a logarithm?

Solution

1. Panel a) uses the original inputs \(1\), \(10\), and \(100\); their gaps are not equal. 2. Panel b) uses \(\log_{10}(x)\), giving horizontal positions \(0\), \(1\), and \(2\). 3. The logarithmic axis in panel b) places successive tenfold increases equally far apart.

Answer

Panel b). Each tenfold increase adds \(1\) to \(\log_{10}(x)\).
Scatterplots and association 40
The two plots show the same four daily temperatures. Which panel makes the rise look steeper? Does either panel show a stronger correlation? Explain.
Figure for problem 563066

Hints

- Compare the labels and ranges of the two vertical axes. - Look for the same input-response pairs in both panels. - Can changing only the drawing scale change a statistic computed from the data?

Solution

1. Panel a) uses a narrow vertical range, so the same increase appears steeper there. 2. Both panels display the same pairs of measurements. Changing the displayed scale does not change their correlation.

Answer

Panel a) looks steeper. Neither has a stronger correlation; both show the same data.
Scatterplots and association 41
A theater group rehearses a short scene several times in one afternoon. Describe how the number of missed cues changes across rehearsals, including the pattern's shape and any unusual points.
Figure for problem 539411

Hints

- Follow the cue counts from the first rehearsal through the last. - Describe any change in direction instead of forcing one trend for all the points. - Judge unusual points relative to the curved pattern, not to an imagined straight line.

Solution

1. Missed cues fall as rehearsals increase from \(1\) to \(3\), then rise again through rehearsal \(7\). 2. The points follow a clear U-shaped, nonlinear pattern rather than one overall increasing or decreasing trend. Fatigue from repeated rehearsals could be one possible explanation for the later rise. 3. No point stands clearly apart from this curved pattern.

Answer

The association is strongly U-shaped rather than linear, with no single overall direction or clear outlier.
Scatterplots and association 42
How do outside temperature and hot chocolate sales relate in the scatterplot? Describe the main pattern and any unusual points.
Figure for problem 539412

Hints

- Look for a direction followed by most observations. - Check whether any point is far from that pattern. - Compare the main pattern with the appearance of all the points together.

Solution

1. For the first six temperatures, from \(20\,^{\circ}\text{F}\) through \(45\,^{\circ}\text{F}\), cups sold fall along a straight-line pattern. 2. At \(50\,^{\circ}\text{F}\), sales are unusually high compared with that pattern. 3. This point weakens the overall negative linear association. The first six points form a strong negative linear trend, but it would be misleading to describe all seven as strongly linear.

Answer

Six points follow a strong negative linear pattern, but the unusually high sales at \(50\,^{\circ}\text{F}\) disrupt it. The overall negative association is weaker.
Scatterplots and association 43
Use the scatterplot of the number of pages in a script and reading time, in minutes. Identify the most important unusual feature and explain why it matters when describing the association.
Figure for problem 539415

Hints

- Temporarily ignore each point in turn and see whether the remaining page-count and reading-time pairs form a clearer pattern. - Compare the reading time for \(5\) pages with the times for nearby page counts. - An isolated point matters because direction and form alone do not describe the entire scatterplot.

Solution

1. Except for one point, reading time increases in a very consistent, roughly linear way as the page count increases. 2. At \(5\) pages, the reading time is \(22.7\) minutes, far above the value suggested by the other points. 3. The point \((5, 22.7)\) is an isolated high point, so it must be included in any complete description of the association.

Answer

The key unusual feature is the point \((5, 22.7)\). It lies well above the otherwise strong positive linear pattern and must be reported.
Scatterplots and association 44
Use the scatterplot of tide height, in feet, and the width of beach covered by water, in yards. Identify the most important unusual feature and explain why it matters when describing the association.
Figure for problem 539416

Hints

- Look for a range of beach-width values that contains no observations. - Compare the first four tide-height pairs with the last four rather than focusing only on the upward direction. - A complete scatterplot description should mention separated groups as well as form and direction.

Solution

1. The points increase overall as tide height increases. 2. The four points with tide heights from \(2\) to \(5\) feet form one group, while the four points from \(6\) to \(9\) feet form a second, higher group. 3. The large gap between the groups creates two clusters, an important feature that an overall positive direction alone would miss.

Answer

The key unusual feature is two separated clusters: one for tide heights from \(2\) to \(5\) feet and another from \(6\) to \(9\) feet. This clustering must be included in the description of the overall positive association.
Scatterplots and association 45
Use the scatterplot of the number of transit stops and trip duration, in minutes. Identify the most important unusual feature and explain why it matters when describing the association.
Figure for problem 539417

Hints

- Focus first on the trips with \(2\) through \(8\) stops and describe their pattern. - Compare the duration at \(9\) stops with the durations for nearby stop counts. - An observation far from the main trend is an important unusual feature even when the remaining points are strongly linear.

Solution

1. For trips with \(2\) through \(8\) stops, duration rises in a very consistent, roughly linear pattern. 2. A trip with \(9\) stops would be expected to take about \(20.2\) minutes from that pattern, but the recorded duration is only \(8.5\) minutes. 3. The point \((9, 8.5)\) is therefore an isolated low point that must be included in the scatterplot description.

Answer

The key unusual feature is the point \((9, 8.5)\). It lies well below the otherwise strong positive linear pattern and must be reported.
Scatterplots and association 46
The scatterplot compares the number of training sessions with skill-assessment scores. The cross is a new observation. How does it change the visual association?
Figure for problem 539424

Hints

- Follow the original points from left to right. - How far is the cross from the direction suggested by those points? - Think about its effect on the overall linear association, not automatically on a fitted line.

Solution

1. The first six points show a very strong positive linear pattern. 2. The new point lies much lower than the continuation of that pattern and farther right than the other points. 3. It sharply weakens the sample's positive linear association, so it is influential for the correlation. This does not by itself tell how much it changes a fitted regression line.

Answer

The new point is far below the trend and at a large training-session value. It greatly weakens the positive linear association and strongly affects the correlation.
Scatterplots and association 47
The scatterplot shows the original relationship between paper-glider launch speed, in feet per second, and flight time, in seconds. The red cross is a new observation. Describe how the new point changes the visual association.
Figure for problem 539427

Hints

- Compare the new launch speed with the original range of \(8\) to \(18\) feet per second. - Extend the original upward trend to \(24\) feet per second and compare it with the new flight time \(1.0\) second. - A point far out in the x-direction can have high leverage and strongly affect the overall pattern.

Solution

1. The original six points show a very strong positive, roughly linear association, with \(r\approx0.996\). 2. The point \((24, 1.0)\) is far to the right of the original launch speeds and far below the continuation of the trend. 3. Because it has high leverage and conflicts with the trend, it reduces the correlation to about \(0.190\) and substantially weakens the apparent positive linear association.

Answer

The point \((24, 1.0)\) is a high-leverage unusual point that substantially weakens the original positive linear association.
Scatterplots and association 48
The scatterplot shows an association between an explanatory variable \(x\) and a response variable \(y\). One point is separated far to the right from the others. Consider the description “an outlier because it is isolated.” Evaluate that description using the point's position relative to the overall pattern.
Figure for problem 547965

Hints

- Distinguish being far from other points horizontally from being far from the overall trend. - Compare the separated point with an extension of the main pattern. - Describe unusual features precisely rather than using one label for every isolated point.

Solution

1. The main group follows a strong negative, roughly linear pattern. 2. The far-right point lies close to the continuation of that same pattern, so its response value is not unusually far from the trend. 3. The point is unusual because its explanatory-variable value is separated from the others, but calling it an outlier from the association is misleading.

Answer

The point is unusual in its \(x\)-value, but it is not far from the overall linear pattern. Its isolation alone does not make it an outlier from the association.
Scatterplots and association 49
Panel a) shows a scatterplot for five observations. Panel b) shows the same five observations plus one additional point. Compare the direction and apparent strength of the linear association in the two panels. Explain the role of the added point.
Figure for problem 547970

Hints

- Compare the overall direction before and after the point is added. - Look at whether the new point supports or conflicts with the existing pattern. - Consider both its position relative to the trend and its separation in the explanatory-variable direction.

Solution

1. Panel a) shows a positive linear association, but the points have noticeable scatter around the increasing trend. 2. In panel b), the added far-right point lies close to an extension of the existing increasing pattern. 3. The direction remains positive, and the linear association appears stronger because the added point extends the trend rather than contradicting it.

Answer

Both panels show a positive association. Panel b) appears more strongly linear because the added far-right point lies along the extension of the existing trend.
Scatterplots and association 50
A data set contains paired measurements of commute distance and commute time for the same \(40\) workers. Before making a scatterplot, suppose the distance column is sorted from smallest to largest and the time column is separately sorted from smallest to largest, then the rows of the two sorted columns are plotted as ordered pairs. Explain why the resulting scatterplot does not represent the original association and what kind of misleading pattern the sorting can create.

Hints

- Ask what one point in the intended scatterplot is supposed to represent. - Track whether the two measurements in a row still belong to the same observational unit after sorting. - Consider what happens when two lists are both placed in increasing order before being paired.

Solution

1. Each original ordered pair must keep the distance and commute time from the same worker together. 2. Sorting the two columns independently breaks those pairings and creates new artificial pairs that were never observed. 3. Pairing small distances with small times and large distances with large times can manufacture a strong positive pattern even when the original paired data do not have one.

Answer

The scatterplot is invalid because independent sorting destroys the original worker-by-worker pairing. It can create an artificial positive association by matching low values with low values and high values with high values.
Scatterplots and association 51
The scatterplot relates input level \(x\) to a sensor reading \(y\). The sensor documentation says it cannot report values above \(100\). Consider the statement: “The scatterplot proves the true response levels off at \(100\).” Evaluate the statement and identify the unusual feature that should be reported.
Figure for problem 548024

Hints

- Compare the high-input readings with the maximum value the device can report. - Ask whether identical maximum readings must represent identical underlying values. - Separate a data-collection limit from the mathematical form of the true association.

Solution

1. The points rise roughly linearly for low and moderate inputs, while several high-input observations pile up exactly at \(y=100\). 2. The pileup coincides with the sensor's upper reporting limit. 3. Values that would exceed \(100\) are censored by the measurement system, so the horizontal band can be a measurement-limit artifact rather than evidence that the true response stops increasing. 4. The scatterplot should be described as showing an increasing pattern at lower inputs with an unusual ceiling or pileup at the response limit.

Answer

The statement is not justified. The horizontal pileup at \(y=100\) is consistent with the sensor's upper limit, so it may reflect censoring rather than a true leveling of the response. The ceiling at \(100\) is an important unusual feature.
Scatterplots and association 52
For each day, a tracker records the percentage of time a device is online, \(x\), and the percentage of time it is offline, \(y\). By definition, \(x+y=100\%\). A scatterplot of the two percentages shows all points on a decreasing straight line. Consider the description “strong evidence of a real-world negative relationship discovered in the data.” Explain why that description is misleading.

Hints

- Write one percentage in terms of the other using the stated total. - Ask whether the plotted relationship could have looked different while the definition still held. - Distinguish an empirical association from a relationship forced by how variables are constructed.

Solution

1. The two variables are complements: \(y=100-x\) for every day by definition. 2. Therefore, the decreasing straight-line pattern is built into how the variables are defined, not discovered from an unconstrained empirical relationship. 3. The plot will show a perfect negative linear association regardless of the day-to-day behavior of the device, as long as online and offline percentages exhaust the full \(100\%\).

Answer

The line is a mathematical consequence of \(x+y=100\%\), not independent evidence of a discovered relationship. The two percentages are complements, so a perfect negative pattern is guaranteed by their definitions.
Scatterplots and association 53
A study records training load and recovery score on five different days for each of \(20\) athletes, giving \(100\) paired observations. A scatterplot is made from all \(100\) pairs. Consider the claim: “The scatterplot is invalid because each athlete appears more than once.” Evaluate the claim, distinguishing descriptive use of the graph from statistical independence.

Hints

- Separate whether a pair of quantitative measurements can be plotted from whether observations are independent. - Identify what the scatterplot describes directly. - Think about what repeated measurements from the same person can share.

Solution

1. Each day provides a legitimate paired training-load and recovery-score observation, so the \(100\) pairs can be displayed in a scatterplot. 2. The scatterplot can describe the visible form, direction, strength, and unusual features of those measurements. 3. However, measurements from the same athlete may be related, so the \(100\) observations should not automatically be treated as \(100\) independent observations for regression inference. 4. Repeated measurements are therefore an inferential design issue, not a reason the descriptive scatterplot itself is invalid.

Answer

The scatterplot is valid as a descriptive display of the \(100\) paired measurements. The concern is that repeated measurements from the same athlete may not be independent, which matters for inference rather than for whether the points can be plotted.
Scatterplots and association 54
The scatterplot shows commute distance and commute time. Trip records indicate that the lower cluster contains mostly subway trips and the upper cluster contains mostly bus trips. What should be said about the association shown by this scatterplot?
Figure for problem 548087

Hints

- Describe the overall direction and any distinct groups visible in the plot. - Compare the pattern inside each cluster with the pattern across the full set of points. - Consider whether transportation mode helps explain why the clusters are separated.

Solution

1. The overall plot has a positive association because observations in the longer-time cluster also tend to have larger distances. 2. The visible clusters indicate that transportation mode is related to the pattern in the two quantitative variables. 3. Within each transportation-mode cluster, the points show little clear direction. 4. The pooled positive trend should therefore not be described as a single uniform relationship applying equally within both groups.

Answer

The pooled scatterplot shows a positive association, but the two transportation-mode clusters are an important feature. The overall trend is largely connected to the separation between the groups, while the within-group association is weak.
Scatterplots and association 55
The scatterplot compares each student's pre-course test score with the same student's post-course test score. The dashed line represents equal pre-course and post-course scores. Consider the claim: “The positive association proves that students improved.” Evaluate the claim using the scatterplot.
Figure for problem 548125

Hints

- First describe what the upward pattern says about students with higher and lower pre-course scores. - Use the dashed line to decide where a point would represent no change. - Determine which side of that line represents a higher post-course score.

Solution

1. The points have a strong positive association: students with higher pre-course scores also tend to have higher post-course scores. 2. Improvement is a different comparison. A student improved only when the point lies above the line \(y=x\). 3. In this scatterplot, the points lie below the line \(y=x\), so the post-course scores are lower than the corresponding pre-course scores despite the positive association. 4. Improvement should be assessed by comparing paired scores directly or by analyzing the post-minus-pre differences.

Answer

The claim is incorrect. The positive association shows that the ranking of students tends to persist, but the points lie below \(y=x\), indicating lower post-course scores. Association direction alone does not establish improvement.
Scatterplots and association 56
A college studies the association between high-school GPA and entrance-exam score, but its data file includes only admitted students, all of whom had entrance-exam scores of at least \(80\). How can this response-based selection affect the scatterplot, and what population should any observed association describe?

Hints

- Identify which observations are systematically absent from the data set. - Removing part of one variable's range can change the visible paired pattern. - State conclusions for the population actually represented by the observations.

Solution

1. Restricting the data to students with exam scores of at least \(80\) removes the lower part of the response distribution. 2. This truncation can change the visible shape and strength of the association compared with the full applicant population. 3. The resulting scatterplot directly describes only admitted students meeting the selection rule, not all applicants.

Answer

The cutoff truncates the scatterplot and can alter the apparent association. Any description should be limited to the selected admitted-student population unless broader data justify generalization.
Scatterplots and association 57
The scatterplot shows paired quantitative observations. Is “no relationship” a complete description of the scatterplot? Explain.
Figure for problem 548150

Hints

- Describe both the center and the vertical spread of the point cloud. - Compare the spread near the ends of the x-axis with the spread near the middle. - A relationship can involve variability even when the average level stays nearly constant.

Solution

1. The center of the response remains about the same across the range of \(x\), so there is little evidence of an increasing or decreasing mean trend. 2. The vertical spread is small for low \(x\), large for middle \(x\), and small again for high \(x\). 3. The variability of \(y\) therefore changes systematically with \(x\). 4. The plot contains meaningful structure even though it has no clear positive or negative trend in its center.

Answer

No. The center shows little upward or downward trend, but the changing vertical spread is a systematic relationship between \(x\) and the variability of \(y\).
Scatterplots and association 58
A survey records satisfaction on the ordered categories \(1=\) “very dissatisfied,” \(2=\) “dissatisfied,” \(3=\) “neutral,” \(4=\) “satisfied,” and \(5=\) “very satisfied.” Consider a scatterplot of satisfaction code versus annual spending. What caution is needed when interpreting a one-unit horizontal difference or a linear association?

Hints

- Separate numerical labels from measured numerical distances. - The categories have a clear order, but decide whether adjacent steps necessarily represent equal amounts. - Linear interpretation requires meaningful numerical spacing, not only ordered codes.

Solution

1. The satisfaction codes preserve order, but they do not guarantee equal quantitative spacing between adjacent categories. 2. Treating \(1, 2, 3, 4, 5\) as an ordinary quantitative scale assumes, for example, that the change from \(1\) to \(2\) has the same size as the change from \(4\) to \(5\). 3. A scatterplot can display the coded groups, but interpreting horizontal distances and linear association as if satisfaction were measured on an equal-interval scale requires justification.

Answer

The codes are ordinal, not automatically an equal-interval quantitative measurement. The plot may display the categories, but a standard quantitative linear-association interpretation is not justified unless equal spacing of the scale is defensible.
Scatterplots and association 59
Panel a) shows monthly rent versus apartment floor area for apartments ranging from about \(500\) to \(2500\) square feet. Panel b) shows only the apartments between \(1400\) and \(1600\) square feet. Does the restricted scatterplot contradict the association seen in the full data set? Explain how restricting the range of the explanatory variable can change the visible association.
Figure for problem 548161

Hints

- Compare the horizontal ranges and overall patterns in the two panels. - Ask how much explanatory-variable variation remains after the subset is selected. - A pattern within a restricted range need not look like the pattern across the entire original range.

Solution

1. Panel a) shows a clear positive, roughly linear association across the full range of apartment floor areas. 2. Panel b) uses only a narrow slice of the explanatory-variable range, so most of the variation in floor area has been removed. Within that slice, the points form a loose cloud with little visible direction. 3. Other sources of rent variation can be large relative to the small remaining differences in floor area, making the restricted association look weak or unclear. 4. The restricted plot therefore does not contradict the positive association in the full data. It describes the association only within the selected range of floor areas.

Answer

No. Restricting floor area to a narrow interval can make the visible association much weaker because little explanatory-variable variation remains. The subset describes only apartments from \(1400\) to \(1600\) square feet and does not erase the positive association across the full range.

All problems may be used, copied and printed free of charge for school and tutoring, including paid tutoring. Commercial adaptations as well as publication or redistribution on the internet are not permitted.

Building free math worksheets