Problem Solving Data Analysis⚡ High Yield (1-3 Questions per Test)
Scatterplots
Digital SAT Math Preparation & Desmos Strategies
4 Concepts20 Practice Qs30 Mock Qs⚡ Desmos Speed Hacks
Key Concepts & Worked Archetypes
Concept 1
Concept 1: Line of Best Fit Interpretation
The line of best fit represents the trend of the data, where the slope indicates the rate of change and the y-intercept represents the initial value.
Slope $m = \frac{y_2 - y_1}{x_2 - x_1}$ represents the predicted change in y per unit change in x.
Y-intercept $(0, b)$ represents the predicted value of y when x is 0.
Residuals are calculated as $e = y_{actual} - y_{predicted}$.
📘 Traditional Algebraic Method
Identify two points on the line of best fit. Calculate the slope using the slope formula. Use the point-slope form to find the equation.
⚡ SAT Speed Trick & Desmos Hack
Use the Desmos regression feature by typing 'y1 ~ mx1 + b' to get the exact line of best fit equation instantly.
💡 Worked SAT Archetype Example
Problem: A scatterplot shows the relationship between hours studied ($x$) and test score ($y$). The line of best fit is $y = 5x + 60$. What is the predicted score for 8 hours of study?
📘 Step-by-Step Textbook Solution:
Step 1
Identify the given equation: $y = 5x + 60$
Step 2
Substitute $x = 8$ into the equation: $y = 5(8) + 60$
Step 3
Calculate the product: $y = 40 + 60$
Step 4
Final result: $y = 100$
⚡ Speed / Desmos Tactic:
Step 1
Type 'y = 5x + 60' into Desmos
Step 2
Click the gear icon or use the table feature to evaluate at $x = 8$
Step 3
Read the result $y = 100$ directly from the graph
Concept 2
Concept 2: Correlation Strength
Correlation describes the direction and strength of the linear relationship between two variables.
Positive correlation: $r > 0$, as x increases, y increases.
Negative correlation: $r < 0$, as x increases, y decreases.
Strength: $|r|$ close to 1 indicates a strong linear relationship, while $|r|$ close to 0 indicates a weak one.
📘 Traditional Algebraic Method
Visually inspect the density of points around the line of best fit. Tighter clusters indicate stronger correlation.
⚡ SAT Speed Trick & Desmos Hack
Look for the 'tightness' of the scatter. If points form a clear, narrow path, the correlation is strong.
💡 Worked SAT Archetype Example
Problem: Which scatterplot shows a strong negative correlation?
📘 Step-by-Step Textbook Solution:
Step 1
Identify the direction: Slope must be downward from left to right.
Step 2
Identify the strength: Points must be closely packed along a line.
Step 3
Select the graph where points are tightly clustered in a downward trend.
⚡ Speed / Desmos Tactic:
Step 1
Scan all options for a downward slope
Step 2
Eliminate options with scattered, random points
Step 3
Choose the option with the narrowest point distribution
Concept 3
Concept 3: Residual Analysis
Residuals measure the vertical distance between an actual data point and the line of best fit.
Residual = $y_{actual} - y_{predicted}$
Positive residual: The actual point is above the line.
Negative residual: The actual point is below the line.
📘 Traditional Algebraic Method
Find the y-coordinate of the data point. Find the y-coordinate of the line at that same x-value. Subtract the line value from the point value.
⚡ SAT Speed Trick & Desmos Hack
Use the vertical distance tool or simply count grid units on the graph to estimate the difference.
💡 Worked SAT Archetype Example
Problem: A point is at $(2, 10)$. The line of best fit at $x=2$ is $y=8$. What is the residual?
📘 Step-by-Step Textbook Solution:
Step 1
Identify $y_{actual} = 10$
Step 2
Identify $y_{predicted} = 8$
Step 3
Calculate $Residual = 10 - 8$
Step 4
Final result: $2$
⚡ Speed / Desmos Tactic:
Step 1
Locate $x=2$ on the x-axis
Step 2
Measure the vertical gap between the point and the line
Step 3
Count the units to find the difference of 2
Concept 4
Concept 4: Extrapolation vs Interpolation
Interpolation is predicting values within the range of the data, while extrapolation is predicting values outside the range.
Interpolation: $x_{min} \leq x \leq x_{max}$
Extrapolation: $x < x_{min}$ or $x > x_{max}$
Extrapolation is generally less reliable than interpolation.
📘 Traditional Algebraic Method
Check if the target x-value falls within the domain of the provided scatterplot data points.
⚡ SAT Speed Trick & Desmos Hack
Check the x-axis bounds. If the x-value is outside the visual range, it is extrapolation.
💡 Worked SAT Archetype Example
Problem: Data ranges from $x=0$ to $x=10$. Is predicting $y$ at $x=15$ interpolation or extrapolation?
📘 Step-by-Step Textbook Solution:
Step 1
Identify the data range: $[0, 10]$
Step 2
Compare target $x=15$ to the range
Result
Since $15 > 10$, it is outside the range
Conclusion
Extrapolation
⚡ Speed / Desmos Tactic:
Step 1
Look at the x-axis scale
Step 2
Observe that 15 is beyond the plotted points
Step 3
Identify as extrapolation
Practice Questions (20)
Question 1Interpreting Trends
Easy
A scatterplot shows the relationship between the number of hours spent studying ($x$) and the test score ($y$) for 15 students. The points generally trend upward from left to right. Which of the following best describes the relationship?
Hint: Look at the direction of the slope of the data points.
📘 Step-by-Step Algebraic Solution
Step 1
Observe the direction of the scatterplot.
Step 2
An upward trend from left to right indicates a positive correlation.
Step 3
A positive correlation means as x increases, y increases.
⚡ Desmos Shortcut / Speed Hack
Step 1
Visualize a line of best fit through the points.
Step 2
The line has a positive slope.
Step 3
Positive slope = positive correlation.
Question 2Interpreting Trends
Easy
A scatterplot displays the age of a car in years ($x$) and its current market value in dollars ($y$). The points show a downward trend. What does this indicate?
Hint: A downward trend means as one variable increases, the other decreases.
📘 Step-by-Step Algebraic Solution
Step 1
Identify the variables: age (x) and value (y).
Step 2
A downward trend indicates a negative correlation.
Step 3
As age increases, value decreases.
⚡ Desmos Shortcut / Speed Hack
Step 1
Look at the slope of the trend.
Step 2
Negative slope implies an inverse relationship.
Step 3
Select the option describing a decrease.
Question 3Interpreting Trends
Medium
A scatterplot shows the relationship between the number of visitors to a park and the daily temperature. The data points are tightly clustered around a line with a positive slope. Which statement is most accurate?
Hint: Closeness to the line indicates the strength of the correlation.
📘 Step-by-Step Algebraic Solution
Step 1
Determine the direction: positive slope means positive correlation.
Step 2
Determine the strength: points tightly clustered mean strong correlation.
Step 3
Combine: strong and positive.
⚡ Desmos Shortcut / Speed Hack
Step 1
Imagine a line of best fit.
Step 2
If points are close to the line, the correlation is strong.
Step 3
If the line goes up, it is positive.
Question 4Interpreting Trends
Medium
A scatterplot shows the relationship between the weight of a backpack and the student's reported back pain. The points are widely scattered, but there is a slight upward trend. How should this be described?
Hint: Wide scattering indicates a weak relationship.
📘 Step-by-Step Algebraic Solution
Step 1
Identify the trend: slight upward means positive.
Step 2
Identify the dispersion: widely scattered means weak.
Step 3
Combine: weak positive correlation.
⚡ Desmos Shortcut / Speed Hack
Step 1
Check the 'tightness' of the points.
Step 2
Wide spread = weak.
Step 3
Upward trend = positive.
Question 5Interpreting Trends
Hard
A scatterplot shows the relationship between $x$ and $y$. The points form a U-shape. Which of the following is true?
Hint: Linear correlation requires a straight-line pattern.
📘 Step-by-Step Algebraic Solution
Step 1
Analyze the shape of the data.
Step 2
A U-shape is a quadratic pattern, not a straight line.
Step 3
Therefore, the relationship is non-linear.
⚡ Desmos Shortcut / Speed Hack
Step 1
Look for a straight line.
Step 2
If the points curve, it is non-linear.
Step 3
Select the non-linear option.
Question 6Line of Best Fit
Easy
A scatterplot has a line of best fit with the equation $y = 2x + 5$. What is the predicted value of $y$ when $x = 10$?
Hint: Substitute $x = 10$ into the equation.
📘 Step-by-Step Algebraic Solution
Step 1
Equation: $y = 2x + 5$
Step 2
Substitute $x = 10$: $y = 2(10) + 5$
Step 3
Calculate: $y = 20 + 5 = 25$
⚡ Desmos Shortcut / Speed Hack
Step 1
Type '2*10+5' into the calculator.
Step 2
Result is 25.
Question 7Line of Best Fit
Easy
In a scatterplot, the line of best fit is $y = -0.5x + 20$. What does the value 20 represent?
Hint: Recall the slope-intercept form $y = mx + b$.
📘 Step-by-Step Algebraic Solution
Step 1
Compare $y = -0.5x + 20$ to $y = mx + b$.
Step 2
$m = -0.5$ (slope), $b = 20$ (y-intercept).
Step 3
The y-intercept is the value of $y$ when $x = 0$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Identify the constant term.
Step 2
The constant term is the y-intercept.
Question 8Line of Best Fit
Medium
A scatterplot shows the relationship between temperature ($x$) and ice cream sales ($y$). The line of best fit is $y = 15x - 100$. What is the meaning of the slope?
Hint: Slope is the change in $y$ per unit change in $x$.
📘 Step-by-Step Algebraic Solution
Step 1
Slope $m = 15$.
Step 2
$m = \Delta y / \Delta x$.
Step 3
This means for every 1 unit increase in $x$, $y$ increases by 15.
⚡ Desmos Shortcut / Speed Hack
Step 1
Look at the coefficient of $x$.
Step 2
It is positive 15.
Step 3
This means an increase of 15 in $y$ for every 1 in $x$.
Question 9Line of Best Fit
Medium
A scatterplot shows the relationship between study time ($x$) and test score ($y$). The line of best fit is $y = 5x + 60$. If a student studies for 4 hours, what is the predicted score?
Hint: Plug $x = 4$ into the equation.
📘 Step-by-Step Algebraic Solution
Step 1
$y = 5(4) + 60$
Step 2
$y = 20 + 60$
Step 3
$y = 80$
⚡ Desmos Shortcut / Speed Hack
Step 1
Calculate $5*4+60$.
Step 2
Result is 80.
Question 10Line of Best Fit
Hard
A scatterplot shows the relationship between $x$ and $y$. The line of best fit passes through $(2, 10)$ and $(6, 22)$. What is the equation of the line?
In a scatterplot of height vs. weight, one point is significantly higher than the others for a given height. What does this point represent?
Hint: Points that deviate significantly from the trend are called outliers.
📘 Step-by-Step Algebraic Solution
Step 1
Observe the point's position relative to the cluster.
Step 2
A point far from the general trend is an outlier.
Step 3
This point is an outlier.
⚡ Desmos Shortcut / Speed Hack
Step 1
Look for the 'odd one out'.
Step 2
That is the outlier.
Question 12Data Point Analysis
Easy
A scatterplot shows the number of hours worked ($x$) and total pay ($y$). If a point is at $(10, 150)$, what does this mean?
Hint: The point is $(x, y)$.
📘 Step-by-Step Algebraic Solution
Step 1
$x$ represents hours, $y$ represents pay.
Step 2
$(10, 150)$ means $x=10$ and $y=150$.
Step 3
10 hours worked results in $150 pay.
⚡ Desmos Shortcut / Speed Hack
Step 1
Match the coordinates to the axes labels.
Step 2
$x$ is horizontal, $y$ is vertical.
Question 13Data Point Analysis
Medium
A scatterplot shows the relationship between $x$ and $y$. If a point $(5, 20)$ is above the line of best fit, what does this mean?
Hint: The line of best fit represents predicted values.
📘 Step-by-Step Algebraic Solution
Step 1
The line represents predicted values.
Step 2
A point above the line has a higher $y$-value than the prediction.
Step 3
Therefore, actual > predicted.
⚡ Desmos Shortcut / Speed Hack
Step 1
Above the line = higher than predicted.
Step 2
Below the line = lower than predicted.
Question 14Data Point Analysis
Medium
In a scatterplot, the residual of a point is defined as $y_{actual} - y_{predicted}$. If a point is at $(4, 10)$ and the line of best fit predicts $y=12$ at $x=4$, what is the residual?
Hint: Residual = Actual - Predicted.
📘 Step-by-Step Algebraic Solution
Step 1
$y_{actual} = 10$.
Step 2
$y_{predicted} = 12$.
Step 3
Residual = $10 - 12 = -2$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Subtract predicted from actual.
Step 2
$10 - 12 = -2$.
Question 15Data Point Analysis
Hard
A scatterplot shows the relationship between $x$ and $y$. The line of best fit is $y = 2x + 3$. A data point is at $(3, 10)$. What is the residual for this point?
Hint: Calculate predicted $y$ first, then find the residual.
📘 Step-by-Step Algebraic Solution
Step 1
Predicted $y = 2(3) + 3 = 9$.
Step 2
Actual $y = 10$.
Step 3
Residual = $10 - 9 = 1$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Calculate $10 - (2*3+3)$.
Step 2
$10 - 9 = 1$.
Question 16Contextual Interpretation
Easy
A scatterplot shows the relationship between the number of umbrellas sold ($y$) and the amount of rainfall ($x$). What is the most likely correlation?
Hint: Does more rain lead to more or fewer umbrellas sold?
📘 Step-by-Step Algebraic Solution
Step 1
More rain usually leads to more umbrella sales.
Step 2
Both variables increase together.
Step 3
This is a positive correlation.
⚡ Desmos Shortcut / Speed Hack
Step 1
Think logically.
Step 2
Rain up, sales up = positive.
Question 17Contextual Interpretation
Easy
A scatterplot shows the relationship between the speed of a car ($x$) and the time it takes to reach a destination ($y$). What is the correlation?
Hint: Does higher speed lead to more or less time?
📘 Step-by-Step Algebraic Solution
Step 1
Higher speed means less time to reach the destination.
Step 2
As $x$ increases, $y$ decreases.
Step 3
This is a negative correlation.
⚡ Desmos Shortcut / Speed Hack
Step 1
Think logically.
Step 2
Speed up, time down = negative.
Question 18Contextual Interpretation
Medium
A scatterplot shows the relationship between the number of years of experience ($x$) and salary ($y$). The line of best fit is $y = 5000x + 40000$. What does 40000 represent?
Hint: The y-intercept represents the value when $x=0$.
📘 Step-by-Step Algebraic Solution
Step 1
The equation is $y = 5000x + 40000$.
Step 2
When $x = 0$ (0 years experience), $y = 40000$.
Step 3
This is the starting salary.
⚡ Desmos Shortcut / Speed Hack
Step 1
Identify the y-intercept.
Step 2
It is the value at $x=0$.
Question 19Contextual Interpretation
Medium
A scatterplot shows the relationship between the number of pages in a book ($x$) and the price ($y$). The line of best fit is $y = 0.1x + 5$. What does 0.1 represent?
Hint: The slope represents the rate of change.
📘 Step-by-Step Algebraic Solution
Step 1
The slope is 0.1.
Step 2
Slope is $\Delta y / \Delta x$ (price change / page change).
Step 3
This is the cost per page.
⚡ Desmos Shortcut / Speed Hack
Step 1
Look at the coefficient of $x$.
Step 2
It represents the rate of change.
Question 20Contextual Interpretation
Hard
A scatterplot shows the relationship between the number of people at a party ($x$) and the total cost of food ($y$). The line of best fit is $y = 15x + 100$. If the cost increases by $300, how many more people were added?
Hint: Use the slope: $\Delta y = m * \Delta x$.
📘 Step-by-Step Algebraic Solution
Step 1
$\Delta y = 300$.
Step 2
$300 = 15 * \Delta x$.
Step 3
$\Delta x = 300 / 15 = 20$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Divide the change in cost by the cost per person.
Step 2
$300 / 15 = 20$.
Official SAT Exam Blueprint & Weightage
Metric
Exam Pattern & Weightage
Question Frequency
1-3 questions per test module
Module 1 Appearance
Medium Frequency (Foundational tests)
Module 2 Appearance
High Frequency (Hard module score differentiator)
Pattern Analysis
College Board frequently tests the interpretation of slope and y-intercept in context, often requiring students to distinguish between correlation and causation.
🏛️
Official SAT PYQ Drill Bank (2023–2026)
We are compiling real verified College Board exam patterns with step-by-step Desmos shortcuts for Scatterplots.
Updated for 2026 Testing Season
Revision, Traps & Desmos Syntax
Slope Formula
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Used to find the rate of change from two points on the line of best fit.
Residual
$e = y - \hat{y}$
Difference between observed value and predicted value.
🚨 Top SAT Traps & Misconceptions
⚠️ SAT Trap: Correlation vs Causation
Just because two variables are correlated does not mean one causes the other. Avoid assuming causal links.
⚠️ SAT Trap: Misinterpreting the Y-intercept
The y-intercept is only meaningful if $x=0$ is within the context of the problem.
⚡ Essential Desmos Cheatsheet
🎯 Regression Analysis
y1 ~ mx1 + b
Use this to find the exact line of best fit for any set of points.
3-Level Mock Test (30 Questions)
🟢 Level 1: Foundation
10 Qs · Sub-600 Score
🟡 Level 2: Target 700+
10 Qs · 600–740 Score
🔴 Level 3: 800-Mastery
10 Qs · 750–800 Score
Question 1Level 1: Foundation
A scatterplot shows the relationship between the number of hours studied and the test score of 10 students. If the points generally trend from the bottom left to the top right, what type of association is shown?
Explanation:
Step 1
Identify the direction of the trend in the scatterplot.
Step 2
A trend from bottom-left to top-right indicates that as one variable increases, the other increases.
Step 3
This is defined as a positive association.
Question 2Level 1: Foundation
A scatterplot displays the age of a car and its current market value. As the age of the car increases, the market value decreases. Which best describes the correlation?
Explanation:
Step 1
Analyze the relationship between the two variables.
Step 2
As age increases, value decreases, which is an inverse relationship.
Step 3
An inverse relationship in a scatterplot represents a negative correlation.
Question 3Level 1: Foundation
In a scatterplot, if the data points are scattered randomly with no discernible pattern, what is the correlation?
Explanation:
Step 1
Observe the distribution of points.
Step 2
Random scattering implies that one variable provides no information about the other.
Step 3
This indicates no correlation.
Question 4Level 1: Foundation
A scatterplot shows the relationship between height and shoe size. Most points fall close to a straight line. This is an example of a:
Explanation:
Step 1
Identify the shape formed by the data points.
Step 2
Data points falling close to a straight line indicate a linear relationship.
Step 3
Therefore, the relationship is linear.
Question 5Level 1: Foundation
Which of the following scatterplots would represent a strong negative correlation?
Explanation:
Step 1
Define negative correlation as a downward trend.
Step 2
Define strong correlation as points being close to the trend line.
Step 3
A tight line going down represents a strong negative correlation.
Question 6Level 1: Foundation
If a scatterplot has an outlier, what does that mean?
Explanation:
Step 1
Recall the definition of an outlier in statistics.
Step 2
An outlier is a data point that differs significantly from other observations.
Step 3
This corresponds to a point far away from the general cluster.
Question 7Level 1: Foundation
A scatterplot shows the number of ice cream cones sold versus the daily temperature. What is the independent variable?
Explanation:
Step 1
Identify the variable that influences the other.
Step 2
Temperature influences sales, not vice versa.
Step 3
The independent variable is the daily temperature.
Question 8Level 1: Foundation
Which correlation coefficient $r$ represents the strongest linear relationship?
Explanation:
Step 1
Recall that the strength of a linear relationship is determined by the absolute value of $r$.
Step 2
Compare the absolute values: $|0.1|=0.1, |-0.5|=0.5, |-0.9|=0.9, |0.8|=0.8$.
Step 3
The largest absolute value is $0.9$, so $r = -0.9$ is the strongest.
Question 9Level 1: Foundation
In a scatterplot, the line of best fit is used to:
Explanation:
Step 1
Understand the purpose of a line of best fit.
Step 2
It models the trend of the data.
Step 3
It is primarily used to make predictions based on the observed trend.
Question 10Level 1: Foundation
If a scatterplot shows a cluster of points, what does this suggest?
Explanation:
Step 1
Define a cluster in a scatterplot.
Step 2
A cluster indicates a concentration of data points in a specific region.
Step 3
This suggests that most observations share similar values for the variables.
Question 11Level 2: Target 700+
A scatterplot models the relationship between $x$ and $y$. The line of best fit is $y = 2.5x + 10$. What is the predicted value of $y$ when $x = 4$?
Explanation:
Step 1
Use the given equation $y = 2.5x + 10$.
Step 2
Substitute $x = 4$ into the equation: $y = 2.5(4) + 10$.
Step 3
Calculate the result: $y = 10 + 10 = 20$.
Question 12Level 2: Target 700+
In a scatterplot, the line of best fit is $y = -1.2x + 50$. What does the slope $-1.2$ represent?
Explanation:
Step 1
Recall the interpretation of slope in a linear equation $y = mx + b$.
Step 2
The slope $m = -1.2$ indicates the rate of change.
Step 3
A negative slope means that as $x$ increases, $y$ decreases by that amount.
Question 13Level 2: Target 700+
A scatterplot shows the relationship between study time and test scores. The line of best fit is $y = 0.8x + 60$. What does the $y$-intercept $60$ represent?
Explanation:
Step 1
Recall the interpretation of the $y$-intercept in a linear model.
Step 2
The $y$-intercept occurs when $x = 0$.
Step 3
Therefore, it represents the predicted score when study time is 0 hours.
Question 14Level 2: Target 700+
A scatterplot has a line of best fit $y = 3x + 5$. A data point exists at $(2, 12)$. What is the residual for this point?
Explanation:
Step 1
Calculate the predicted value $\hat{y}$ for $x = 2$: $\hat{y} = 3(2) + 5 = 11$.
Step 2
The residual is the difference between the actual value and the predicted value: $y - \hat{y}$.
Step 3
Calculate the residual: $12 - 11 = 1$.
Question 15Level 2: Target 700+
Which of the following best describes the effect of an outlier on the line of best fit?
Explanation:
Step 1
Consider how the line of best fit is calculated (minimizing squared residuals).
Step 2
An outlier far from the trend line will have a large squared residual.
Step 3
To minimize this, the line will shift toward the outlier, significantly affecting the slope and intercept.
Question 16Level 2: Target 700+
A scatterplot shows a non-linear pattern where data points increase rapidly. Which model is most appropriate?
Explanation:
Step 1
Analyze the growth pattern.
Step 2
Rapid increase suggests a multiplicative or exponential growth rather than a constant additive one.
Step 3
Therefore, an exponential model is most appropriate.
Question 17Level 2: Target 700+
If the correlation coefficient $r$ is $0.95$, what can be concluded?
Explanation:
Step 1
Evaluate the sign of $r$ (positive) and the magnitude (close to 1).
Step 2
A positive sign indicates a positive relationship.
Step 3
A value close to 1 indicates a strong linear relationship.
Question 18Level 2: Target 700+
A scatterplot shows the relationship between $x$ and $y$. If the data points are very tightly clustered around the line of best fit, what does this imply about the residual values?
Explanation:
Step 1
Recall that residuals measure the distance from the points to the line.
Step 2
Tightly clustered points mean they are very close to the line.
Step 3
Therefore, the distances (residuals) must be small.
Question 19Level 2: Target 700+
Which of the following is true regarding the line of best fit?
Explanation:
Step 1
Recall the definition of the least-squares regression line.
Step 2
The line of best fit is defined as the line that minimizes the sum of the squared vertical distances (residuals) from the data points to the line.
Step 3
This confirms option C.
Question 20Level 2: Target 700+
A scatterplot shows a U-shaped pattern. Which model is best?
Explanation:
Step 1
Identify the shape of the data.
Step 2
A U-shape or parabolic shape is characteristic of a quadratic function.
Step 3
Therefore, a quadratic model is the best fit.
Question 21Level 3: 800 Mastery
A researcher finds that the correlation between two variables $X$ and $Y$ is $r = 0.85$. If $X$ is transformed to $Z = 2X + 5$, what is the new correlation between $Z$ and $Y$?
Explanation:
Step 1
Recall that the correlation coefficient is invariant under linear transformations of the form $aX + b$ where $a > 0$.
Step 2
Since $Z = 2X + 5$ is a positive linear transformation, the strength and direction of the relationship remain unchanged.
Step 3
The correlation remains $0.85$.
Question 22Level 3: 800 Mastery
In a study of 50 points, the sum of the residuals is found to be 0. What does this imply?
Explanation:
Step 1
Recall the mathematical properties of the least-squares regression line.
Step 2
By definition, the sum of the residuals $\sum(y_i - \hat{y}_i)$ for a least-squares line is always zero.
Step 3
This is a standard property, not an indicator of perfect fit.
Question 23Level 3: 800 Mastery
A scatterplot shows a strong non-linear relationship. Which statistic is most appropriate to describe the strength of this relationship?
Explanation:
Step 1
Note that Pearson's $r$ only measures linear relationships.
Step 2
For non-linear monotonic relationships, Spearman's rank correlation is more appropriate.
Step 3
Therefore, Spearman's rank correlation is the correct choice.
Question 24Level 3: 800 Mastery
If the coefficient of determination $r^2$ is $0.64$, what percentage of the variation in $y$ is explained by the variation in $x$?
Explanation:
Step 1
Recall that $r^2$ represents the proportion of variance in the dependent variable explained by the independent variable.
Step 2
Convert the decimal $0.64$ to a percentage.
Step 3
$0.64 \times 100 = 64\%$.
Question 25Level 3: 800 Mastery
A scatterplot of residuals vs. $x$ shows a clear curved pattern. What does this indicate about the original linear model?
Explanation:
Step 1
Understand that a residual plot should show random scatter if a linear model is appropriate.
Step 2
A pattern (like a curve) in the residual plot indicates that the linear model failed to capture the structure of the data.
Step 3
Therefore, the linear model is inappropriate.
Question 26Level 3: 800 Mastery
Which of the following scenarios describes a lurking variable?
Explanation:
Step 1
Define a lurking variable in the context of correlation.
Step 2
A lurking variable is an unobserved variable that affects the association between the variables of interest.
Step 3
This can lead to a spurious correlation.
Question 27Level 3: 800 Mastery
In a regression analysis, what is the effect of increasing the sample size on the standard error of the slope?
Explanation:
Step 1
Consider the formula for the standard error of the slope, which involves $1/\sqrt{n}$.
Step 2
As the sample size $n$ increases, the denominator increases.
Step 3
Therefore, the standard error of the slope decreases.
Question 28Level 3: 800 Mastery
If the slope of the regression line is 0, what is the correlation $r$?
Explanation:
Step 1
Recall the relationship between slope $b$ and correlation $r$: $b = r(s_y / s_x)$.
Step 2
If $b = 0$, then $r(s_y / s_x) = 0$.
Step 3
Since standard deviations $s_y$ and $s_x$ are positive, $r$ must be 0.
Question 29Level 3: 800 Mastery
What is the primary risk of extrapolation in a scatterplot model?
Explanation:
Step 1
Define extrapolation as predicting values outside the range of the original data.
Step 2
There is no guarantee that the observed trend continues beyond the data range.
Step 3
Therefore, predictions made via extrapolation are unreliable.
Question 30Level 3: 800 Mastery
If a scatterplot shows heteroscedasticity, what does this mean?
Explanation:
Step 1
Define heteroscedasticity.
Step 2
It refers to the condition where the spread (variance) of the residuals is not constant across the range of $x$.
Step 3
This often appears as a 'fan' shape in the residual plot.