AP Statistics Quiz: Correlation

What this quiz covers

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

How to use this quiz

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

Question 1

1. The correlation is probably close to $+1$ because the trend is upward.
2. The correlation is probably small and positive because there is an upward tendency but lots of scatter. (correct answer)
3. The correlation is negative because some people with high sodium have low blood pressure.
4. A weak correlation implies sodium has no effect on blood pressure.
5. Correlation cannot be used because the variables have different units (mg vs mmHg).

Explanation: This question examines weak positive correlation. With an upward tendency but substantial scatter, the correlation will be positive but small in magnitude, making choice B correct. Choice A overestimates correlation strength based solely on direction, while choice C misinterprets individual exceptions as determining the overall correlation sign. Weak correlation doesn't imply no effect (choice D) - it just means the linear relationship isn't strong. Correlation is unitless and can be calculated regardless of different measurement units (choice E).

Question 2

1. Removing the two outliers would definitely make the correlation closer to $0$.
2. The correlation describes the strength and direction of the linear association, and outliers can change it substantially. (correct answer)
3. The correlation is unaffected by outliers as long as the slope is positive.
4. Because speed and heart rate both increase, the correlation must be exactly $+1$.
5. A positive correlation proves that increasing speed causes heart rate to rise for all cyclists.

Explanation: This AP Statistics question explores how outliers affect correlation, which measures linear association's strength and direction. Choice B correctly notes that outliers, especially influential ones, can substantially alter the correlation coefficient. The upward trend suggests positive correlation, but the two outliers could skew it. Distractor A assumes removal always weakens correlation, but it depends on the outliers' position. Correlation doesn't prove causation, countering E. In essence, correlation calculates average product of z-scores, sensitive to extreme points disrupting the linear pattern.

Question 3

1. The correlation could be close to $0$ even though there is a strong nonlinear association. (correct answer)
2. The correlation must be close to $+1$ because commute time and price are related.
3. The correlation must be close to $-1$ because the pattern eventually goes downward.
4. The correlation equals the slope of the curve at the center of the plot.
5. A correlation near $0$ would prove that commute time and home price are unrelated.

Explanation: Correlation in AP Statistics is specifically for linear associations, so nonlinear patterns like curves can yield correlations near 0 despite strong relationships. Choice A accurately states this, referencing the curved pattern where prices peak at moderate commutes. The eventual downward part doesn't force negative correlation, as in distractor C. Correlation near 0 doesn't prove no relationship, just no linear one, countering E. It's not the slope of any curve, debunking D. Thus, correlation measures linear predictability, potentially missing curved associations.

Question 4

1. The correlation is strongly negative because the line of best fit slopes downward.
2. The correlation is near $0$ because the points are widely scattered even though there is a slight downward trend. (correct answer)
3. The correlation is positive because exam scores are larger numbers than absences.
4. The correlation must be exactly $-1$ because more absences always lower scores.
5. The correlation is undefined because the units are different (days vs points).

Explanation: Correlation in AP Statistics quantifies linear association, and a value near 0 indicates weak or no linear relationship, even if a trend exists amid scatter. The weak downward trend with substantial scatter points to a correlation near 0, as choice B states. This references the pattern where more absences loosely associate with lower scores, but the wide scatter weakens the linear strength. Choice A is a distractor, overstating the negativity by ignoring the scatter's impact on magnitude. Units don't affect correlation, countering choice E. Fundamentally, correlation measures how predictably one variable changes with another in a linear fashion, with scatter reducing its absolute value.

Question 5

1. The correlation is close to $+1$ because the points follow a strong upward linear pattern. (correct answer)
2. The correlation is close to $-1$ because plant growth increases as sunlight increases.
3. The correlation is $0$ because the slope is not exactly $1$.
4. The correlation is positive only if sunlight causes growth.
5. The correlation must be small because the units (hours and centimeters) are different.

Explanation: In AP Statistics, correlation assesses the linear relationship between variables, with positive values indicating that as one variable increases, the other tends to increase. The strong upward linear pattern with little scatter suggests a correlation close to +1, as described in choice A. This references the pattern of increasing plant growth with more sunlight, showing a tight positive association. A distractor like choice B incorrectly assumes a negative correlation based on the direction of increase, but positive means both rise together. Correlation is unitless and unaffected by different units, debunking choice E. Overall, correlation measures linear strength and direction, not causation, so even a strong positive r doesn't prove sunlight causes growth.

Question 6

1. The correlation is positive because humidity is a percentage and temperature is in degrees.
2. The correlation is negative because higher humidity tends to be associated with lower high temperatures in this data set. (correct answer)
3. The correlation is $0$ because humidity and temperature are measured in different units.
4. The correlation must be exactly $-1$ because the points trend downward.
5. The correlation shows that increasing humidity causes the temperature to drop.

Explanation: Correlation in AP Statistics is negative when higher values of one variable pair with lower values of the other, as in choice B's description of humidity and temperature. The strong downward pattern supports this inverse association. Units differ but don't impact correlation, debunking C. It's not exactly -1 unless perfect linearity, countering D. Positivity isn't from measurement types, as in distractor A. Correlation shows association, not causation, so it doesn't prove humidity causes temperature drops.

Question 7

1. Multiplying concentration by 1000 will multiply the correlation by 1000.
2. Changing units by multiplying one variable by a positive constant does not change the correlation (it stays the same). (correct answer)
3. Changing units will reverse the sign of the correlation because the scale is different.
4. After converting units, the correlation becomes 0 because the variables are no longer comparable.
5. Correlation equals slope, so converting mol/L to mmol/L makes the correlation smaller in magnitude.

Explanation: This question tests understanding that correlation is invariant under linear transformations. Multiplying one variable by a positive constant (like converting mol/L to mmol/L by multiplying by 1000) doesn't change the correlation coefficient - it remains exactly the same. Choice B correctly states this property. Choice A wrongly assumes correlation scales with the variable, while choice C incorrectly claims the sign changes. The correlation doesn't become zero (choice D) or change based on slope considerations (choice E). This invariance property makes correlation a standardized measure of linear association.

Question 8

1. Since rent is measured in dollars, the correlation must be larger than 1 in magnitude.
2. The correlation is negative because higher distance values tend to be paired with lower rent values. (correct answer)
3. The correlation is positive because rent and distance are both increasing variables.
4. A negative correlation means moving farther from downtown causes rent to drop for any apartment.
5. The correlation is the same as the slope of the best-fit line, so changing miles to kilometers changes $r$.

Explanation: This question tests understanding correlation's sign and interpretation. As distance from downtown increases, rent tends to decrease, creating a negative linear association. Choice B correctly identifies this negative correlation. Choice A misunderstands that correlation is always between -1 and +1 regardless of variable units. Choice C incorrectly assumes both variables increasing means positive correlation - it's about how they vary together. Correlation describes association, not causation (choice D), and unlike slope, correlation is unitless (choice E).

Question 9

1. If all battery capacities were converted from mAh to Ah, the correlation would change because the units changed.
2. The correlation would stay the same if battery capacity were converted from mAh to Ah (a positive linear rescaling). (correct answer)
3. The correlation would become negative if battery life were measured in minutes instead of hours.
4. The correlation equals the slope, so changing units must change the correlation.
5. A positive correlation proves that increasing capacity causes longer life for every phone model.

Explanation: AP Statistics teaches that correlation is invariant under linear transformations of variables, like unit conversions. Choice B correctly explains that converting mAh to Ah (dividing by 1000) won't change the correlation, as it's a positive linear rescaling. The positive linear association remains, unaffected by units. Distractor A wrongly suggests units alter correlation, but it's standardized. Time units like minutes vs. hours wouldn't flip the sign, countering C. Correlation indicates association strength, not causation or slope equality.

Question 10

1. Because the correlation is negative, the slope of the least-squares line must be positive.
2. A negative correlation means higher sleep tends to be associated with lower stress scores, but it does not by itself establish causation. (correct answer)
3. A negative correlation means sleep and stress are independent.
4. The correlation must be $-1$ because stress decreases as sleep increases.
5. The correlation must be $0$ because stress is measured on a scale rather than in physical units.

Explanation: This AP Statistics question addresses correlation's interpretation, emphasizing it shows association but not causation. Choice B rightly states the negative correlation means higher sleep associates with lower stress, without proving cause. The moderate strength implies not -1, countering D. Negative doesn't mean independence, debunking C. Slope sign matches correlation sign, so negative r means negative slope, not positive as in A. Correlation isn't zero due to scales, countering E; it's about linear linkage.

Question 11

1. Because the relationship is increasing, the correlation must be close to $+1$ even if it is curved.
2. Correlation describes linear association, so a strong curved pattern can still have a correlation that is not close to $1$. (correct answer)
3. If the pattern is curved, the correlation must be negative.
4. A curved pattern proves that length causes speed to increase.
5. Correlation depends on the units, so using cm instead of m would change the sign of $r$.

Explanation: This question highlights that correlation specifically measures linear association. A curved pattern, even if consistently increasing, will not produce a correlation close to +1 because correlation only captures straight-line relationships. Choice B correctly explains this limitation. Choice A wrongly assumes any increasing pattern yields high positive correlation, while choice C incorrectly links curved patterns to negative correlation. Correlation is about association strength, not causation (choice D), and it's unitless - changing measurement units doesn't affect r (choice E).

Question 12

1. Because the points slope upward, the correlation must be exactly $r=1$.
2. The correlation is positive and likely moderate to strong, since the pattern is roughly linear and increasing. (correct answer)
3. The correlation is negative because some students study a lot but do not get perfect scores.
4. A positive correlation proves that studying more causes higher test scores for these students.
5. Correlation measures the steepness of the trend line, so a steeper upward trend always means a larger $r$.

Explanation: This question tests understanding of correlation as a measure of linear association strength. The scatterplot shows an upward trend with moderate scatter, indicating a positive correlation that is not perfect (r=1 would require all points to fall exactly on a line). Choice B correctly identifies this as a positive correlation that is moderate to strong. Choice A incorrectly assumes any upward pattern must have r=1, while choice D confuses correlation with causation. Correlation measures how closely points follow a linear pattern, not the steepness of the trend (choice E is wrong).

Question 13

1. The correlation is positive because both variables are measured in hours.
2. The correlation is negative and likely strong because the points cluster around a decreasing line. (correct answer)
3. The correlation must be $r=-1$ because the trend is downward.
4. A strong negative correlation proves that more screen time causes less sleep.
5. Correlation measures the change in sleep per hour of screen time, so it equals the slope.

Explanation: This question tests recognizing strong negative correlation. A fairly tight downward linear pattern indicates negative correlation with high magnitude (close to -1 but not exactly -1), making choice B correct. Choice A incorrectly links correlation sign to unit similarity, while choice C wrongly assumes any downward trend must have r=-1. Strong correlation describes association strength, not causation (choice D). Correlation and slope are different concepts - correlation is standardized and unitless while slope has units (choice E).

Question 14

1. Because the points slope downward, training time causes faster 5K times.
2. The correlation is negative because larger training times tend to be associated with smaller 5K times. (correct answer)
3. The correlation must be close to $0$ because the relationship is not perfectly linear.
4. The correlation is positive because both variables are measured on ratio scales.
5. The correlation is negative only if training time and 5K time have the same units.

Explanation: This question tests understanding of correlation in AP Statistics, which measures the strength and direction of the linear association between two quantitative variables, ranging from -1 to +1. The scatterplot shows a downward linear trend, indicating that as training time increases, 5K race time decreases, resulting in a negative correlation. Choice B correctly identifies this negative association without implying causation. A common distractor, like choice A, confuses correlation with causation by suggesting training time 'causes' faster times, but correlation does not prove causality. Remember, correlation quantifies how closely points follow a straight line, with the sign reflecting the slope's direction and the magnitude indicating the tightness of the fit. The presence of a high-leverage point might influence the correlation value, but it doesn't change the overall negative direction here.

Question 15

1. The correlation is negative and likely strong because the points follow a fairly straight decreasing pattern. (correct answer)
2. The correlation is zero because temperature and electricity use have different units.
3. The correlation is positive because electricity use is always positive.
4. A negative correlation means temperature causes electricity use to decrease in every situation.
5. A steeper downward trend guarantees $|r|$ is smaller because the slope is more extreme.

Explanation: This question examines negative correlation in a real-world context. When temperature increases, electricity use decreases (likely due to less heating needed), creating a downward linear pattern. Choice A correctly identifies this as a negative correlation that is likely strong due to the fairly straight pattern. Choice B incorrectly claims different units prevent correlation calculation, while choice C misunderstands that correlation sign depends on the relationship direction, not variable values. Correlation measures association strength and direction, not causation (choice D) or slope steepness (choice E).

Question 16

1. The correlation is positive because rent is measured in dollars, which are always positive.
2. The correlation is negative because larger distances tend to be associated with lower rents. (correct answer)
3. The correlation is negative only if distance causes rent to decrease.
4. The correlation is $0$ because a negative slope means no correlation.
5. The correlation must be exactly $-1$ because the points slope downward.

Explanation: AP Statistics correlation evaluates the direction and strength of linear relationships, negative when variables move oppositely. The downward association means higher distances link to lower rents, yielding negative correlation per choice B. This pattern shows inverse movement, not requiring causation as in distractor C. Choice A wrongly ties positivity to positive measurements, but correlation depends on association, not scales. A negative slope doesn't mean zero correlation, debunking D. Correlation is a standardized measure, focusing on relative deviations, not absolute values or units.

Question 17

1. The unusual month cannot affect the correlation because correlation only depends on the overall slope.
2. The correlation could change noticeably if the unusual month were removed, because influential points can affect $r$. (correct answer)
3. The correlation must be negative because one point has high advertising and low revenue.
4. A strong positive correlation proves that increasing advertising will cause revenue to increase.
5. The correlation is measured in thousands of dollars, so changing units would change the sign of $r$.

Explanation: In AP Statistics, influential points can significantly affect correlation by altering the linear fit. Choice B correctly notes that removing the unusual month could change the correlation noticeably. The strong positive association might weaken or strengthen without it. Correlation depends on all points, not just slope, countering A. One low-revenue point doesn't force negativity, debunking C. Correlation proves association, not causation, and units don't affect its sign or value.

Question 18

1. The overall correlation could be positive even if the within-group correlations are weak, because combining groups can create an apparent linear association. (correct answer)
2. If there are two clusters, correlation is undefined and cannot be computed.
3. A positive overall correlation proves advertising causes higher sales for both small and large stores.
4. Because sales and spending use the same units (thousands of dollars), the correlation must be exactly $+1$.
5. The overall correlation must be near 0 because each cluster has weak association.

Explanation: This question explores Simpson's paradox in correlation. When data contains distinct groups (clusters), the overall correlation can differ dramatically from within-group correlations. Combining small and large stores creates an overall positive trend even if each group shows weak association internally, making choice A correct. Choice B wrongly claims correlation can't be computed with clusters, while choice E incorrectly assumes weak within-group correlations guarantee weak overall correlation. Correlation measures association, not causation (choice C), and having same units doesn't force r=1 (choice D).

Question 19

1. The outlier likely weakens the magnitude of the negative correlation compared with the rest of the pattern. (correct answer)
2. Because most points go down, the correlation cannot be affected by a single unusual point.
3. The correlation must become positive because of the one high-grade point.
4. The negative correlation proves that absences cause lower grades for every student.
5. Correlation measures the steepness of the trend, so the outlier changes only the slope, not $r$.

Explanation: This question explores how outliers affect correlation. The single student with many absences but high grades goes against the overall negative trend, which will pull the correlation coefficient closer to zero, weakening its magnitude. Choice A correctly identifies this effect. Choice B wrongly assumes correlation is unaffected by outliers, while choice C overestimates the outlier's impact. Correlation measures association, not causation (choice D), and it's not the same as slope (choice E). Outliers can substantially influence correlation values.