Correlation
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AP Statistics › Correlation
A nutritionist compares daily sodium intake (mg) and systolic blood pressure (mmHg) for adults. The scatterplot shows a weak positive linear trend with substantial scatter. Which statement about correlation is correct?
The correlation is probably close to $+1$ because the trend is upward.
The correlation is negative because some people with high sodium have low blood pressure.
A weak correlation implies sodium has no effect on blood pressure.
Correlation cannot be used because the variables have different units (mg vs mmHg).
The correlation is probably small and positive because there is an upward tendency but lots of scatter.
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).
A sports scientist records, for 13 cyclists, average speed (miles per hour) and heart rate (beats per minute) during a steady ride. The scatterplot shows an upward trend, but two points are far from the pattern (possible outliers). Which statement about correlation is correct?
Removing the two outliers would definitely make the correlation closer to $0$.
A positive correlation proves that increasing speed causes heart rate to rise for all cyclists.
The correlation describes the strength and direction of the linear association, and outliers can change it substantially.
The correlation is unaffected by outliers as long as the slope is positive.
Because speed and heart rate both increase, the correlation must be exactly $+1$.
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.
A city planner records, for 20 neighborhoods, average commute time (minutes) and average home price (thousands of dollars). The scatterplot appears to have a curved (nonlinear) pattern: home prices are highest for moderate commute times and lower for very short or very long commutes. Which statement about correlation is correct?
A correlation near $0$ would prove that commute time and home price are unrelated.
The correlation must be close to $+1$ because commute time and price are related.
The correlation equals the slope of the curve at the center of the plot.
The correlation could be close to $0$ even though there is a strong nonlinear association.
The correlation must be close to $-1$ because the pattern eventually goes downward.
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.
A teacher compares, for 14 students, number of absences (days) and final exam score (points out of 100). The scatterplot shows a weak downward trend with substantial scatter. Which statement about correlation is correct?
The correlation is positive because exam scores are larger numbers than absences.
The correlation is near $0$ because the points are widely scattered even though there is a slight downward trend.
The correlation must be exactly $-1$ because more absences always lower scores.
The correlation is strongly negative because the line of best fit slopes downward.
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.
An ecologist measures daily sunlight (hours per day) and plant growth (centimeters per week) for 15 plants. The scatterplot shows a strong positive linear association with little scatter. Which statement about correlation is correct?
The correlation is close to $+1$ because the points follow a strong upward linear pattern.
The correlation is $0$ because the slope is not exactly $1$.
The correlation is positive only if sunlight causes growth.
The correlation is close to $-1$ because plant growth increases as sunlight increases.
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.
A meteorologist records, over 18 days, humidity (percent) and high temperature (°F). The scatterplot shows a fairly strong downward linear association. Which statement about correlation is correct?
The correlation is negative because higher humidity tends to be associated with lower high temperatures in this data set.
The correlation is $0$ because humidity and temperature are measured in different units.
The correlation shows that increasing humidity causes the temperature to drop.
The correlation must be exactly $-1$ because the points trend downward.
The correlation is positive because humidity is a percentage and temperature is in degrees.
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.
A lab group measures the concentration of a solution (mol/L) and the reaction rate (mL/min). The scatterplot shows a strong positive linear association. One student suggests converting concentration from mol/L to mmol/L (multiplying by 1000). Which statement about correlation is correct?
Correlation equals slope, so converting mol/L to mmol/L makes the correlation smaller in magnitude.
Multiplying concentration by 1000 will multiply the correlation by 1000.
Changing units by multiplying one variable by a positive constant does not change the correlation (it stays the same).
After converting units, the correlation becomes 0 because the variables are no longer comparable.
Changing units will reverse the sign of the correlation because the scale is different.
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.
A city planner records distance from downtown (miles) and monthly rent (dollars) for apartments. The scatterplot shows a fairly strong negative linear association. Which statement about correlation is correct?
Since rent is measured in dollars, the correlation must be larger than 1 in magnitude.
A negative correlation means moving farther from downtown causes rent to drop for any apartment.
The correlation is negative because higher distance values tend to be paired with lower rent values.
The correlation is positive because rent and distance are both increasing variables.
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).
A student collects data on 10 phones: battery capacity (mAh) and battery life (hours). The scatterplot shows a positive linear association. Which statement about correlation is correct?
A positive correlation proves that increasing capacity causes longer life for every phone model.
The correlation equals the slope, so changing units must change the correlation.
The correlation would become negative if battery life were measured in minutes instead of hours.
The correlation would stay the same if battery capacity were converted from mAh to Ah (a positive linear rescaling).
If all battery capacities were converted from mAh to Ah, the correlation would change because the units changed.
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.
A psychologist studies, for 25 adults, hours of sleep per night (hours) and stress score (on a 0–50 scale). The scatterplot shows a moderate negative linear association. Which statement about correlation is correct?
A negative correlation means higher sleep tends to be associated with lower stress scores, but it does not by itself establish causation.
The correlation must be $0$ because stress is measured on a scale rather than in physical units.
The correlation must be $-1$ because stress decreases as sleep increases.
A negative correlation means sleep and stress are independent.
Because the correlation is negative, the slope of the least-squares line must be positive.
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.