Introducing Statistics: Are Variables Related
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AP Statistics › Introducing Statistics: Are Variables Related
A gym manager records, for 28 members, the number of minutes they exercise per workout (x) and their resting heart rate (y, beats per minute). The manager is interested in whether exercise time and resting heart rate appear to be related. Do the data suggest the variables are related?
No; the points form a perfect horizontal line, so there is no relationship.
Yes; the points show a generally negative association, so longer workouts tend to go with lower resting heart rates.
Yes; the points show a generally positive association, so longer workouts tend to go with higher resting heart rates.
Yes; because longer workouts cause resting heart rate to be exactly the same for everyone.
No; since there are a few points that don’t follow the trend, there is no association.
Explanation
This question examines the relationship between exercise duration and resting heart rate. The correct answer identifies a negative association - longer workouts tend to go with lower resting heart rates, which makes biological sense. Choice C incorrectly claims exercise causes everyone to have the same heart rate. Choice D wrongly concludes that a few exceptions eliminate any association. When analyzing scatterplots, look for the overall direction of the point cloud: negative associations show a downward trend from left to right. Individual variations don't negate the overall pattern.
A teacher records, for 20 students, the number of absences during a semester (x) and the student’s final course percentage (y). The teacher is interested in whether absences and final grade appear to be related. Do the data suggest the variables are related?
No; the points show no association because grades vary at each absence count.
Yes; the points show a generally negative association, so more absences tend to go with lower final grades.
Yes; the points show a generally positive association, so more absences tend to go with higher grades.
No; since one student with many absences still earned a high grade, there is no relationship.
Yes; because absences directly cause every student’s grade to drop by the same amount.
Explanation
This question asks about the relationship between student absences and final grades. The correct answer identifies a negative association - more absences tend to go with lower grades. Choice C wrongly claims direct causation and identical effects for every student. Choice E incorrectly reasons that one exception (high grade despite many absences) eliminates any relationship. When analyzing educational data, negative associations are common between absences and performance. Focus on the overall downward trend rather than expecting every student to follow the pattern exactly.
A real estate agent samples 26 homes and records the home’s size (x, in square feet) and its sale price (y, in thousands of dollars). The agent wants to know whether size and price appear to be related. Do the data suggest the variables are related?
Yes; because increasing square footage causes the sale price to increase by a fixed amount.
Yes; the points show a generally positive association, so larger homes tend to sell for more.
No; the points show no clear pattern, so size and price are unrelated.
No; because there is at least one expensive small home, there is no relationship.
Yes; the points show a generally negative association, so larger homes tend to sell for less.
Explanation
This question tests recognition of positive association between home size and sale price. The correct answer identifies that larger homes tend to sell for more, showing a positive association. Choice B incorrectly states causation and claims a fixed increase amount. Choice D commits the error of thinking one exception (expensive small home) disproves the entire relationship. When examining real estate data, we expect positive associations between size and price, but remember that association allows for variability - it describes tendency, not absolute rules. Look for the overall upward trend in the data.
A botanist measures, for 24 plants of the same species, the amount of fertilizer applied each week (x, in grams) and the plant’s height after 6 weeks (y, in centimeters). The botanist wants to know whether fertilizer amount and height appear to be related. Do the data suggest the variables are related?
No; since some plants with low fertilizer are tall, there cannot be any relationship.
No; the points show no clear pattern, suggesting little to no association.
Yes; because fertilizer causes height to increase for every plant by the same amount.
Yes; there is a generally positive association, so more fertilizer tends to go with taller plants.
Yes; the points show a generally negative association, so more fertilizer tends to go with shorter plants.
Explanation
This question asks about the relationship between fertilizer amount and plant height. The correct answer states there is no clear pattern, indicating little to no association between the variables. Choice D incorrectly claims causation and suggests every plant responds identically. Choice E makes the common error of thinking that any exception (tall plants with low fertilizer) disproves a relationship entirely. When data points are scattered without a clear upward or downward trend, we conclude the variables show little to no association. Remember that "no association" means no consistent pattern, not that every point must be identical.
An environmental scientist records, for 22 days, the day’s high temperature (x, in °F) and the amount of electricity used in a small office building (y, in kWh). The scientist wants to know whether temperature and electricity use appear to be related. Do the data suggest the variables are related?
Yes; the points show a generally positive association, so higher temperatures tend to go with greater electricity use.
No; since a few cool days still had high usage, there can’t be any association.
Yes; the points prove a perfect linear relationship with no variability.
Yes; because higher temperature causes electricity use to increase on every day with no exceptions.
No; the points are widely scattered with no trend, so temperature and electricity use are unrelated.
Explanation
This question examines temperature and electricity use in an office building. The correct answer recognizes a positive association - higher temperatures tend to go with greater electricity use, likely due to air conditioning. Choice C incorrectly states causation and claims no exceptions exist. Choice E wrongly concludes that a few cool days with high usage eliminate any association. When analyzing environmental data, look for overall trends while understanding that other factors (like special events) can create exceptions. Association describes the general pattern, not a rule without exceptions.