- To test a new fertilizer, 100 identical plots of land are used. 50 plots are chosen at random to receive the new fertilizer, while the other 50 receive a standard fertilizer. All other conditions are kept the same. The plots with the new fertilizer show significantly higher crop yields. What is a reasonable conclusion?
Answer: The evidence suggests that the new fertilizer causes an increase in crop yield.
Because this is a controlled experiment with random assignment, it is designed to establish causality. The random selection of plots helps to ensure that other factors (like soil quality) are not systematically biased between the two groups. Therefore, a conclusion about causation is warranted. Distractor A is incorrect because this experimental design allows for causal inference. Distractor C is contradicted by the results. Distractor D is unlikely because the random assignment is meant to control for such factors.
- A pediatrician tracks children's height from age 2 to age 10 and finds a strong, positive linear association between age and height. Based only on this trend, what can be concluded about a child's height at age 25?
Answer: The child's height at age 25 cannot be reliably predicted from this data.
A trend found within a specific range of data (ages 2-10) cannot be assumed to continue outside that range. This error is known as extrapolation. Human growth slows significantly after puberty, so the linear trend will not continue. Therefore, predicting height at age 25 from this data is not reliable.
- The population of a species of fish in a lake was recorded for ten years. For the first five years, the population increased. For the next five years, the population decreased. Which statement best describes the association between time and the fish population over the entire ten-year period?
Answer: The association is non-linear.
Since the trend changes direction (from increasing to decreasing), a single straight line cannot describe the relationship across the entire period. This indicates a non-linear association. It is not consistently positive or negative, but there is a clear pattern, so 'no association' is incorrect.
- A student's scores on five successive quizzes were recorded as ordered pairs (quiz number, score): (1, 65), (2, 72), (3, 78), (4, 85), and (5, 91). Which of the following best describes the trend in the student's scores over time?
Answer: There is a positive association between the quiz number and the score.
As the first number in each pair (the quiz number) increases, the second number (the score) also consistently increases. This is the definition of a positive association. Distractor A describes the opposite trend. Distractor C is incorrect as there is a clear pattern. Distractor D confuses a measure of central tendency (the average) with an association between two variables.
- A researcher studying the relationship between exercise frequency and mood ratings finds a correlation of r = 0.45. She then discovers that 30% of her participants were taking antidepressant medications, which she had not initially considered. If she were to analyze the data separately for medicated and non-medicated participants, which outcome would be most likely?
Answer: The non-medicated group would likely show a stronger correlation since their mood is more directly influenced by exercise
The correct answer is C. For non-medicated individuals, exercise is likely a more direct influence on mood, creating a stronger correlation. For medicated individuals, the antidepressant effect may mask or reduce the apparent relationship between exercise and mood. Choice A is incorrect because medication status is likely a confounding variable that affects the exercise-mood relationship. Choice B is wrong because antidepressants would more likely stabilize mood and reduce the variability that creates correlation. Choice D focuses on sample size effects but ignores the more important issue of how medication affects the underlying relationship.
- A town observes that as the number of public parks increases, the average property value also increases. A council member concludes that to raise property values, the town must build more parks. Why might this conclusion be premature?
Answer: A third factor, like neighborhood wealth, could be causing both more parks and higher property values.
This is an issue of correlation versus causation. While the two variables are associated, a third (confounding) variable might be the actual cause. Wealthier neighborhoods may have the resources to both build more parks and have higher property values. This makes the conclusion that parks cause the increase in value premature. Distractor D suggests reverse causation, which is also possible, but C points to a classic confounding variable.
- A sociologist's survey finds that individuals who report regularly eating breakfast have, on average, lower body weights than those who report skipping breakfast. What is the most appropriate conclusion from this observational study?
Answer: There is an observed association between the habit of eating breakfast and lower body weight.
Observational studies can identify associations (correlations) but cannot prove causation due to potential confounding variables (e.g., people who eat breakfast might also have other healthier habits). Choice B correctly states the finding as an association. Distractor A incorrectly claims causation. Distractor C makes an absolute statement that isn't supported by an average trend. Distractor D contradicts the study's findings.
- A researcher investigates if there is a relationship between the price of a bottle of shampoo and its customer satisfaction rating. After analyzing data for 200 products, she finds that both cheap and expensive shampoos have a full range of high and low ratings, with no clear pattern emerging. What is the most likely conclusion?
Answer: There is likely no association.
The description of the data—where high and low values of one variable are paired with both high and low values of the other—is characteristic of a data set with no discernible association or trend. A non-linear association would still show a clear, predictable pattern (e.g., a U-shape), which is not described here.
- Marketplace data shows a strong negative association between a car's age and its resale value. Given this trend, what is a reasonable expectation when comparing a 3-year-old car to a 5-year-old car of the same make and model?
Answer: The 3-year-old car will likely have a higher resale value than the 5-year-old car.
A negative association means that as the value of one variable (age) increases, the value of the other variable (resale value) tends to decrease. Therefore, the younger car is expected to have a higher value. The word 'likely' is used because this is a general trend, not an absolute rule for every single car. Distractor D claims a precise mathematical relationship that cannot be determined from a general association.
- A school administrator notices that students who eat breakfast tend to have higher test scores than those who don't. However, when she examines the data more closely by family income level, she finds that the breakfast-test score relationship disappears within each income group. This pattern suggests that:
Answer: Family income is a confounding variable that explains the apparent relationship between breakfast and test scores
When you encounter questions about relationships between variables that change when additional factors are considered, you're dealing with confounding variables—factors that influence both the independent and dependent variables, creating misleading correlations.
In this scenario, the administrator initially observes that breakfast eaters have higher test scores. However, when she controls for family income (examines each income group separately), this relationship disappears. This is a classic sign of confounding. Higher-income families are more likely to provide breakfast AND have children with higher test scores due to other advantages (tutoring, educational resources, stable home environments). The breakfast itself isn't causing the higher scores—income is driving both variables.
Choice D correctly identifies family income as the confounding variable that explains the apparent breakfast-test score relationship. When you control for the true underlying factor (income), the spurious relationship vanishes.
Choice A is wrong because measurement error would typically add noise to data rather than create systematic patterns that disappear when controlling for income. Choice B incorrectly suggests the relationship is stronger but masked—actually, controlling for income reveals there's no real relationship. Choice C assumes breakfast does have a causal effect for certain groups, but the data shows the relationship disappears entirely within income groups, indicating no causal effect.
Remember this pattern: when an apparent relationship between two variables disappears after controlling for a third variable, that third variable is likely a confounder explaining the original correlation. Always look for underlying factors that could influence both variables simultaneously.
- A biologist observes that as the amount of a specific nutrient added to a plant's soil increases from zero, the plant's growth rate increases. However, after a certain point, adding more of the nutrient causes the growth rate to slow down and eventually decrease. Which term best describes this relationship?
Answer: A non-linear association
The relationship changes direction—it is positive at first and then becomes negative. A single straight line cannot model this pattern. Therefore, the association is non-linear. It is not a simple positive or negative linear association, and there is clearly a relationship, so 'no association' is incorrect.
- A teacher finds that students who missed more assignments generally scored lower on the final exam. The median number of missed assignments was 3, and the median exam score was 75. Which piece of information describes the association between the two variables?
Answer: Students who missed more assignments generally scored lower.
An association describes how two variables relate to each other. The statement that 'students who missed more assignments generally scored lower' does exactly that. The median values for each variable (75 and 3) are measures of central tendency, not measures of association. The relationship described is negative (more missed assignments, lower score), so D is incorrect.
- A city official observes that monthly ice cream sales are positively associated with the monthly number of residents reporting heat-related illnesses. The official concludes that eating ice cream causes heat-related illnesses. Why is this conclusion likely flawed?
Answer: An unobserved variable, such as high temperatures, is likely causing both to increase.
This is a classic example of a confounding (or lurking) variable. High temperatures cause people to both eat more ice cream and suffer more heat-related illnesses. The ice cream itself is not causing the illness. Therefore, the conclusion of causality is flawed. Distractor A incorrectly defines a negative association. Distractor C is too absolute; controlled experiments can strongly suggest causation. Distractor D raises a different statistical issue (sample size) but doesn't address the logical flaw in the causal conclusion.
- Researchers found a strong negative association between the number of hours a person spends watching television per day and the number of hours they spend exercising per week. Which statement is the best interpretation of this finding?
Answer: People who watch more television tend to be the same people who exercise less.
A negative association means that as one variable increases, the other tends to decrease. Choice C accurately describes this relationship without making a claim of causation. Distractor A incorrectly assumes causation. Distractor B incorrectly describes a positive association. Distractor D makes an unsubstantiated prediction and assumes causation.
- An observational study reveals a positive association between an individual's self-reported happiness level and the number of close friends they have. Which of the following represents a plausible alternative interpretation to 'having more friends makes people happier'?
Answer: People who are inherently happier may find it easier to make and keep friends.
Correlation does not imply a specific direction of causation. While it's possible that friends cause happiness, it's also plausible that happiness is a trait that makes it easier to attract friends. This is called reverse causation and is a valid alternative interpretation of an association found in an observational study. Distractor D is incorrect because it ignores this possibility. Distractor A is too dismissive, and C contradicts the premise.
- A fitness researcher finds a strong association where individuals who exercise more minutes per week tend to have lower resting heart rates. What type of association does this describe?
Answer: A negative association
A negative association occurs when one variable increases while the other variable tends to decrease. In this case, as exercise (the first variable) increases, resting heart rate (the second variable) decreases. Students might mistakenly choose 'positive' because both outcomes are generally considered good for health, but the term refers to the mathematical direction of the relationship.
- A researcher studies the link between daily caffeine intake and hours of sleep. The data show that people who consume more caffeine tend to sleep slightly less, but there are many exceptions and the data points are widely scattered. Which is the best description of the association?
Answer: A weak negative association
The direction of the trend is negative because as caffeine intake increases, sleep tends to decrease. The strength of the association is weak because the data points are 'widely scattered' and there are 'many exceptions'. Therefore, the best description is a weak negative association.
- A researcher collected data on the relationship between hours of sleep and test scores for 20 students. The correlation coefficient was calculated to be r = 0.72. However, upon reviewing the data more carefully, the researcher discovered that one student who reported 12 hours of sleep and scored 95% was actually an outlier due to a data entry error. When this outlier is removed, what is the most likely effect on the correlation coefficient?
Answer: The correlation coefficient will decrease because the outlier was reinforcing the positive trend in the data
The correct answer is B. The outlier (12 hours, 95%) reinforces the positive correlation between sleep and test scores. Removing a point that supports the trend will weaken the correlation, causing r to decrease from 0.72. Choice A is wrong because removing outliers doesn't always strengthen correlations—it depends on whether the outlier supports or contradicts the trend. Choice C is incorrect because outliers can significantly impact correlation coefficients, especially in small samples (n=20). Choice D is wrong because removing a single outlier supporting a positive trend won't reverse the correlation to negative.
- A company tracks monthly sales data over two years. In Year 1, sales showed a steady increasing trend. In Year 2, sales initially continued increasing but then plateaued at a high level for several months before declining. If you were to compare the correlation coefficients between month number and sales for Year 1 versus Year 2, which statement is most accurate?
Answer: Year 1 would have a stronger positive correlation because the increasing trend was more consistent throughout
The correct answer is A. Year 1 shows a consistent linear increasing trend, which produces a strong positive correlation. Year 2's pattern (increase, plateau, decline) is non-linear and less consistent, resulting in a weaker correlation coefficient even though it starts with increases. Choice B is incorrect because correlation measures the strength of linear relationship, not the magnitude of values. Choice C is wrong because the consistency of the trend matters more than the general direction. Choice D is incorrect because more variation typically leads to weaker, not stronger, correlations.
- In a dataset, as the value of Variable A increases, the value of Variable B also consistently increases. The data points form a pattern that closely follows an upward-sloping straight line. Which of the following accurately describes the relationship?
Answer: A strong positive association
The relationship is positive because both variables increase together. The relationship is strong because the points 'closely follow' a line, indicating very little scatter. The relationship is linear because the pattern follows a 'straight line'. Therefore, it is a strong positive association.
